From 1ff52d52bb6f7e512345385cf884f5436047546d Mon Sep 17 00:00:00 2001
From: elemel <elise.lemeledo@math.uzh.ch>
Date: Wed, 13 May 2020 17:53:27 +0200
Subject: [PATCH] added continuous galerkin -- fixed some misdistribution

---
 LocalLib/BarycentricBasisFunctions.py         | 150 ++++++++
 LocalLib/BarycentricTools.py                  |  69 ++++
 LocalLib/LobattoQuadratureNodes.py            |  10 +-
 LocalLib/ReversedFunctools.py                 |  18 +-
 .../BarycentricBasisFunctions.cpython-38.pyc  | Bin 0 -> 4948 bytes
 .../BarycentricTools.cpython-38.pyc           | Bin 0 -> 2288 bytes
 .../LobattoQuadratureNodes.cpython-38.pyc     | Bin 2318 -> 2325 bytes
 .../ReversedFunctools.cpython-38.pyc          | Bin 447 -> 821 bytes
 .../AnnularCylinder_o3_CG_res_20.pyMsh        | Bin 591163 -> 0 bytes
 .../BigSimpleRectangle_o1_CG_res_20.pyMsh     | Bin 465106 -> 0 bytes
 .../BigSimpleRectangle_o1_CG_res_50.pyMsh     | Bin 2384397 -> 0 bytes
 .../BigSimpleSquare_o1_CG_res_50.pyMsh        | Bin 3174260 -> 0 bytes
 ...Msh => BigSimpleSquare_o2_CG_res_20.pyMsh} | Bin 508601 -> 576028 bytes
 .../BigSimpleSquare_o2_CG_res_30.pyMsh        | Bin 0 -> 1274467 bytes
 ...eWithHoleAndRoundCorner_o1_CG_res_50.pyMsh | Bin 1777078 -> 0 bytes
 ...eWithHoleAndRoundCorner_o5_DG_res_10.pyMsh | Bin 200800 -> 0 bytes
 .../SimpleRectangle_o1_CG_res_50.pyMsh        | Bin 2437620 -> 0 bytes
 .../SimpleSquare_o1_CG_res_20.pyMsh           | Bin 508598 -> 0 bytes
 .../SimpleSquare_o1_CG_res_50.pyMsh           | Bin 3174257 -> 0 bytes
 .../Advection_Rotation_1Fluid.cpython-38.pyc  | Bin 817 -> 817 bytes
 ...dvection_Translation_1Fluid.cpython-38.pyc | Bin 828 -> 835 bytes
 .../CustomSettingsTemplate.txt                |   6 +-
 RunPostProcess.py                             |   4 +-
 RunProblem.py                                 |   4 +-
 SourceCode/Mappers.py                         |  10 +-
 SourceCode/Meshing/MeshStructure.py           |   7 +-
 SourceCode/Meshing/Meshing_DualMesh.py        |   3 +-
 SourceCode/Meshing/Meshing_PrimalMesh.py      | 181 ++++++---
 .../Meshing_Table_ElementDiameterIndex.txt    |   0
 .../Meshing_DualMesh.cpython-38.pyc           | Bin 5486 -> 5502 bytes
 .../Meshing_PrimalMesh.cpython-38.pyc         | Bin 23634 -> 26619 bytes
 .../Meshing_VariationalInfo.cpython-38.pyc    | Bin 0 -> 5697 bytes
 SourceCode/PostProcessing/Plots_Matplotlib.py |  48 ++-
 SourceCode/PostProcessing/Plots_Plotly.py     |   2 +-
 .../Plots_Matplotlib.cpython-38.pyc           | Bin 35242 -> 36910 bytes
 .../__pycache__/Plots_Plotly.cpython-38.pyc   | Bin 38190 -> 38190 bytes
 .../LinearAdvectionRotation.py                |   5 +-
 .../LinearAdvectionTranslation.py             |   9 +-
 .../__pycache__/EulerEquations.cpython-38.pyc | Bin 19574 -> 19574 bytes
 .../LinearAdvectionRotation.cpython-38.pyc    | Bin 13159 -> 13150 bytes
 .../LinearAdvectionTranslation.cpython-38.pyc | Bin 13312 -> 13290 bytes
 SourceCode/Solve.py                           |  14 +-
 .../NaiveLS_ContinuousGalerkin.py             | 358 ++++++++++++++++++
 .../_Solver/QuadratureRules/HandBased.py      | 101 +++++
 .../__pycache__/HandBased.cpython-38.pyc      | Bin 192 -> 2770 bytes
 SourceCode/_Solver/Solver_Initialisations.py  |  19 +-
 SourceCode/_Solver/Solver_Spatial.py          |  43 ++-
 SourceCode/_Solver/Solver_Temporal.py         |  33 +-
 .../SpatialSchemes/CellCentredFV_LFX.py       | 172 ++++-----
 .../SpatialSchemes/CellCentredFV_LFXorig.py   | 203 ++++++++++
 .../SpatialSchemes/CellCentredFV_Natural.py   | 191 ++++++++++
 .../SpatialSchemes/ContinuousGalerkin.py      | 300 +++++++++++++++
 .../SpatialSchemes/VertexCentredFV_LFX.py     | 181 +++++----
 .../SpatialSchemes/VertexCentredFV_Natural.py | 204 ++++++++++
 .../CellCentredFV_LFX.cpython-38.pyc          | Bin 5903 -> 6588 bytes
 .../CellCentredFV_LFXorig.cpython-38.pyc      | Bin 0 -> 5945 bytes
 .../CellCentredFV_Natural.cpython-38.pyc      | Bin 0 -> 6099 bytes
 .../ContinuousGalerkin.cpython-38.pyc         | Bin 0 -> 8125 bytes
 .../VertexCentredFV_Fn.cpython-38.pyc         | Bin 0 -> 6279 bytes
 .../VertexCentredFV_LFX.cpython-38.pyc        | Bin 5381 -> 6642 bytes
 .../VertexCentredFV_LFXorig.cpython-38.pyc    | Bin 0 -> 5373 bytes
 .../VertexCentredFV_Natural.cpython-38.pyc    | Bin 0 -> 6663 bytes
 SourceCode/_Solver/TemporalSchemes/DeC.py     |   6 +-
 SourceCode/_Solver/TemporalSchemes/RK.py      |   2 -
 .../__pycache__/DeC.cpython-38.pyc            | Bin 6449 -> 6444 bytes
 .../Solver_Initialisations.cpython-38.pyc     | Bin 13629 -> 13689 bytes
 .../__pycache__/Solver_Spatial.cpython-38.pyc | Bin 2568 -> 2567 bytes
 .../Solver_Temporal.cpython-38.pyc            | Bin 2412 -> 2768 bytes
 SourceCode/__pycache__/Mappers.cpython-38.pyc | Bin 4884 -> 5115 bytes
 .../__pycache__/PostProcess.cpython-38.pyc    | Bin 8691 -> 8691 bytes
 SourceCode/__pycache__/Solve.cpython-38.pyc   | Bin 4976 -> 5024 bytes
 71 files changed, 2085 insertions(+), 268 deletions(-)
 create mode 100644 LocalLib/BarycentricBasisFunctions.py
 create mode 100644 LocalLib/BarycentricTools.py
 create mode 100644 LocalLib/__pycache__/BarycentricBasisFunctions.cpython-38.pyc
 create mode 100644 LocalLib/__pycache__/BarycentricTools.cpython-38.pyc
 delete mode 100644 PredefinedTests/PredefinedMeshes/AnnularCylinder_o3_CG_res_20.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/BigSimpleRectangle_o1_CG_res_20.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/BigSimpleRectangle_o1_CG_res_50.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/BigSimpleSquare_o1_CG_res_50.pyMsh
 rename PredefinedTests/PredefinedMeshes/{BigSimpleSquare_o1_CG_res_20.pyMsh => BigSimpleSquare_o2_CG_res_20.pyMsh} (66%)
 create mode 100644 PredefinedTests/PredefinedMeshes/BigSimpleSquare_o2_CG_res_30.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o1_CG_res_50.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o5_DG_res_10.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/SimpleRectangle_o1_CG_res_50.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/SimpleSquare_o1_CG_res_20.pyMsh
 delete mode 100644 PredefinedTests/PredefinedMeshes/SimpleSquare_o1_CG_res_50.pyMsh
 delete mode 100644 SourceCode/Meshing/Meshing_Table_ElementDiameterIndex.txt
 create mode 100644 SourceCode/Meshing/__pycache__/Meshing_VariationalInfo.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/FluidSelectors/NaiveLS_ContinuousGalerkin.py
 create mode 100644 SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFXorig.py
 create mode 100644 SourceCode/_Solver/SpatialSchemes/CellCentredFV_Natural.py
 create mode 100644 SourceCode/_Solver/SpatialSchemes/ContinuousGalerkin.py
 create mode 100644 SourceCode/_Solver/SpatialSchemes/VertexCentredFV_Natural.py
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/CellCentredFV_LFXorig.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/CellCentredFV_Natural.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/ContinuousGalerkin.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/VertexCentredFV_Fn.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/VertexCentredFV_LFXorig.cpython-38.pyc
 create mode 100644 SourceCode/_Solver/SpatialSchemes/__pycache__/VertexCentredFV_Natural.cpython-38.pyc
 delete mode 100644 SourceCode/_Solver/TemporalSchemes/RK.py

diff --git a/LocalLib/BarycentricBasisFunctions.py b/LocalLib/BarycentricBasisFunctions.py
new file mode 100644
index 0000000..0e2543a
--- /dev/null
+++ b/LocalLib/BarycentricBasisFunctions.py
@@ -0,0 +1,150 @@
+ 
+'''
+################################################################################
+#	Module that implements the basis functions given a reference shape         #
+#   (its number of edges basically), the order and the barycentric             #
+#   coordinates of the considered point with respect to the given triangle     #
+################################################################################
+
+-----------------------------------------------------------------------------'''
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+# ==============================================================================
+#	Basis functions for triangular shape
+# ==============================================================================
+
+#### Simplicial basis function from barycentric coordinates ####
+def SimplicialBasis(Type, Order, points):
+    """ Determines the values of the basis functions according to barycentric coordinates.
+        The order of the basis function is given by the order of the vertices upon which 
+        are defined the points. 
+        
+        Input:  Type,   string,           type of wished basis functions
+                Order,  integer,          order of the wished basis functions
+                points, 2D numpy array,   the points at which to evaluate the basis functons (NbPoints, 3)
+                
+        Output: EvaluatedBase, 2D numpy array,  the value of all the basis functions at each given point,
+                                                ordered accordingly to the vertices from which the barycentic 
+                                                coordinates have been generated (NbBasisFunc x NbPoints).
+                                                
+        Note, in all the graphics below the sample vertex ordering is given by:       1
+                                                                                    /   \
+                                                                                   2 ___ 3
+    --------------------------------------------------------------------------------------- """
+    
+    # Bernstein basis function
+    if Type == "B":
+        # ------------------------------------------------------------------------
+        if Order == 1:
+            """
+                Gives the basis functions in the order  1
+                                                      /   \
+                                                     2 ___ 3
+            """
+            EvaluatedBase = points
+        
+        # ------------------------------------------------------------------------
+        if Order == 2:
+            """
+                Gives the basis functions in the order  1
+                                                      /   \
+                                                     2     6
+                                                    /       \
+                                                   3 __ 4 __ 5
+            """
+            
+            # Prepares the basis in the Gmsh format
+            EvaluatedBase = np.zeros((points.shape[0], 6))
+            EvaluatedBase[:, 0:3] = points**2
+            EvaluatedBase[:, 3]   = 2*points[:,0]*points[:,1]
+            EvaluatedBase[:, 4]   = 2*points[:,1]*points[:,2]
+            EvaluatedBase[:, 5]   = 2*points[:,2]*points[:,0]
+        
+            # Map it to the custom MeshStructure
+            EvaluatedBase = EvaluatedBase[:, [0, 3, 1, 4, 2, 5]]
+    
+    # Return the considered basis
+    return(EvaluatedBase)
+    
+#### Simplicial gradient of basis function from barycentric coordinates ####    
+def SimplicialGradients(Type, Order, points, n, vol):
+    """ Determines the values of the basis functions gradient according to barycentric coordinates.
+        The order of the basis function is given by the order of the vertices upon which 
+        are defined the points. 
+        
+        Input:  Type,   string,           type of wished basis functions
+                Order,  integer,          order of the wished basis functions
+                points, 2D numpy array,   the points at which to evaluate the basis functons (NbPoints, 3)
+                n,      2D numpy array,   the normal vector to each face, in the order corresponding to the
+                                          vertices furnishing the barycentric coordinates referential
+                vol,    float,            the volume of the element furnishing the barycentric coordinates referential
+                
+        Output: EvaluatedGradient, 3D numpy array,  the value of all the x and y gradient's coordinates at each 
+                                                    given point, ordered accordingly to the vertices from which 
+                                                    the barycentic coordinates have been generated 
+                                                    (NbBasisFunc x NbPoints x 2).
+                                                
+        Note, in all the graphics below the sample vertex ordering is given by:       1
+                                                                                    /   \
+                                                                                   2 ___ 3
+    --------------------------------------------------------------------------------------- """
+    
+    # Flippling the orientation of the normal as required in the gradient's formula
+    n = -n
+   
+    # Bernstein basis function
+    if Type == "B":
+        # ------------------------------------------------------------------------
+        if Order == 1:
+            """
+                Gives the basis functions in the order  1
+                                                      /   \
+                                                     2 ___ 3
+            """
+            
+            # Initilising the gradient array
+            Grad = np.zeros((points.shape[0], 2, 3))
+            
+            # Performing the triangle and order dependent computations
+            EvaluatedGradient = np.zeros((points.shape[0], 3, 3))
+            EvaluatedGradient[:, :, :] = np.array([[1.0, 0.0, 0.0],
+                                                   [0.0, 1.0, 0.0],
+                                                   [0.0, 0.0, 1.0]])
+        
+        # ------------------------------------------------------------------------
+        if Order == 2: 
+            """
+                Gives the basis functions in the order  1
+                                                      /   \
+                                                     2     6
+                                                    /       \
+                                                   3 __ 4 __ 5
+            """
+            # Initilising the gradient array
+            Grad = np.zeros((points.shape[0], 2, 6))
+            
+            # Performing the triangle and order dependent computations
+            EvaluatedGradient = np.zeros((points.shape[0], 6, 3))
+            for i in range(points.shape[0]):
+                EvaluatedGradient[i, :, :] = np.array([[2*points[i,0], 0.0, 0.0],
+                                                       [0.0, 2*points[i,1], 0.0],
+                                                       [0.0, 0.0, 2*points[i,2]],
+                                                       [2*points[i,1], 2*points[i,0], 0.0],
+                                                       [0., 2*points[i,2], 2*points[i,1]],
+                                                       [2*points[i,2], 0., 2*points[i,0]]])
+            
+            # Ordering it according to the MeshStructure rather than to the Gmsh format
+            EvaluatedGradient = EvaluatedGradient[:, [0, 3, 1, 4, 2, 5], :]
+           
+        # ------------------- Computing the gradient --------------------------------------------
+        Grad = np.dot(np.swapaxes(np.array([EvaluatedGradient[:,:,0]-EvaluatedGradient[:,:,2],
+                                            EvaluatedGradient[:,:,1]-EvaluatedGradient[:,:,2]]),\
+                                  0,2),\
+                      n[1:,:])/(2.*vol)
+        
+    # Returning the computed quantity
+    return(Grad)
diff --git a/LocalLib/BarycentricTools.py b/LocalLib/BarycentricTools.py
new file mode 100644
index 0000000..ae21f16
--- /dev/null
+++ b/LocalLib/BarycentricTools.py
@@ -0,0 +1,69 @@
+"""
+#####################################################################################
+#		File that procures the necessary tools to handle barycentric coordinates    #
+#####################################################################################
+
+Implemented features:
+	->
+	->
+
+Usage:
+	
+Note:
+
+----------------------------------------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+# ==============================================================================
+#	GetBarycentricCoordinates handling
+# ==============================================================================
+
+#### Get the barycentric coordinates in their classical definiton ####
+def GetBarycentricCoordinates(points, vertices):
+    """ Barycentric coordinates from the given vertices (forming a triangles)
+    
+    Input: Points,    2D numpy array,  Cartesian coordinates of the points to convert (NbPoints x 2)
+           vertices,  2D numpy array,  Cartesian coordinates of the points taken as reference, ordered. (3 x 2)
+           
+    Output: bcoors,  2D numpy array,  Barycentric coordinates of points with respect to the vertices (NbPoints x 3)
+    ----------------------------------------------------------------------"""
+    
+    # Performing the computations of the barycentric coordinates
+    hyp1 = vertices[1,:]-vertices[0,:]
+    hyp2 = vertices[2,:]-vertices[0,:]
+    dist = points-vertices[0,:]
+    
+    dhyp11 = np.dot(hyp1, hyp1)
+    dhyp12 = np.dot(hyp1, hyp2)
+    dhyp22 = np.dot(hyp2, hyp2)
+    
+    dhyp31 = np.dot(dist, hyp1)
+    dhyp32 = np.dot(dist, hyp2)
+    
+    quotu = dhyp11*dhyp22-dhyp12**2
+    
+    b1 = (dhyp22*dhyp31-dhyp12*dhyp32)/quotu
+    b2 = (dhyp11*dhyp32-dhyp12*dhyp31)/quotu
+    bcoords = np.array([1-b1-b2, b1, b2]).T
+
+    # Returning the barycentric coordinates
+    return(bcoords)
+
+#### Retrieving the cartesian coordinates ####
+def GetCartesianCoordinates(bcoords, vertices):
+    """ Barycentric coordinates from the given vertices (forming a triangles)
+    
+    Input:  bcoors,    2D numpy array,  Barycentric coordinates of points with respect to the vertices (NbPoints x 3)
+            vertices,  2D numpy array,  Cartesian coordinates of the points taken as reference, ordered. (3 x 2)
+           
+    Output: Points,    2D numpy array,  Cartesian coordinates of the points to convert (NbPoints x 2)
+    ----------------------------------------------------------------------"""
+    
+    # Computing back the cartesian points and returning them
+    points = np.dot(bcoords, vertices)
+    return(points)
+    
diff --git a/LocalLib/LobattoQuadratureNodes.py b/LocalLib/LobattoQuadratureNodes.py
index 8a327bf..0ca101a 100644
--- a/LocalLib/LobattoQuadratureNodes.py
+++ b/LocalLib/LobattoQuadratureNodes.py
@@ -69,11 +69,11 @@ def GaussLobatto(n,eps):
         P[:,0] = 1.0
         P[:,1] = x
 
-        for k in range(2,n+1):
-            P[:,k] = ( (2*k-1)*x*P[:,k-1] - (k-1)*P[:,k-2] ) / k
-        x = xold - ( x*P[:,n] - P[:,n-1] )/( (n+1)*P[:,n])
+        for k in range(2,n+1): P[:,k] = ((2*k-1)*x*P[:,k-1] - (k-1)*P[:,k-2])/k
+        x = xold - (x*P[:,n] - P[:,n-1])/((n+1)*P[:,n])
 
         if (np.max(np.abs(x - xold).flatten()) < eps): break
 
-    w = 2.0 / ( (n*(n+1))*(P[:,n]**2))
-    return x, w
+    w = 2.0/((n*(n+1))*(P[:,n]**2))
+    
+    return(x, w)
diff --git a/LocalLib/ReversedFunctools.py b/LocalLib/ReversedFunctools.py
index f560564..8d34962 100644
--- a/LocalLib/ReversedFunctools.py
+++ b/LocalLib/ReversedFunctools.py
@@ -1,4 +1,18 @@
+"""
+################################################################################
+#			File that procures simple pythonic hacks            			   #
+################################################################################
 
+Implemented features:
+	->
+	->
+
+Usage:
+	
+Note:
+
+----------------------------------------------------------------------------------------------------------"""
+
+#### Partial function applying the last arguments ####
 def partial(func, *args):
-    """Partially applies last arguments."""
-    return lambda *a: func(*(a + args))
+    return(lambda *a: func(*(a + args)))
diff --git a/LocalLib/__pycache__/BarycentricBasisFunctions.cpython-38.pyc b/LocalLib/__pycache__/BarycentricBasisFunctions.cpython-38.pyc
new file mode 100644
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diff --git a/LocalLib/__pycache__/BarycentricTools.cpython-38.pyc b/LocalLib/__pycache__/BarycentricTools.cpython-38.pyc
new file mode 100644
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diff --git a/PredefinedTests/PredefinedMeshes/BigSimpleRectangle_o1_CG_res_50.pyMsh b/PredefinedTests/PredefinedMeshes/BigSimpleRectangle_o1_CG_res_50.pyMsh
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diff --git a/PredefinedTests/PredefinedMeshes/BigSimpleSquare_o1_CG_res_20.pyMsh b/PredefinedTests/PredefinedMeshes/BigSimpleSquare_o2_CG_res_20.pyMsh
similarity index 66%
rename from PredefinedTests/PredefinedMeshes/BigSimpleSquare_o1_CG_res_20.pyMsh
rename to PredefinedTests/PredefinedMeshes/BigSimpleSquare_o2_CG_res_20.pyMsh
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diff --git a/PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o1_CG_res_50.pyMsh b/PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o1_CG_res_50.pyMsh
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diff --git a/PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o5_DG_res_10.pyMsh b/PredefinedTests/PredefinedMeshes/LShapeWithHoleAndRoundCorner_o5_DG_res_10.pyMsh
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diff --git a/PredefinedTests/PredefinedMeshes/SimpleSquare_o1_CG_res_50.pyMsh b/PredefinedTests/PredefinedMeshes/SimpleSquare_o1_CG_res_50.pyMsh
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diff --git a/PredefinedTests/PredefinedProblems/__pycache__/Advection_Rotation_1Fluid.cpython-38.pyc b/PredefinedTests/PredefinedProblems/__pycache__/Advection_Rotation_1Fluid.cpython-38.pyc
index 649fc723b23c0bae8dc191dfce842bff0848b7ca..bc51712f1e104bfe6b227856bdfa552e2138d514 100644
GIT binary patch
delta 113
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delta 113
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diff --git a/PredefinedTests/PredefinedProblems/__pycache__/Advection_Translation_1Fluid.cpython-38.pyc b/PredefinedTests/PredefinedProblems/__pycache__/Advection_Translation_1Fluid.cpython-38.pyc
index e2bf59d6287fa4333e0841e2e3074f97b2b4a550..d4ea580459ac5708d2361fde9191558631e3cfe7 100644
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diff --git a/PredefinedTests/PredefinedSettings/CustomSettingsTemplate.txt b/PredefinedTests/PredefinedSettings/CustomSettingsTemplate.txt
index 62c1f5c..2639067 100644
--- a/PredefinedTests/PredefinedSettings/CustomSettingsTemplate.txt
+++ b/PredefinedTests/PredefinedSettings/CustomSettingsTemplate.txt
@@ -1,6 +1,6 @@
 # Spatial scheme properties
-1			# Index of the scheme
-1 0		    # Scheme properties (see doc for more details, if any, the order should be first)
+0			# Index of the scheme
+2 B		    # Scheme properties (see doc for more details, if any, the order should be first)
 0			# Index of limiter (0=None)
 0 0		    # Limiter properties (see doc for more details)
 1			# Type of fluid spotter (1=LS)
@@ -13,7 +13,7 @@
 # Temporal scheme properties
 1			# Time scheme
 2 2 		# Properties (RK order, Dec order, as many integers as needed, in the order required by the scheme (see docs))
-0.4    	# CFL
+0.2         # CFL
 1			# Tmax, in seconds
 
 # Export properties
diff --git a/RunPostProcess.py b/RunPostProcess.py
index 05d6593..35b35ef 100644
--- a/RunPostProcess.py
+++ b/RunPostProcess.py
@@ -29,9 +29,9 @@ if __name__ == '__main__':
     # "Advection_Translation_2Fluids.py"
     # "Advection_Translation_1Fluid.py" 
     # "Advection_Rotation_1Fluid.py"
-    ProblemDefinition = "Advection_Rotation_1Fluid"
+    ProblemDefinition = "Euler_StationaryVortex_1Fluid"
     SettingsChoice    = "CustomSettingsTemplate"
-    MeshResolution    = 50
+    MeshResolution    = 20
 
     # Post - processing tasks
     #PostProcess.Plot_DebugMatplotlib(ProblemDefinition, SettingsChoice, [0.0], MeshResolution)
diff --git a/RunProblem.py b/RunProblem.py
index 88d2715..999cf2d 100644
--- a/RunProblem.py
+++ b/RunProblem.py
@@ -26,9 +26,9 @@ if __name__ == '__main__':
     # "Advection_Translation_2Fluids.py"
     # "Advection_Translation_1Fluid.py" 
     # "Advection_Rotation_1Fluid.py"
-    ProblemDefinition = "Advection_Rotation_1Fluid"
+    ProblemDefinition = "Advection_Translation_1Fluid"
     SettingsChoice    = "CustomSettingsTemplate"
-    MeshResolution    = 50
+    MeshResolution    = 20
 
     # Solve the problem
     Solve.Solve(ProblemDefinition, SettingsChoice, MeshResolution)
diff --git a/SourceCode/Mappers.py b/SourceCode/Mappers.py
index 0faed09..88fd984 100644
--- a/SourceCode/Mappers.py
+++ b/SourceCode/Mappers.py
@@ -99,8 +99,13 @@ def Spatial_Scheme(id):
     """ Mapper that defines the temporal scheme to use.
     Returns the spatial scheme module"""
 
-    if   id==1:  return(SC.CellCentredFV_LFX)
-    elif id==2:  return(SC.VertexCentredFV_LFX)
+    if   id==0:  return(SC.ContinuousGalerkin)
+    if   id==1:  return(SC.VertexCentredFV_LFX)
+    elif id==2:  return(SC.CellCentredFV_Natural)
+    elif id==3:  return(SC.VertexCentredFV_Natural)
+    elif id==4:  return(SC.CellCentredFV_LFX)
+    if   id==5:  return(SC.VertexCentredFV_LFXorig)
+    if   id==6:  return(SC.CellCentredFV_LFXorig)
 
     else: raise ValueError("Error in the problem definition:\n\
                             Wrong index for the wished spatial scheme.\n\
@@ -135,6 +140,7 @@ def Fluid_Selector(id):
     if   id == 1: return(FS.NaiveLS_CellCentredLFX)
     elif id == 2: return(FS.NaiveLS_VertexCentredLFX)
 
+
     else: raise ValueError("Error in the problem definition:\n\
                             Wrong index for the wished fluid selector.\n\
                             Check the Mapper and ProblemSettings files.\n\
diff --git a/SourceCode/Meshing/MeshStructure.py b/SourceCode/Meshing/MeshStructure.py
index da528ad..ca81d21 100644
--- a/SourceCode/Meshing/MeshStructure.py
+++ b/SourceCode/Meshing/MeshStructure.py
@@ -31,9 +31,10 @@ class MeshStructure():
 		self.BoundaryVerticesMap  = []		# Dictionary that maps directly the vertices number to the corresponding tag
 
 		self.ElementsBoundaryWiseDofs = []	# 3-level nested list giving the Dofs that are lying on each boundary of each inner element: [element][boundary] -> [dof list]
-		self.ElementsBoundaryDofs     = []	# 2-level nested list giving the Dofs lying on the boundary of an element [element] -> [dof list]
-		self.ElementsInnerDofs        = []	# 2-level nested list giving the Dofs living strictly within the element [element] -> [dof list]
-
+		self.ElementsBoundaryDofs     = []	# 1-level list giving the Dofs lying on the boundary of an element [element] -> [dof list]
+		self.ElementsInnerDofs        = []	# 1-level list giving the Dofs living strictly within the element [element] -> [dof list]
+		self.Mesh.Domain              = []  # 1-level list giving the Dofs living within the element [element] -> [dof list]
+		
 		self.BoundaryDofs    = []			# List of all the domain's boundary Dofs (warning: the list index is meaningless here)
 		self.BoundaryDofsTag = []			# Corresponding labelling of the boundary (warning: the list index has to be mapped with the index of self.BoundaryDofs, it does not relate to the Dof number directly)
 		self.BoundaryDofsMap = []			# Dictionary creatng a direct access to the tag given a boundary Dof index
diff --git a/SourceCode/Meshing/Meshing_DualMesh.py b/SourceCode/Meshing/Meshing_DualMesh.py
index 65bf7e3..250ced8 100644
--- a/SourceCode/Meshing/Meshing_DualMesh.py
+++ b/SourceCode/Meshing/Meshing_DualMesh.py
@@ -135,4 +135,5 @@ class DualMesh(MeshHandling):
                 self.Mesh.DualAreas[i][f] = Pp.area
 
                 # Filling the area of the dual control cell
-                self.Mesh.DualAreasTot[self.Mesh.ElementsBoundaryDofs[i][f]] += self.Mesh.DualAreas[i][f]
+                if self.Mesh.ElementsBoundaryDofs[i]:
+                    self.Mesh.DualAreasTot[self.Mesh.ElementsBoundaryDofs[i][f]] += self.Mesh.DualAreas[i][f]
diff --git a/SourceCode/Meshing/Meshing_PrimalMesh.py b/SourceCode/Meshing/Meshing_PrimalMesh.py
index 24f1eb4..39dedfb 100644
--- a/SourceCode/Meshing/Meshing_PrimalMesh.py
+++ b/SourceCode/Meshing/Meshing_PrimalMesh.py
@@ -179,7 +179,8 @@ class MeshHandling():
 		if len(args) == 1:
 			self.Mesh = args[0]
 			self.ExtractMeshProperties()
-
+			self.ExtractVariationalProperties()
+			
 		# If a file is to be loaded, filling the above created elements
 		elif len(args) == 4:
 			self.Mesh.MeshName       = args[0]
@@ -191,13 +192,15 @@ class MeshHandling():
 			try:
 				self.LoadMesh()
 				self.ExtractMeshProperties()
-
+				self.ExtractVariationalProperties()
+				
 			# If no binary file, try to load it from text file and generate binary file
 			except:
 				try:
 					self.ReadTxtMesh()
 					self.ExportBinaryMesh()
 					self.ExtractMeshProperties()
+					self.ExtractVariationalProperties()
 
 				# If neither files were found
 				except:
@@ -254,46 +257,46 @@ class MeshHandling():
 		# registered in the structure (during the init phase), convenient
 		# for reloading a new mesh when changing only one parameter
 		------------------------------------------------------------------------"""
-
+		
 		# ----------- Registering the target mesh options ----------------------
 		# Saving the preregistered values
 		Name  = self.Mesh.MeshName
 		Order = self.Mesh.MeshOrder
 		VType = self.Mesh.MeshDofsType
 		Resolution = self.Mesh.MeshResolution
-
+		
 		# Overriting depending on the optional items desired by the user
 		for key, value in kwargs.items():
 			if key == "MeshName": Name = value
 			if key == "MeshResolution": Resolution = value
 			if key == "MeshDofsType": VType = value
-
+		
 		# ---------- Reloading a new mesh --------------------------------------
 		# Constructing the mesh file name and opening the file
 		FileName = os.path.join(Pathes.MeshPath, Name + "_o{0:1d}_{1!s}_res_{2:02d}.Msh".format(Order, VType, Resolution))
 		MeshFile = open(FileName, "r")
-
+		
 		# Loading the full file content in a single list of strings
 		MeshContent = list(map(str.strip, MeshFile.readlines()))
-
+		
 		# ---------- Reading and splitting the mesh-properties line-wise -------
 		# Initialise the mesh structure
 		Mesh = MeshStructure()
-
+		
 		# Mesh name
 		Mesh.MeshName =  Name
-
+		
 		# Mesh element size parameters (the hmax will be overwriten when exporting the mesh properties)
 		specs = list(filter(None, MeshContent[2].split("\t")))
 		Mesh.MeshResolution, self.Mesh.hmax = int(specs[0]), float(specs[1])
-
+		
 		# Getting the global properties of the mesh
 		Dofs = list(filter(None,MeshContent[3].split("\t")))
 		Mesh.MeshOrder, self.Mesh.MeshDofsType = int(Dofs[0]), Dofs[1]
 		Mesh.MeshType     = list(map(int, filter(None,MeshContent[4].split(" "))))
 		Mesh.Convexity    = int(MeshContent[5])
 		Mesh.BdConformity = int(MeshContent[6])
-
+		
 		# Getting the number of the various quantities
 		CurrLine = 7
 		Mesh.NbMeshPoints       = int(MeshContent[CurrLine + 2])
@@ -301,33 +304,33 @@ class MeshHandling():
 		Mesh.NbBoundaryElements = int(MeshContent[CurrLine + 4])
 		Mesh.NbDofs             = int(MeshContent[CurrLine + 5])
 		Mesh.NbBoundaryDofs     = int(MeshContent[CurrLine + 6])
-
+		
 		# Extracting the coordinates of the vertices
 		CurrLine = CurrLine + 9
 		NextLine = CurrLine + self.Mesh.NbMeshPoints 
 		Mesh.CartesianPoints      = np.zeros((self.Mesh.NbMeshPoints, 2))
 		Mesh.CartesianPoints[:,:] = [list(map(float, a.split(" "))) for a in MeshContent[CurrLine:NextLine]]
-
+		
 		# Extracting the coordinates of the dofs
 		CurrLine = CurrLine + 2
 		NextLine = CurrLine + self.Mesh.NbDofs
 		Mesh.DofsXY      = np.zeros((self.Mesh.NbDofs, 2))
 		Mesh.DofsXY[:,:] = [list(map(float, a.split(" "))) for a in MeshContent[CurrLine:NextLine]]
-
+		
 		# Retrieving the 2D elements and their neighbors
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbInnerElements
 		buff = [list(map(int, filter(None,re.split(' |\\|',a)))) for a in MeshContent[CurrLine:NextLine]]
 		Mesh.InnerElements = [a[1:a[0]+1] for a in buff]
 		Mesh.Neighborhoods = [a[a[0]+1:2*a[0]+1] for a in buff]
-
+		
 		# Retrieving the domain's boundary elements (1D) and their neigbors
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbBoundaryElements
 		buff = [list( map(int, filter(None,re.split(' |\\|',a)))) for a in MeshContent[CurrLine:NextLine]]
 		Mesh.BoundaryElements    = [a[0:2] for a in buff]
 		Mesh.BoundaryElementsTag = [a[2] for a in buff]
-
+		
 		# Retrieving the vertices lying on the domain's boundary
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbBoundaryElements
@@ -335,7 +338,7 @@ class MeshHandling():
 		Mesh.BoundaryVertices    = [a[0] for a in buff]
 		Mesh.BoundaryVerticesTag = [a[1] for a in buff]
 		Mesh.BoundaryVerticesMap = dict(zip(self.Mesh.BoundaryVertices, self.Mesh.BoundaryVerticesTag))
-
+		
 		# Retrieving the Dofs of each 2D element
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbInnerElements
@@ -343,13 +346,17 @@ class MeshHandling():
 		Mesh.ElementsInnerDofs        = [list(map(int, filter(None,re.split(' ',a[1])))) for a in buff]
 		Mesh.ElementsBoundaryWiseDofs = [[list(map(int, filter(None, re.split(' ', b)))) for b in a[2:]] for a in buff ]
 		Mesh.ElementsBoundaryDofs     = [np.unique(list(chain(*a))).tolist() for a in self.Mesh.ElementsBoundaryWiseDofs]
-
+		
 		# Retrieiving the Dofs lying on the boundary elements
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbBoundaryElements
 		buff = [list( map(int, filter(None,re.split(' ',a)))) for a in MeshContent[CurrLine:NextLine]]
 		Mesh.BoundaryElementsDofs = [a for a in buff]
-
+		
+		# Retrieve a direct access to the dofs, element-wise
+		Mesh.ElementsDofs = [[]]*self.Mesh.NbInnerElements
+		for i in range(Mesh.NbInnerElements): Mesh.ElementsDofs[i] = Mesh.ElementsBoundaryDofs[i] + Mesh.ElementsInnerDofs[i]
+		
 		# Retrieving the global access to the boundary Dofs, not binded by elements
 		CurrLine = NextLine + 2
 		NextLine = CurrLine + self.Mesh.NbBoundaryDofs
@@ -357,7 +364,7 @@ class MeshHandling():
 		Mesh.BoundaryDofs    = [a[0] for a in buff]
 		Mesh.BoundaryDofsTag = [a[1] for a in buff]
 		Mesh.BoundaryDofsMap = dict(zip(self.Mesh.BoundaryDofs, self.Mesh.BoundaryDofsTag))
-
+		
 		# Saving it in the internal Structure
 		self.Mesh = Mesh
 		self.ExtractMeshProperties()
@@ -366,40 +373,39 @@ class MeshHandling():
 	def ExtractMeshProperties(self):
 		"""# Input: A freshly loaded mesh in the mesh structure
 		# Output: The same mesh, with the three additional fields InnerElementsVolume,
-		#         InnerElementsDiameter, ElementsBoundaryWiseNormals filled
+		#         InnerElementsDiameter, ElementsBoundaryWiseNormals, ElementsDofs  filled
 		# WARNING: The boundaryElements list should not be empty
 		-----------------------------------------------------------------------"""
-
+		
 		# ------------- Initilising the four new fields -----------------------
 		self.Mesh.InnerElementsVolume   = np.zeros(self.Mesh.NbInnerElements)
 		self.Mesh.InnerElementsDiameter = np.zeros(self.Mesh.NbInnerElements)
 		self.Mesh.ElementsBoundaryWiseNormals = [[]]*self.Mesh.NbInnerElements
 		self.Mesh.ElementsBoundaryWiseWidth   = [[]]*self.Mesh.NbInnerElements
-		self.Mesh.Domain = []
-
+		
 		# ------------- Computing the mesh properties element-wise -------------
 		# Spanning the inner elements to get their volume and their diameter
 		for (i, ele) in enumerate(self.Mesh.InnerElements):
 			# Get the coordinates of the vertices
 			xy = self.Mesh.CartesianPoints[ele,:]
-
+			
 			# Assigning the flip parameter to compute the outer normal regardless the element's orientation
 			ori = 1
 			if not ShapelyLinearRing([tuple(a) for a in xy]).is_ccw: ori = -1
-
+		
 			# Computing the area of the element either by the shoelace formula or the shapely tools
 			#area = 0.5*np.abs(np.dot(xy[:-1,0], xy[1:,1]) - np.dot(xy[:-1,1], xy[1:,0]) + xy[-1,0]*xy[0,1] - xy[-1,1]*xy[0,0])
 			#self.Mesh.InnerElementsVolume[i] = area
 			Pp = ShapelyPolygon([tuple(a) for a in xy]).area
 			self.Mesh.InnerElementsVolume[i] = Pp
-
+			
 			# Computing the maximum spacing between two vertices of the element
 			diam = 0
 			for j in range(len(xy[:,0])):
 				for k in range(0,j):
 					diam = max(diam, np.linalg.norm(xy[j,:]-xy[k,:],2))
 			self.Mesh.InnerElementsDiameter[i] = diam
-
+			
 			# Computing the outward normal of the element
 			XY = np.append(xy,[xy[0,:]],axis=0)
 			normals = np.zeros((len(xy[:,0]),2))
@@ -412,30 +418,30 @@ class MeshHandling():
 			
 		# --------- Computing the boundary rings and sorting them to find the outer and inners boundary rings -------------
 		# This does not support completely disconnected domains, and is not optimized
-	
+		
 		# --- Creating the linear rings
 		# Gathering all the boundary elements (and take care of python closures)
 		BdElements = copy.deepcopy(self.Mesh.BoundaryElements)
-
+		
 		# Check if we can actually perform the extraction of the boundary rings or not
 		if not BdElements:	raise(ValueError("No boundary element registered. Cannot find the outer boundary ring. Abortion."))
-
+		
 		# Initialising the local rings list and buffers
 		LinearRingsList   = []
 		CollectedVertices = []
 		RingVertices      = []
-
+		
 		# Initialise the first ring by the first vertex of the boundary element
 		CurrentElement = BdElements[0]
 		CurrentVertex  = CurrentElement[0]
 		BdElements.remove(BdElements[0])
-
+		
 		# Span the boundary elements while all the boundary points are not collected until all the
 		# rings are formed. WARNING: if the boundary elements are not forming close lines then it will crash
 		j=0
 		while j<self.Mesh.NbBoundaryElements:		# Just for safety, not practically doing anything
 			j=j+1
-
+			
 			# If the construction of the current ring is now closed and there is no more
 			# non-assigned boundary elements, then perform the end operations
 			if not BdElements and \
@@ -446,67 +452,146 @@ class MeshHandling():
 				# Enrich the list of already collected vertices
 				CollectedVertices = CollectedVertices + RingVertices
 				break
-
+			
 			# Check if the BdElement is valid, and if so continuing the ring construction
 			if np.shape(np.array(BdElements).T)[0] == 2:
 				# Get the next boundary element connected to this one
 				Idx, Idy = np.where(np.array(BdElements) == CurrentVertex)
-
+				
 				# There should be only one or none (end of the loop) index
 				if len(Idx)>1:
 					raise(ValueError("Boundary vertex shared by more than two boundary elements. Abortion"))
-
+					
 				# Case where the ring is to be continued
 				elif len(Idx)==1:
 					# Register the connected vertex to the current ring
 					RingVertices = RingVertices + [CurrentVertex]
-
+					
 					# Locate the Cursor to the next element (the connected vertex: complement to Idx)
 					# And flipping the current element to follow the spanning order
 					CurrentElement = BdElements[Idx[0]]
 					CurrentVertex  = CurrentElement[1-Idy[0]]
-
+					
 					# Remove the element that is currently spanned from the BdElements
 					BdElements.remove(CurrentElement)
-
+					
 				# If none index, check that indeed, we are closing the loop
 				elif len(Idx)==0:
 					# If we can indeed close the ring (the last vert would form an element with the first registered vertex)
 					if (   [RingVertices[0], CurrentVertex] in self.Mesh.BoundaryElements \
 						or [CurrentVertex, RingVertices[0]] in self.Mesh.BoundaryElements):
-
+						
 						# Close the ring and register it as a boundary
 						LinearRingsList   = LinearRingsList   + [RingVertices]
 						# Enrich the list of already collected vertices
 						CollectedVertices = CollectedVertices + RingVertices
-
+						
 						# Jump to the next arbitrary beginning of the ring
 						CurrentElement = BdElements[0]
 						CurrentVertex = CurrentElement[0]
-
+						
 						# Jump to another ring
 						RingVertices = []
-
+						
 						# Remove the element that is currently spanned from the BdElements
 						BdElements.remove(CurrentElement)
-
+						
 					else:
 						raise(ValueError("Error in closing the boundary ring. Abortion."))
 			else:
 				raise(ValueError("Encountered an invalid boundaryElement in the boundary ring creation. Abortion."))
-
+			
 		# --- Determining whose linear ring is the outerboundary (nested domains are not supported)
 		# Construct the polygons list out of the rings
 		BoundaryPolygonList =  [self.Mesh.CartesianPoints[Ring, :].tolist() for Ring in LinearRingsList]
-
+		
 		# Sorting the polygons by their areas: by assumptions, the polygon defining the bigger aera
 		# will define the outer boundary. It is assumed that the smaller ones form holes (not checked)
 		PolygonAreas = [ShapelyPolygon([tuple(Point) for Point in Poly]).area for Poly in BoundaryPolygonList]
 		Index        = np.argsort(PolygonAreas)[::-1]
-
+		
 		# Sorting the list of domains in the order [external, biggest internal hole, ..., smallest internal hole]
 		self.Mesh.Domain = np.array(BoundaryPolygonList)[Index].tolist()
 
+	### Extracting the variational properties of the mesh ####
+	def ExtractVariationalProperties(self):
+		"""# Input: A freshly loaded mesh in the mesh structure
+		# Output: The same mesh, with the three additional fields
+        #         Dof2Vertex,		 correspondance table from global Dof numbering to the corresponding Vertex global index
+		#         Vertex2Dofs,       correspondance table from global Vertex numbering to the corresponding
+		#                            Dofs global indices (in case of DG, mapping to several Dofs may occur)
+		#         TabulatedVertex,   Gives locally the correspondance table from Vertex to Dof
+		#         VertexLocation,    Give locally the position of the vertices in the list of Dofs
+		#         TabulatedDofs,     Gives locally the correspondance table from Dof to Vertex
+		#         Dof2VertexLoc,     Same as Dof2Vertex, but locally to an element
+		#         Vertex2DofLoc,     Same as Vertex2Dofs, but locally to an element
+		#
+		# WARNING: The boundaryElements list should not be empty
+		-----------------------------------------------------------------------"""
+		
+		# --------------- Initialisations --------------------------------------------
+		# Shortcutting the cartesian coordinates of the physical vertices and the Dofs
+		DofsXY = self.Mesh.DofsXY[:,:]
+		VertXY = self.Mesh.CartesianPoints[:,:]
+		
+		# Creating the new fields
+		self.Mesh.Vertex2Dofs = [[]]*self.Mesh.NbMeshPoints
+		self.Mesh.Dofs2Vertex = [[]]*self.Mesh.NbDofs
+
+		self.Mesh.TabulatedVertex = [[]]*self.Mesh.NbInnerElements
+		self.Mesh.VertexLocation  = [[]]*self.Mesh.NbInnerElements
+		self.Mesh.TabulatedDofs   = [[]]*self.Mesh.NbInnerElements
+		self.Mesh.Dof2VertexLoc   = [[]]*self.Mesh.NbInnerElements
+		self.Mesh.Vertex2DofLoc   = [[]]*self.Mesh.NbInnerElements
+		
+		# --------------- Computations of the global mapping -------------------------
+		# Computing the global mapping between the Dofs index and the vertex ones.
+		# NOTE: The two mappings have been separated as it is possible to have high order schemes 
+		# where internal Dofs are not associated to vertices and vertices not corresponding to Dofs
+		
+		# --- Mapping the dof to the vertex
+		for i in range(self.Mesh.NbDofs):
+			# Retrieve the correspondance map w.r.t. the cartesian xy location
+			Coin = np.all(np.isclose(DofsXY[i,:], VertXY), axis=1)
+			vertex = np.where(Coin)[0]
+			
+			# Check if only one vertex got selected
+			if   len(vertex) == 1: self.Mesh.Dofs2Vertex[i] = vertex[0]
+			elif len(vertex) == 0: self.Mesh.Dofs2Vertex[i] = None
+			else: raise ValueError("""Mesh error: a same Dof has been assigned to several vertices.\n
+					                            Invalid Mesh. Abortion.""")
+		
+		# --- Mapping the vertex to dofs
+		for i in range(self.Mesh.NbMeshPoints):
+			# Retrieve the correspondance map w.r.t. the cartesian xy location
+			Coin = np.all(np.isclose(VertXY[i,:], DofsXY), axis=1)
+			dof = np.where(Coin)[0]
+			
+			# In case of discontinuous variational setting, it is possible to have several Dofs
+			# at one vertex. The above check is thus not necessary and we register the output  as a list.
+			self.Mesh.Vertex2Dofs[i] = np.array(dof)
+		
+		# --------------- Computations of the local mapping ------------------------------------
+		# For each element, furnish the local mapping between dofs and vertex
+		# and furnish the vertices in the order associated to the dof ordering (in view of computing
+		# the barycentric coordinates from them)
+		for e in range(self.Mesh.NbInnerElements):
+			# Create the local correspondance table from the previous exported global one
+			self.Mesh.Dof2VertexLoc[e] = np.array([self.Mesh.Dofs2Vertex[p]   for p in self.Mesh.ElementsDofs[e]])
+			self.Mesh.Vertex2DofLoc[e] = np.array([self.Mesh.Vertex2Dofs[p] for p in self.Mesh.InnerElements[e]])
+		
+			# Select the location of the vertices depending on the type of the element
+			if self.Mesh.MeshDofsType in ['DG', 'CG'] and self.Mesh.MeshOrder>0:
+				self.Mesh.VertexLocation[e] = np.array([i*(self.Mesh.MeshOrder) for i in range(len(self.Mesh.InnerElements[e]))])
+			else:
+				self.Mesh.VertexLocation[e] = np.array([0, 1, 2])
+			
+			# (Crucial for CG/DG) Tabulate the Dofs from their local index to make the order of the Dofs correspond to the order
+			# of the physical vertices
+			self.Mesh.TabulatedDofs[e]   = np.array(list([np.where(self.Mesh.Dof2VertexLoc[e][self.Mesh.VertexLocation[e]] == p)[0][0]\
+				                                        for p in self.Mesh.InnerElements[e]]))
+			self.Mesh.TabulatedVertex[e] = np.argsort(self.Mesh.TabulatedDofs[e])
+	
 	#### Export the obtained the best mesh file ####
 	def ExportTextMesh(self):
 		"""# Input:  The mesh structure stored within the instance of mesh
diff --git a/SourceCode/Meshing/Meshing_Table_ElementDiameterIndex.txt b/SourceCode/Meshing/Meshing_Table_ElementDiameterIndex.txt
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diff --git a/SourceCode/PostProcessing/Plots_Matplotlib.py b/SourceCode/PostProcessing/Plots_Matplotlib.py
index f2f941a..a83425b 100644
--- a/SourceCode/PostProcessing/Plots_Matplotlib.py
+++ b/SourceCode/PostProcessing/Plots_Matplotlib.py
@@ -11,7 +11,8 @@ Contains (in scrolling order):
     -> Plotting 3D solutions
 
     -> Animation routines
-
+    
+    -> Checkup routines
 Usage:
 
 Note: -> The module is made to be used on-line, that is either while
@@ -946,6 +947,7 @@ def PlotVariableOnWholeDomain(Problem, Mesh, Solution, VariableId, ParametersId)
 def PlotVariableOneEachSubDomain(Problem, Mesh, Solution, VariableId, ParametersId):
     """ Plot one variable with a color scale depending of the subdomain
     Input:   Problem,        Problem instance,          the considered problem and its properties
+             Mesh,           MeshStructure,             the considered mesh
              Solution,       Solution instance,         the solution of the considered problem
              variableId,     integer,                   index of the desired variable
              ParametersId,   string,                    name of the export folder (TestCase_TestSettings)
@@ -1020,3 +1022,47 @@ def PlotVariableOneEachSubDomain(Problem, Mesh, Solution, VariableId, Parameters
     FileName = "Mplt_3DSol_OnSubDomains_{0!s}_t{1:5.3f}".format(variableName, Solution.t).replace(".","_")
     plt.savefig(os.path.join(FolderName, FileName)+".png", transparent=True, dpi=300)
     plt.close()
+
+
+# ==============================================================================
+#  Checkup tools
+# ==============================================================================
+
+#### Plot the normal and dual information of an element ###
+def PlotNormalDualInfo(Mesh, idx):
+    """ Plots the normals and dual information of an element (normals, dual normals, barycentric point)
+    for the considered element.
+    Input:  Mesh,           MeshStructure,             the considered mesh
+    Output: None, Runtime-plotted plot
+    --------------------------------------------------------------------------------------------"""
+
+    # Select the physical element to plot
+    ele = self.Mesh.InnerElements[idx]
+    
+    # Retrieve its physical properties
+    coords = self.Mesh.CartesianPoints[ele+[ele[0]],:]
+    a = self.Mesh.DualCenters[idx]
+    
+    n = self.Mesh.ElementsBoundaryWiseNormals[idx]
+    v = self.Mesh.DualNormals[idx]
+        
+    # Loop on each face to plot the associated properties
+    for f in range(len(ele)):
+        
+        # Retrieve the mid-point of the face
+        md = a[f]
+        
+        # Retrieve the mid-point of the associated dual segment
+        midpoint = 0.5*(a[f]+a[-1])
+        
+        # Plot the associated dual segment in blue and the face segment in red
+        plt.plot([a[f][0], a[-1][0]], [a[f][1], a[-1][1]], "-pb")
+        plt.plot(coords[f:f+2,0], coords[f:f+2,1], "*r-")
+        
+        # Plot the normal information
+        plt.quiver(midpoint[0], midpoint[1], v[f][0], v[f][1])
+        plt.quiver(md[0], md[1], n[f][0], n[f][1])
+    
+    # Show the element
+    plt.show()
+
diff --git a/SourceCode/PostProcessing/Plots_Plotly.py b/SourceCode/PostProcessing/Plots_Plotly.py
index 3fbb007..46c0f7b 100644
--- a/SourceCode/PostProcessing/Plots_Plotly.py
+++ b/SourceCode/PostProcessing/Plots_Plotly.py
@@ -707,7 +707,7 @@ def GenerateFrameSolutionOnEachSubdomain(Problem, Mesh, Solution, cbounds, Varia
     Y  = Mesh.CartesianPoints[:,1]
     Z  = Solution.RSol[VariableId,:]
     LS = Solution.RLSValues[:,:]
-
+    
     # -------------------- Customisation properties --------------------------------------
     # Retrieve the variables names
     variableName      = Problem.GoverningEquations.VariablesNames[VariableId]
diff --git a/SourceCode/PostProcessing/__pycache__/Plots_Matplotlib.cpython-38.pyc b/SourceCode/PostProcessing/__pycache__/Plots_Matplotlib.cpython-38.pyc
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diff --git a/SourceCode/PostProcessing/__pycache__/Plots_Plotly.cpython-38.pyc b/SourceCode/PostProcessing/__pycache__/Plots_Plotly.cpython-38.pyc
index 338960367f5f5710506cae55f075836b5b078989..86a17bb39cc3e0144c53dee9a9e47d24c543484d 100644
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diff --git a/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionRotation.py b/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionRotation.py
index c2d7151..1f4b44f 100644
--- a/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionRotation.py
+++ b/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionRotation.py
@@ -291,12 +291,13 @@ class InitialConditions():
 
 		# Retrieving the xy values of the points and the geometrical informations from the subdomain
 		xy = Mesh.DofsXY[PointsID,:]
-		xc = subdomain.centroid.x-2.5
-		yc = subdomain.centroid.y-2.5
+		xc = subdomain.centroid.x-3
+		yc = subdomain.centroid.y-3
 
 		# Defining the space-dependent quantities
 		r2 = ((xy[:,0]-xc)**2+(xy[:,1]-yc)**2)
 		co = 3*np.exp(-1/(1-r2))*(r2<1)
+		
 		Init[0,:] = co
 
 		# Returning the computed values
diff --git a/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionTranslation.py b/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionTranslation.py
index 3439ec7..905782b 100644
--- a/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionTranslation.py
+++ b/SourceCode/ProblemDefinition/GoverningEquations/LinearAdvectionTranslation.py
@@ -98,7 +98,7 @@ class Equations():
 
 		# --------- Computing the Euler flux directly in conservative variables  ------------------------
 		# Giving the flux furnishing the roation
-		Fx =   np.array([Var[0,:]])         # Displacement in x direction
+		Fx = np.array([Var[0,:]])         # Displacement in x direction
 		Fy = 0*np.array([Var[0,:]]) 		# Displacement in y direction
 
 		# Returning the flux
@@ -320,8 +320,11 @@ class InitialConditions():
 		
 		# Defining the space-dependent quantities
 		r2 = ((xy[:,0]-xc)**2+(xy[:,1]-yc)**2)
-		co = 3*np.exp(-1/(1-r2))*(r2<1)
-		Init[0,:] = co-2.
+		co = np.exp(-1/(1-r2))*(r2<1)
+		#Init[0,:] = co-2.
+		
+		#co = (r2<1)
+		Init[0,:] = co
 
 		# Returning the computed values
 		return(Init)
diff --git a/SourceCode/ProblemDefinition/GoverningEquations/__pycache__/EulerEquations.cpython-38.pyc b/SourceCode/ProblemDefinition/GoverningEquations/__pycache__/EulerEquations.cpython-38.pyc
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diff --git a/SourceCode/Solve.py b/SourceCode/Solve.py
index dbe00d2..54a40bf 100644
--- a/SourceCode/Solve.py
+++ b/SourceCode/Solve.py
@@ -46,7 +46,7 @@ import PostProcessing.Exports     as EX
 import Meshing
 import _Solver
 
-
+import matplotlib.pyplot as plt
 # ==============================================================================
 # Actual solving routine
 # ==============================================================================
@@ -109,11 +109,6 @@ def Solve(*argv):
 	 		SolverParams.MeshType, argv[2]).Mesh
 	print("--> Done.")
 	
-	# Loading the mesh depending on the Solver's requirements
-	print("Generating the mass lumping coefficients")
-	_Solver.Solver_Temporal.GetLumpingCoefficients(Mesh)
-	print("--> Done.")
-
 	# ----------- Perform the initialisation tasks ------------------------------------------------
 	# Initialise the variables to work with shortcut the mapped scheme routines
 	print("Initialising the problem")
@@ -124,6 +119,11 @@ def Solve(*argv):
 	print("--> Initialising the solution")
 	Solution = _Solver.Solver_Initialisations.Solution(Problem, Mesh)
 	print("--> Done.")
+	
+	# Generating the mass lumping coefficients
+	print("Generating the mass lumping coefficients")
+	_Solver.Solver_Temporal.GetLumpingCoefficients(Mesh, Solver)
+	print("--> Done.")
 
 	# Initialise the level set on the data at the Dof points
 	print("Initialising the Fluid selector")
@@ -173,7 +173,7 @@ def Solve(*argv):
 		
 		# ----- Inform the user and export the solution when relevant
 		print("Computing for the time {0!s}".format(Solution.t))
-		if np.mod(k,10)==0:
+		if np.mod(k,10)==1:
 			print(np.max(Solution.Sol))
 			print(np.min(Solution.Sol))
 			EX.ExportSolution(TestName, Mesh, ProblemData, Problem, Solution)
diff --git a/SourceCode/_Solver/FluidSelectors/NaiveLS_ContinuousGalerkin.py b/SourceCode/_Solver/FluidSelectors/NaiveLS_ContinuousGalerkin.py
new file mode 100644
index 0000000..7d32553
--- /dev/null
+++ b/SourceCode/_Solver/FluidSelectors/NaiveLS_ContinuousGalerkin.py
@@ -0,0 +1,358 @@
+"""
+################################################################################
+# 		Module implementing a naive level set with redistancing	  	           #
+################################################################################
+
+Contains (in scrolling order):
+
+Usage:
+
+Note:
+-----------------------------------------------------------------------------"""
+
+# ==============================================================================
+# Importing python libraries
+# ==============================================================================
+from shapely.geometry import MultiLineString as ShapelyMultiLineString
+from shapely.geometry import MultiPolygon    as ShapelyMultiPolygon
+from shapely.geometry import Polygon         as ShapelyPolygon
+from shapely.geometry import Point           as ShapelyPoint
+from shapely.ops      import unary_union
+import matplotlib.pyplot   as plt
+import matplotlib.path     as PH
+import numpy               as np
+
+import Plots_Matplotlib as PP
+# ==============================================================================
+#    Defining the actuall class gathering the externally accessible functions
+# ==============================================================================
+class FS():
+	""" Classical level set approach, initialised according to a signed distance
+	and benefiting from a redistancing/initialisation (one LS per fluid registered
+	in the domain)
+	----------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, Problem, Mesh, Params = []):
+		"""Input: 
+			Mesh,      MeshStructure,       the mesh on which the problem is solved
+			Params,    list of strings,     list of parameters passed to the LS"""
+	
+		# Setting the name of the class to be accessible from outside
+		self.FluidSelectorName = "Naive Level Set"
+		# Setting the mesh and parameters for an internal access 
+		self.Problem = Problem
+		self.Mesh    = Mesh
+		self.Params  = Params
+
+	#### External initialisation routine ####
+	def Initialiser(self, Solution):
+		""" External initialisation routine.
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+		Output: LSValues,  float numpy array,   the level set values at the MeshVertex and Dofs,
+		                                        to be copied to the Solution structure (NbSubdomains x (NbMeshPoints+NbDofs))
+		------------------------------------------------------------------------------------------------------------------"""
+
+		# --------------- Shortcutting the required values ----------------------------------------------
+		# Getting the subdomains list
+		Subdomains = Solution.Subdomains
+
+		# Recompute the number of dofs to drop the dependency on the mesh
+		NbFluids = len(Subdomains)									# Valid because straight after initialisation
+		NbDofs   = len(Solution.FluidFlag)									# Only dofs are there
+
+		# Initialising the Level set values
+		LSValues = np.zeros((len(Subdomains), len(Solution.FluidFlag)))
+
+		# ------------- Get the distance from the interfaces --------------------------------------------
+		# Evaluate the distance from Dofs to each boundary, and mark them by +-
+		# depending on their location with respect to the fluid's boundary
+		# Return the LS value for each fluid and for each Dof # may lead to non-precise determination
+		# in case of multiple junctions (side effect of the classicall method.)
+
+		# Extract the boundary of the domain
+		Borders = []
+		for sub in self.Problem.StudyDomain:
+			Borders += [tuple(tuple(a) for a in np.array(sub.exterior.coords.xy).T)]\
+					  +[[tuple(tuple(a) for a in np.array(interior.coords.xy).T)] for interior in sub.interiors]
+		DomainBorder = ShapelyMultiLineString(Borders)
+
+		# Looping on the intial number of fluids present in the computational domain
+		for i in range(NbFluids):
+			# Get the subdomain as a shapely multipolygon
+			FluidArea = Subdomains[i]
+			
+			# Extract the boundary of the fluid
+			Borders = []
+			for sub in FluidArea:
+				Borders += [ tuple(tuple(a) for a in np.array(sub.exterior.coords.xy).T)]\
+						  +[[tuple(tuple(a) for a in np.array(interior.coords.xy).T)] for interior in sub.interiors]
+			FluidBorder = ShapelyMultiLineString(Borders)
+			
+			# Mask the parts that coincide with the domain's boundary itself
+			FluidBorder = FluidBorder.difference(DomainBorder)
+			
+			# Convert to MultiLineString in any case
+			
+			# Computing the distance and location w.r.t. the subdomain through the shapely information
+			Value = np.array([[ShapelyPoint(tuple(pt)).distance(FluidBorder),                 \
+								int(2*FluidArea.intersects(ShapelyPoint(tuple(pt)).buffer(1e-14))-1)] \
+								for pt in self.Mesh.DofsXY]).T
+			
+			# Computing the signed distance
+			LSValues[i,:] = np.prod(Value, axis=0)
+
+		# Returning the fresh LS values
+		return(LSValues)
+
+
+
+	#### External initialisation routine ####
+	def Initialiser2(self, Solution):
+		""" External initialisation routine.
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+		Output: LSValues,  float numpy array,   the level set values at the MeshVertex and Dofs,
+		                                        to be copied to the Solution structure (NbSubdomains x (NbMeshPoints+NbDofs))
+		------------------------------------------------------------------------------------------------------------------"""
+
+		# --------------- Shortcutting the required values ----------------------------------------------
+		# Getting the subdomains list
+		Subdomains = Solution.Subdomains
+
+		# Recompute the number of dofs to drop the dependency on the mesh
+		NbFluids = len(Subdomains)									# Valid because straight after initialisation
+		NbDofs   = len(Solution.FluidFlag)									# Only dofs are there
+
+		# Initialising the Level set values
+		LSValues = np.zeros((len(Subdomains), len(Solution.FluidFlag)))
+
+		# ------------- Get the distance from the interfaces --------------------------------------------
+		# Evaluate the distance from Dofs to each boundary, and mark them by +-
+		# depending on their location with respect to the fluid's boundary
+		# Return the LS value for each fluid and for each Dof # may lead to non-precise determination
+		# in case of multiple junctions (side effect of the classicall method.)
+
+		# Looping on the intial number of fluids present in the computational domain
+		for i in range(NbFluids):
+			# Computing the distance and location w.r.t. the subdomain through the shapely information
+			FluidArea = Subdomains[i]
+			Value = np.array([[FluidArea.boundary.distance(ShapelyPoint(tuple(pt))),                 \
+			                   int(2*FluidArea.intersects(ShapelyPoint(tuple(pt)).buffer(1e-14))-1)] \
+							   for pt in self.Mesh.DofsXY]).T
+			# Computing the signed distance
+			LSValues[i,:] = np.prod(Value, axis=0)
+
+		# Returning the fresh LS values
+		return(LSValues)
+
+	#### Determining the fluid flags depending on the various values of the level set #####
+	def FlagDofs(self, Solution):
+		""" Flags the degrees of freedom according to the different values of the 
+			associated level set.
+		
+			Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+			Output: None,                           updating directly the value of FluidFlag in the
+			                                        solution instance
+		--------------------------------------------------------------------------------------- """
+		
+		# ---------------- Updating the flags at Dofs ------------------------------
+		Solution.FluidFlag = np.argmax(Solution.LSValues, axis=0)
+	
+	#### Recomputing the subdomains according to the new values of the level set #####^
+	def UpdateSubdomains(self, Solution):
+		""" Method modifying the subdomains associated to the Solution upon the values of 
+		LSValues.
+		
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+			    Output:    None,                updating directly the subdomains in the
+			                                    solution instance
+		
+		WARNING: Not safe yet if the first path given out is internal 
+		---------------------------------------------------------------------------- """
+		
+		# ------- Updating the subdomains from the countour values -----------------
+		# Matplotlib initialisation to get back the countours
+		fig = plt.figure()
+		ax = fig.add_subplot(2, 1, 1)
+		
+		# Update each of the fluid's locations
+		for i in range(self.Problem.NbFluids):
+			# ---------- Initialisations --------------------------------------------
+			# Buffering the LSvalues to modify them on the edges upon our wishes to infer closed contours
+			Buffer = Solution.LSValues[:,:]
+		
+			# ----------- Editing the LS value with a pythonic hack -----------------
+			# Force the boundary of the domain to be part of the contour if relevant
+			ind = np.where(Solution.LSValues[i,self.Mesh.BoundaryDofs]>=0)[0]
+			Buffer[i,np.array(self.Mesh.BoundaryDofs)[ind]]=0
+			
+			cs = ax.tricontour(self.Mesh.DofsXY[:,0], self.Mesh.DofsXY[:,1],\
+					           self.Mesh.ElementsBoundaryDofs, Buffer[i,:], [0])
+		
+			# ----------- Extracting the contour of the level set -------------------
+			# And transforming into a shapely mutlipolygon
+			
+			# Extracting the 0 level of the function and creating a polygon out of it,
+			# making sure that it is not self-intersecting
+			LV0 = cs.allsegs[0]
+			polygon = ShapelyPolygon([tuple(a) for a in np.array(LV0[0])]).buffer(0)
+			
+			# Spanning over the possibly disconnected area encompassing the target fluid
+			for contour in LV0[1:]:
+				# Determining whether the patch is out or in the previously registered polygon 
+				# and adding/removing it upon
+				patch = ShapelyPolygon([tuple(a) for a in np.array(contour)]).buffer(0)
+				if patch.difference(polygon).is_empty == False:	polygon = polygon.union(patch)
+				else: polygon = polygon.difference(patch)
+			
+			# Converting it to multipolygon for compatibility with other parts of the code (esp. plots)
+			if not (polygon.geom_type == 'MultiPolygon'): polygon = ShapelyMultiPolygon([polygon])
+
+			# Registering the new shape
+			Solution.Subdomains[i] = polygon
+		
+		plt.close(fig)
+
+	#### Redistancing the LSvalues according to the freshly computed level set values ####
+	def Redistancing(self, Solution):
+		""" Redistances the Level set values upon the computed subdomains.
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+			    Output:    None,                updating directly the LSValues in the
+			                                    solution instance
+		----------------------------------------------------------------------------------"""
+
+		# Redistancing the points with the nearest
+		# Looping on the intial number of fluids present in the computational domain
+		for i in range(self.Problem.NbFluids):
+			# Computing the distance and location w.r.t. the subdomain through the shapely information
+			FluidArea = Solution.Subdomains[i]
+			Value = np.array([[FluidArea.boundary.distance(ShapelyPoint(tuple(pt))),                 \
+			                   int(2*FluidArea.intersects(ShapelyPoint(tuple(pt)).buffer(1e-14))-1)] \
+							   for pt in self.Mesh.DofsXY]).T
+			
+			# Computing the signed distance
+			Solution.LSValues[i,:] = np.prod(Value, axis=0)
+
+	def ComputeFlux(self, U, LS):
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.NbFluids, 2))
+		for i in range(self.Mesh.NbInnerElements):
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			
+			# Get the average state of the considered element
+			LS1  = np.mean(LS[:,dele], axis = 1)
+			Val1 = np.mean(U[:,dele],      axis = 1)
+			res = np.zeros((1,self.Problem.NbFluids))
+
+			# Looping on each edge
+			for f in range(len(ele)):
+
+				# Get the physical vertices of the considered edge
+				n = self.Mesh.ElementsBoundaryWiseNormals[i][f]
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				Nei  = self.Mesh.Neighborhoods[i][f]
+				Edge = Ele[f:f+2]
+				
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				
+				# Get the average state of the neighboring element
+				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
+				LS2   = np.mean(LS[:,Dele2], axis=1)
+				Val2  = np.mean(U [:,Dele2], axis=1)
+				
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
+				UV = self.Problem.GoverningEquations.GetUV(np.array([Val1,Val2]).T, EdgeMidPoint)
+				MidFluxes = np.array([UV[0,:]*np.array([LS1,LS2]).T,\
+								      UV[1,:]*np.array([LS1,LS2]).T])
+				
+				fluxes[i,f,:,:,:] = MidFluxes
+		
+		return(fluxes)
+		
+	#### (Crucial routine) Spatial scheme to update the level set ####
+	def UpdateFSValues(self, Solution, fluxes, du, dt=1):
+		"""  Spatial scheme to update the level set:
+		     Performs the level set update from stupid LFX centred finite volume scheme
+		
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+			    Output:    Res,                 the residuals after one iteration
+		--------------------------------------------------------------------------------- """
+		
+		# ---------------- Updating the level set values ---------------------------
+		# Initialising the mollification vector (residuals)
+		Res = np.zeros(np.shape(Solution.LSValues))
+		
+		# Looping on each element
+		for i in range(self.Mesh.NbInnerElements):
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			
+			# Get the average state of the considered element
+			LS1  = np.mean(Solution.LSValues[:,dele], axis = 1)
+			Val1 = np.mean(Solution.Sol[:,dele],      axis = 1)
+			res = np.zeros((1,self.Problem.NbFluids))
+
+			# Looping on each edge
+			for f in range(len(ele)):
+
+				# Get the physical vertices of the considered edge
+				n = self.Mesh.ElementsBoundaryWiseNormals[i][f]
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				Nei  = self.Mesh.Neighborhoods[i][f]
+				Edge = Ele[f:f+2]
+				
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				
+				# Get the average state of the neighboring element
+				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
+				LS2   = np.mean(Solution.LSValues[:,Dele2], axis=1)
+				Val2  = np.mean(Solution.Sol[:,Dele2],      axis=1)
+				
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
+				UV = self.Problem.GoverningEquations.GetUV(np.array([Val1,Val2]).T, EdgeMidPoint)
+				MidFluxes = fluxes[i,f,:,:,:]
+				
+				# Computing the Jacobian for the second coordinate
+				J = np.zeros((self.Problem.NbFluids, self.Problem.NbFluids, 2, 2))
+				for nf in range(self.Problem.NbFluids):
+					J[nf, nf, 0, :] = np.array([UV[0,:]])
+					J[nf, nf, 1, :] = np.array([UV[1,:]])
+				sprctrad = np.max(np.abs([np.sum(J[i,i,:,:]*n.T, axis = 0) for i in range(self.Problem.NbFluids)]), axis=0)
+		
+				# Compute the flux contribution
+				MidFluxes[:,:,1] = -MidFluxes[:,:,1]
+				avrg = 0.5*np.sum(MidFluxes, axis=2)
+				FluxN = np.dot(avrg.T,n.T)
+				
+				# Add the pondered contribution
+				res[:] += L*FluxN + L*np.max(sprctrad)*(LS1[:]-LS2[:])
+
+			# Distribute it equally to each submit ( the square is from the ponderation of the residual
+			# by the control volume's area
+			Res[:,dele] += (res[:].T)/(3)
+			
+		# Updating each nodal value (here representing the average) 
+		Res[:, :] = dt*Res[:, :]+du[:,:]/(3)
+		
+		# Returning the residuals
+		return(Res)
+
+	#### Wrapper routine to update the quantitites that depend on the UpdateFSValues output ####
+	def UpdateFSVAttributes(self, Solution):
+		""" Wrapper routine to update the quantitites that depend on the UpdateFSValues fresh output 
+		Input:  Solution,  Solution structure,  the solution instance the scheme is working on
+			    Output:    None,                updating directly the subvalues in the
+			                                    solution instance
+		--------------------------------------------------------------------------------- """
+		
+		#self.UpdateSubdomains(Solution)
+		#self.Redistancing(Solution)
+		self.FlagDofs(Solution)
diff --git a/SourceCode/_Solver/QuadratureRules/HandBased.py b/SourceCode/_Solver/QuadratureRules/HandBased.py
index e69de29..e206c29 100644
--- a/SourceCode/_Solver/QuadratureRules/HandBased.py
+++ b/SourceCode/_Solver/QuadratureRules/HandBased.py
@@ -0,0 +1,101 @@
+"""
+################################################################################
+#			File that procures some quadrature points and weights 			   #
+################################################################################
+
+Implemented features:
+	-> 1D and 2D quadrature for simplicial shape, exact up to order 5 
+	->
+
+Usage:
+	
+Note:
+
+----------------------------------------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+
+# ==============================================================================
+#	Definition of quadrature points and weights
+# ==============================================================================
+
+#### Definition of inner quadrature points and weights ###
+def InnerQuadrature():
+	""" Hardly coded definition of inner quadrature points and weights, given in barycentric coordinnates.
+		Exact up to order 5.
+
+	Input: None
+	Output: points,   2D numpy float array,  the 12 quadrature points in barycentric coordinates (12 x 3)
+			weights,  numpy float array,     the associated quadrature weights
+	--------------------------------------- """
+
+	# Initialising the export quantities
+	weights = np.zeros(12)
+	points  = np.zeros((12,3))
+	
+	# Defining the quadrature points and weights
+	weights[0:3]  = 0.116786275726379
+	weights[3:6]  = 0.050844906370207
+	weights[6:12] = 0.082851075618374
+	
+	xo=0.501426509658179
+	zo=0.5*(1.0-xo)
+	points[0,:] = np.array([xo, zo, zo])
+	points[1,:] = np.array([zo, xo, zo])
+	points[2,:] = np.array([zo, zo, xo])
+	
+	xo=0.873821971016996
+	zo=0.5*(1.0-xo)
+	points[3,:] = np.array([xo, zo, zo])
+	points[4,:] = np.array([zo, xo, zo])
+	points[5,:] = np.array([zo, zo, xo])
+	
+	w  = 0.082851075618374
+	xo = 0.053145049844816
+	yo = 0.310352451033785
+	zo = 1.0-xo-yo
+	points[6,:] = np.array([xo, yo, zo])
+	points[7,:] = np.array([xo, zo, yo])
+	points[8,:] = np.array([yo, xo, zo])
+	points[9,:] = np.array([yo, zo, xo])
+	points[10,:] = np.array([zo, xo, yo])
+	points[11,:] = np.array([zo, yo, xo])
+    
+	# Just to be sure, up to overriding
+	points[:,2] = 1.- np.sum(points[:,0:2], axis = 1)
+	weights    /= np.sum(weights)
+	
+	# Giving back the values
+	return(points, weights)
+
+#### Definition of boundary quadrature points and weights ###
+def BoundaryQuadrature(f):
+	""" Hardly coded definition of boundary quadrature points and weights, given in barycentric coordinnates.
+		Exact up to order 5.
+
+	Input:  f,        integer,               consider triangle face
+	Output: points,   2D numpy float array,  the 3 quadrature points in barycentric coordinates (3 (NbPoints) x 3)
+			weights,  numpy float array,     the associated quadrature weights
+	--------------------------------------- """
+
+   # Initialising the export quantities
+	weights = np.array([5., 8., 5.])/18.
+	points  = np.zeros((3,3))
+	
+	# Selecting the considered face and prescribing the non-null coefficients 
+	# on the target vertices
+	sb = [f, int(np.mod(f+1,3))]
+	
+	# Defining the quadrature points and weights
+	s=np.sqrt(0.6)
+	points[0,sb] = np.array([0.5*(1.0 - s), 0.5*(1.0 + s)])
+	points[1,sb] = np.array([0.5, 0.5])
+	points[2,sb] = np.array([0.5*(1.0 + s), 0.5*(1.0 - s)])
+	
+	# Returning the desired quantitites
+	return(points, weights)
+	
diff --git a/SourceCode/_Solver/QuadratureRules/__pycache__/HandBased.cpython-38.pyc b/SourceCode/_Solver/QuadratureRules/__pycache__/HandBased.cpython-38.pyc
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diff --git a/SourceCode/_Solver/Solver_Initialisations.py b/SourceCode/_Solver/Solver_Initialisations.py
index 425f8d0..d3bdf9d 100644
--- a/SourceCode/_Solver/Solver_Initialisations.py
+++ b/SourceCode/_Solver/Solver_Initialisations.py
@@ -80,12 +80,12 @@ class Parameters():
 		self.Tmax = float(re.split(" |\t", Params[16])[0])
 
 		# Retrieve the time scheme and its parameters
-		self.TimeSchemeId = int(re.split(" |\t" , Params[13])[0])
-		self.TimerParams  = list(map(int, filter(None,re.split(",|\t", re.sub("\D", ",", Params[14])))))
+		self.TimeSchemeId = int(re.split("\t| " , Params[13])[0])
+		self.TimerParams  = list(map(int, filter(None,re.split(",|\t| ", re.sub("\D", ",", Params[14])))))
 
 		# Retrieve the spatial scheme and limiter to use
-		self.LimiterParams = list(filter(None,re.split(",|\t", re.sub("\D", ",", Params[4]))))
-		self.SchemeParams  = list(filter(None,re.split(",|\t", re.sub("\D", ",", Params[2]))))
+		self.LimiterParams = list(filter(None,re.split(",|\t| ", re.sub("\D", ",", Params[4]))))
+		self.SchemeParams  = list(filter(None,re.split(",|\t| ", re.sub("\D", ",", Params[2]))))
 		self.SchemeId      = int(re.split(" |\t", Params[1])[0])
 		self.LimiterId     = int(re.split(" |\t", Params[3])[0])
 
@@ -102,7 +102,7 @@ class Parameters():
 		self.Verbose        = re.split(" |\t", Params[19])[0]
 
 		# Retrieve the type of required Mesh
-		MeshProp = Mappers.Spatial_Scheme(self.SchemeId).AssociatedMeshType()
+		MeshProp = Mappers.Spatial_Scheme(self.SchemeId).AssociatedMeshType(self.SchemeParams)
 		self.MeshType  = MeshProp.MeshType
 		self.MeshOrder = MeshProp.MeshOrder
 
@@ -281,12 +281,13 @@ class Solver():
 		self.FluidSelector = Mappers.Fluid_Selector(SolverParams.FluidSpotterId).FS(Problem, Mesh, SolverParams.FluidSpotterParams)
 
 		# Map the schemes functions to the structure
-		Scheme = Mappers.Spatial_Scheme(SolverParams.SchemeId).Scheme(Problem, Mesh, SolverParams.SchemeParams)
-		self.SpatialScheme = Scheme
-		self.Limiter       = Mappers.Limiter(SolverParams.LimiterId, SolverParams.LimiterParams)
+		Scheme = Mappers.Spatial_Scheme(SolverParams.SchemeId)
+		self.ControlVolumes = Scheme.AssociatedMeshType().ControlVolumes
+		self.SpatialScheme  = Scheme.Scheme(Problem, Mesh, SolverParams.SchemeParams)
+		self.Limiter        = Mappers.Limiter(SolverParams.LimiterId, SolverParams.LimiterParams)
 
 		# Map the solution's reconstructor to the structure
-		self.Reconstructor = Scheme.ReconstructSolutionAtVertices
+		self.Reconstructor = self.SpatialScheme.ReconstructSolutionAtVertices
 		
 		# Set up the time scheme and relocate the time parameters variables
 		self.TimeStep = Mappers.Temporal_Scheme(SolverParams.TimeSchemeId, SolverParams.TimerParams)
diff --git a/SourceCode/_Solver/Solver_Spatial.py b/SourceCode/_Solver/Solver_Spatial.py
index 20a22c9..fb19600 100644
--- a/SourceCode/_Solver/Solver_Spatial.py
+++ b/SourceCode/_Solver/Solver_Spatial.py
@@ -1,7 +1,28 @@
+"""
+################################################################################
+#			File that procures interfaces from the scheme to the DeC		   #
+#			when used within a residual distribution framework				   #
+################################################################################
+
+Implemented features:
+	->
+	->
+
+Usage:
+	
+Note:
+
+----------------------------------------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
 import numpy as np
 import copy 
 
-
+# ==============================================================================
+#	Core routines
+# ==============================================================================
 
 #### Integrate the RD formulation in the Dec setting ####
 def RD(Theta, m, M, dt, schemes, Mesh, Solution, up, lp):
@@ -19,29 +40,39 @@ def RD(Theta, m, M, dt, schemes, Mesh, Solution, up, lp):
             lp,       float numpy array,   time-local increment in the FluidSpotter values coming from DeC
     """
     
+    # --------------------- Initialisations -----------------------------------------------
     # Intialising the value which will be considered at this time step
     SolBuff = copy.deepcopy(Solution)
     
+    # ------------------ Incrementing according to the DeC approximation ------------------
     # Computing the temporal increment coming from dec
     dup = up[:,:,m-1]-up[:,:,0]
     dlp = lp[:,:,m-1]-lp[:,:,0]
     
     # Computing the temporally local flux and the infinitesimal update to the solution
-    fincr = np.sum(np.array([Theta[r,m]*schemes[0].ComputeFlux(Solution.FluidFlag, up[:,:,r]) for r in range(M+1)]), axis = 0)
+    fincr = np.sum(np.array([Theta[r,m]*schemes[0].ComputeFlux(Solution.FluidFlag, up[:,:,r], lp[:,:,r]) for r in range(M+1)]), axis = 0)
     sincr = np.sum(np.array([Theta[r,m]*up[:,:,r] for r in range(M+1)]), axis = 0)
     
     # Computing the temporally local flux and the correction to the function
-    lincr = np.sum(np.array([Theta[r,m]*schemes[1].ComputeFlux(up[:,:,r], lp[:,:,r]) for r in range(M+1)]), axis = 0)
-    vincr = np.sum(np.array([Theta[r,m]*lp[:,:,r] for r in range(M+1)]), axis = 0)
+    #lincr = np.sum(np.array([Theta[r,m]*schemes[1].ComputeFlux(up[:,:,r], lp[:,:,r]) for r in range(M+1)]), axis = 0)
+    #vincr = np.sum(np.array([Theta[r,m]*lp[:,:,r] for r in range(M+1)]), axis = 0)
     
+    # ------------------ Applying the dynamic to the iterated values ------------------
     # Update fluid flag upon the new value of the level sets and filling a buffer structure
-    SolBuff.LSValues = vincr
+    #SolBuff.LSValues = vincr
     SolBuff.Sol      = sincr
     schemes[1].FlagDofs(SolBuff)
     
     # Applying the LS and Dynamic schemes upon the time-locally iterated solution
     upres = schemes[0].Iteration(SolBuff, fincr, dup, dt)[:]
-    lsres = schemes[1].UpdateFSValues(SolBuff, lincr, dlp, dt)[:]
+    #lsres = schemes[1].UpdateFSValues(SolBuff, lincr, dlp, dt)[:]
     
+    # ------------------ Applying the dynamic to the iterated values ------------------
+    # Updating each Dof (here representing the average) by the residual
+    # pondered (and lumped) by the control volume's area
+    upres = upres*Mesh.Lumping
+    #lsres = lsres*Mesh.Lumping
+    lsres = 0
+
     # Retuning the residuals'increments (not the new value of the solution)
     return([upres, lsres])
diff --git a/SourceCode/_Solver/Solver_Temporal.py b/SourceCode/_Solver/Solver_Temporal.py
index 6467948..b79e7c3 100644
--- a/SourceCode/_Solver/Solver_Temporal.py
+++ b/SourceCode/_Solver/Solver_Temporal.py
@@ -22,17 +22,35 @@ import numpy as np
 # ==============================================================================
 
 #### Computing the mass lumping coefficients ####
-def GetLumpingCoefficients(Mesh):
+def GetLumpingCoefficients(Mesh, Solver):
     """ Computes the mass lumping coefficients. 
     Input:  Mesh,      MeshStructure,       the instance containing the mesh information
+            Solver,    Solver instance,     instance containing all the solver's parameters
     Output: None,                           fills directly the "Lumping" field of the mesh structure   
     ---------------------------------------------"""
     
-    # Compute the mass lumping coefficients by agglomerating the pondered volume of each cell
+    # ------------ Initialisations ------------------------------------
+    # Getting the type of the control volume 
+    typ = Solver.ControlVolumes
+    # Initialising the 
     Mesh.Lumping = np.zeros(Mesh.NbDofs)
-    for i in range(Mesh.NbInnerElements):
-        Mesh.Lumping[Mesh.ElementsBoundaryDofs[i]] += Mesh.InnerElementsVolume[i]/len(Mesh.ElementsBoundaryDofs[i])
-    Mesh.Lumping = 1./Mesh.Lumping
+    
+    # ------------ Actual mass lumping computations -------------------
+    # In the case of a dual mesh based control volume, compute the mass lumping coefficients
+    # by agglomerating the pondered volume of each cell (which is the information contained in the DualAreasTot)
+    if typ == "Dual":
+        for i in range(Mesh.NbInnerElements):
+            Mesh.Lumping[Mesh.ElementsBoundaryDofs[i]] += Mesh.InnerElementsVolume[i]/len(Mesh.ElementsBoundaryDofs[i])
+        
+    # In the case of a primal mesh based control volume, (warning: only cell-centred FV in the RD setting)
+    # register the elment's volume as the control one
+    elif typ == "Primal":
+        for i in range(Mesh.NbInnerElements):
+            Mesh.Lumping[Mesh.ElementsInnerDofs[i]] += Mesh.InnerElementsVolume[i]/len(Mesh.ElementsInnerDofs[i])
+    
+    # Safe exception case
+    else:
+        raise ValueError("Error: Unknown type of control volume. Check you scheme definition. Abortion.")
 
 #### Computing the time step upon the solution and the CFL ####
 def GetTimeStep(CFL, Problem, Mesh, Solution):
@@ -51,11 +69,12 @@ def GetTimeStep(CFL, Problem, Mesh, Solution):
     for i in range(Mesh.NbInnerElements):
         # Extracting the physical element to get the number of edges
         ele = Mesh.InnerElements[i]
-       
+        dele = Mesh.ElementsInnerDofs[i]
+        
         # Looping on each edge
         for f in range(len(ele)):
             # Getting the variational information
-            dele  = Mesh.ElementsBoundaryWiseDofs[i][f]
+            dele  += Mesh.ElementsBoundaryWiseDofs[i][f]
             Flags = Solution.FluidFlag[dele]
             Vars  = Solution.Sol[:,dele]
             
diff --git a/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFX.py b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFX.py
index bdc1025..baea08f 100644
--- a/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFX.py
+++ b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFX.py
@@ -19,12 +19,13 @@ Note:
 #	Preliminary imports
 # ==============================================================================
 import numpy as np
+import copy 
 
 # ==============================================================================
 #	Parameter (and reader) class of scheme properties and runtime informations
 # ==============================================================================
 # Name of the module to be externally accessible
-SchemeName = "Finite Volumes -- Cell centred with LFX flux"
+SchemeName = "Finite Volumes -- Vertex centred with natural flux"
 
 # Mapping to the mesh that should be loaded
 class AssociatedMeshType():
@@ -43,8 +44,9 @@ class AssociatedMeshType():
 		""" Automatic initialisation routine defining the type of mesh
         to be loaded for this solver."""
 
-		self.MeshType = "CG"
-		self.MeshOrder = 1	# Or dependent on VariationalOrder
+		self.MeshType = "DG"
+		self.MeshOrder = 0	# Or dependent on VariationalOrder
+		self.ControlVolumes = "Primal"
 
 
 # ==============================================================================
@@ -63,115 +65,106 @@ class Scheme():
 		self.Params  = Params
 		self.Mesh    = Mesh
 
-	def ComputeFlux(self, FluidFlag, U):
-		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables, 2))
+    #### Emulates a flux computation as done in the iteration routine ####
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		""" Emulates a flux computation as done in the iteration routine in order
+			to be used in the DeC subtimestep evaluation.
+			
+			Input: 
+				FluidFlag,   numpy float array,  buffer vector containing the current fluid flags at each dof (NbDofs)
+				U,           numpy float array,  buffer containing the current state values at each dof (NbVars   x NbDofs)
+				LSValues,    numpy float array,  buffer containing the current FluidSpotters values at each dof(NbFluids x NbDofs)
+		
+			Output:
+				fluxes,     numpy float (multiD)array,   fluxes in the format requires by the Iteration routine
+		---------------------------------------------------- """
+	
+        # --------------------------  Initialisations -------------------------------------------------
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables,2))
 		
+		# ------------- Flux computation at the control's volume interface ----------------------------
 		for i in range(self.Mesh.NbInnerElements):
-			dele = self.Mesh.ElementsBoundaryDofs[i]
-			ele  = self.Mesh.InnerElements[i]
-			Dele = np.array(dele + [dele[0]], dtype=int)
-			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			# Retrieving the information on the local degree of freedom
+			dele  = self.Mesh.ElementsInnerDofs[i][0]
+			ele   = self.Mesh.InnerElements[i]
+			Eele  = self.Mesh.InnerElements[i] + [self.Mesh.InnerElements[i][0]]
 			
-			# Get the average state of the considered element
-			Var1 = np.mean(U[:,dele], axis = 1)
-			res = np.zeros((1,self.Problem.GoverningEquations.NbVariables))
-
 			# Looping on each edge
 			for f in range(len(ele)):
-
-				# Get the physical vertices of the considered edge
-				n = self.Mesh.ElementsBoundaryWiseNormals[i][f]
-				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
-				Nei  = self.Mesh.Neighborhoods[i][f]
-				Edge = Ele[f:f+2]
-				
-				# Get the variational information at each Dof lying on the edge
-				DEdge = Dele[f:f+2]
-				Flags = FluidFlag[DEdge]
-
-				# Get the average state of the neighboring element
-				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
-				Var2  = np.mean(U[:,Dele2], axis=1)
+				# Getting the variational info of the immediate neighbor
+				ala  = self.Mesh.Neighborhoods[i][f]
+				dala = self.Mesh.ElementsInnerDofs[ala][0]
 				
-				# 
-				Vars = np.array([Var1, Var2]).T
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Flags = FluidFlag[np.array([dele,dala])]
+				Var   = np.array([U[:,dele], U[:,dala]])
 				
 				# Compute the input exchange from the neighboring element at the edge midpoint
-				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
-				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Vars, EdgeMidPoint, Flags)
+				EdgeMidPoint = np.array(2*[0.5*np.sum(self.Mesh.CartesianPoints[Eele[f:f+2],:], axis=0)])
+				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Var, EdgeMidPoint, Flags)
+				
 		
+		# Returning the full array
 		return(fluxes)
 
-    #### Main routine, defininin the scheme ####
-	def Iteration(self,  Solution, fluxes, du, dt=1):
+    #### Main routine, definining the scheme ####
+	def Iteration(self,  Solution, fluxes, du=0, dt=1):
 		""" Main iteration of the scheme, implementing the most stupid scheme
 		you can think of.
 		
 		Args:
-			SolverSpecs (string):  the name of the setting file
+			Solution,   solution structure,   structure containing the current solution's values to iterate
+			fluxes,     numpy (multiD)array,  pre-computed fluxes at the points of interest. For this scheme, 
+											  access with fluxes[element, face, coordinate(fx or fy), variable, pointindex]
+			du,         float numpy array,    (optional) when using DeC, the difference in the time iteration 
+			dt,         float,                (optional) when using DeC, the full time step
+			
 		Returns:
-			None                :  fills direclty the Parameters data structure
+			Resu,       float numpy array,    the computed residuals (NbVars x NbDofs)
 		---------------------------------------------------------------------"""
 
-		# Initialising the mollification vector (residuals)
+		# -------- Initialising the mollification vector (residuals) -------------
 		Resu = np.zeros(np.shape(Solution.Sol))
-		Res = np.zeros(np.shape(Solution.Sol))
 		
-		# Looping on each element
+		# ------- Getting the residuals for each element -------------------------
 		for i in range(self.Mesh.NbInnerElements):
-			dele = self.Mesh.ElementsBoundaryDofs[i]
-			ele  = self.Mesh.InnerElements[i]
-			Dele = np.array(dele + [dele[0]], dtype=int)
-			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			# Retrieving the information on the local degree of freedom
+			dele  = self.Mesh.ElementsInnerDofs[i][0]
+			ele   = self.Mesh.InnerElements[i]
 			
-			# Get the average state of the considered element
-			Var1 = np.mean(Solution.Sol[:,dele], axis = 1)
-			res = np.zeros((1,self.Problem.GoverningEquations.NbVariables))
-
 			# Looping on each edge
 			for f in range(len(ele)):
-
-				# Get the physical vertices of the considered edge
-				n = self.Mesh.ElementsBoundaryWiseNormals[i][f]
-				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
-				Nei  = self.Mesh.Neighborhoods[i][f]
-				Edge = Ele[f:f+2]
 				
-				# Get the variational information at each Dof lying on the edge
-				DEdge = Dele[f:f+2]
-				Flags = Solution.FluidFlag[DEdge]
-
-				# Get the average state of the neighboring element
-				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
-				Var2  = np.mean(Solution.Sol[:,Dele2], axis=1)
-
-				# Compute the input exchange from the neighboring element at the edge midpoint
+				ala  = self.Mesh.Neighborhoods[i][f]
+				dala = self.Mesh.ElementsInnerDofs[ala][0]
+				
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Flags = FluidFlag[np.array([dele,dala])]
+				Var   = np.array([U[:,dele], U[:,dala]])
+				
+				# Get the physical properties of the considered edge
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				n = self.Mesh.ElementsBoundaryWiseNormals[i][f,:]
 				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
-				MidFluxes    = fluxes[i,f]
-				sprctrad     = self.Problem.GoverningEquations.SpectralRadius(np.array([Var1, Var2]).T, Flags, np.array(2*[n]), EdgeMidPoint)
+				
+				# Retrieving the flux computed through DeC
+				MidFluxes = fluxes[i,f,:,:,:]
+				MidFluxes[:,:,1] = -MidFluxes[:,:,1]
+				sprctrad  = self.Problem.GoverningEquations.SpectralRadius(Var.T, Flags, np.array(2*[n]), EdgeMidPoint)
 
 				# Compute the pondered contribution
-				MidFluxes[:,:,1] = -MidFluxes[:,:,1]
-				avrg  = 0.5*np.sum(MidFluxes, axis=2)
-				#avrg = MidFluxes[:,:,0] 
-				FluxN = np.dot(avrg.T,n.T)
+				FluxN = np.dot(0.5*np.sum(MidFluxes, axis=2).T,n)
+				quant = L*FluxN - L*np.max(sprctrad)*(U[:,dele]-U[:,dala])
 				
-				# Add the contribution
-				res[:] += L*FluxN #- L*np.max(sprctrad)*(Var1[:]-Var2[:])
-
-			# Distribute it equally to each submit
-			#Res[:,dele] += res[:].T*self.Mesh.DualAreas[i][:].T/self.Mesh.InnerElementsVolume[i]
-			Resu[:,dele] += (du[:,dele]/3\
-							+dt*(res[:].T)/3)
-					
-		# Updating each nodal value (here representing the average) by the residual
-		# pondered by the control volume's area
-		for i in range(self.Mesh.NbInnerElements):
-			for e in self.Mesh.ElementsBoundaryDofs[i]:
-				Res[:, e] = Res[:, e] - Resu[:,e]*self.Mesh.Lumping[e]
+				# Distribute it equally to each submit
+				Resu[:,dele] = Resu[:,dele] + (quant)
+				
+		#----- Modifying the residual on each Dof by addin the DeC increment --------
+		# Adapting the residual to Dec
+		Resu += du[:,:]/self.Mesh.Lumping
 		
 		# Returning the result
-		return(Res)
+		return(Resu)
 
 	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
 	def ReconstructSolutionAtVertices(self, Solution):
@@ -184,20 +177,27 @@ class Scheme():
 		Note: This is only for later plotting purposes.
 		----------------------------------------------------------------------------------- """
 		
-		# Stupid reconstruction at the closest value for Dofs #TODO better
+		# -------------- Initialising the reconstructed values ----------------------------
+		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
 		Solution.RLSValues  = np.zeros(np.shape(Solution.LSValues))
 		Solution.RSol       = np.zeros(np.shape(Solution.Sol))
-		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
 		
+		# ------------- Reconstructing the values -----------------------------------------
+		# Extract the cartesian coordinates of the Dofs
+		xyDofs = self.Mesh.DofsXY[:,:]
+		
+		# Spanning all the mesh points where to fill the values
 		for i in range(self.Mesh.NbMeshPoints):
+            
+            # Retrieve the catesian coordinates of the considered point
 			xy = self.Mesh.CartesianPoints[i,:]
-			xyDofs = self.Mesh.DofsXY[:,:]
-			
+					
+			# Stupid reconstruction at the closest value for Dofs #TODO better
 			norm = np.linalg.norm(xy-xyDofs,2, axis=1)
 			index = np.argsort(norm)[0]
 			
+			# Fill the reconstructed solution upon the found index
 			Solution.RLSValues[:,i] = Solution.LSValues[:,index]
 			Solution.RFluidFlag[i]  = Solution.FluidFlag[index]
 			Solution.RSol[:,i]      = Solution.Sol[:,index]
 
-	
diff --git a/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFXorig.py b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFXorig.py
new file mode 100644
index 0000000..50a816f
--- /dev/null
+++ b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_LFXorig.py
@@ -0,0 +1,203 @@
+"""
+################################################################################
+#	Module that defines a Vertex centred finite volume scheme                  #
+################################################################################
+
+Contains (in scrolling order):
+	-> User defined data reader tools
+
+	-> Structure definition
+
+	-> Initialisation routines
+
+Usage:
+
+Note:
+-----------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+# ==============================================================================
+#	Parameter (and reader) class of scheme properties and runtime informations
+# ==============================================================================
+# Name of the module to be externally accessible
+SchemeName = "Finite Volumes -- Cell centred with LFX flux"
+
+# Mapping to the mesh that should be loaded
+class AssociatedMeshType():
+	""" Class that only gives out the mesh type that should be loaded when tackling the problem
+	with this scheme, given the wished variational order. Access the associated mesh properties
+	through the inner variables
+
+	MeshType,   string,  the mesh type according to fenics's classification
+	MeshOrder,  integer, the mesh order
+
+	Note: This is only a product-type class: no method is available
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, VariationalOrder = 0):
+		""" Automatic initialisation routine defining the type of mesh
+        to be loaded for this solver."""
+
+		self.MeshType = "CG"
+		self.MeshOrder = 1	# Or dependent on VariationalOrder
+		self.ControlVolumes = "Dual"
+
+
+# ==============================================================================
+#	Class furnishing all the scheme tools
+# ==============================================================================
+class Scheme():
+	""" Class furnishing all the schemes tools and registers the scheme's wished
+	parameters. The main iteration is accessible with the method "Iteration".
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, Problem, Mesh, Params = []):
+		""" Automatic initialisation routine. Just registers the user defined scheme parameters
+		within the instance"""
+		self.Problem = Problem
+		self.Params  = Params
+		self.Mesh    = Mesh
+
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables,2))
+		
+		# Initialising the mollification vector (residuals)
+		Res = np.zeros(np.shape(U))
+		
+		# Looping on each element
+		for i in range(self.Mesh.NbInnerElements):
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			
+			# Get the average state of the considered element
+			Var1 = np.mean(U[:,dele], axis = 1)
+			res = np.zeros((1,self.Problem.GoverningEquations.NbVariables))
+
+			# Looping on each edge
+			for f in range(len(ele)):
+
+				# Get the physical vertices of the considered edge
+				n = self.Mesh.ElementsBoundaryWiseNormals[i][f]
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				Nei  = self.Mesh.Neighborhoods[i][f]
+				Edge = Ele[f:f+2]
+				
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				Flags = FluidFlag[DEdge]
+
+				# Get the average state of the neighboring element
+				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
+				Var2  = np.mean(U[:,Dele2], axis=1)
+
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
+				fluxes[i,f,:,:] = self.Problem.GoverningEquations.Flux(np.array([Var1, Var2]).T, EdgeMidPoint, Flags)
+		return(fluxes)
+
+
+    #### Main routine, defininin the scheme ####
+	def Iteration(self,  Solution,  fflx, du=0, dt=1):
+		""" Main iteration of the scheme, implementing the most stupid scheme
+		you can think of.
+		
+		Args:
+			SolverSpecs (string):  the name of the setting file
+		Returns:
+			None                :  fills direclty the Parameters data structure
+		---------------------------------------------------------------------"""
+
+		# Initialising the mollification vector (residuals)
+		Res = np.zeros(np.shape(Solution.Sol))
+		
+		# Looping on each element
+		for i in range(self.Mesh.NbInnerElements):
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			Ele  = np.array(ele  + [ele[0]],  dtype=int)
+			
+			# Get the average state of the considered element
+			Var1 = np.mean(Solution.Sol[:,dele], axis = 1)
+			res = np.zeros((1,self.Problem.GoverningEquations.NbVariables))
+			
+
+			# Looping on each edge
+			for f in range(len(ele)):
+
+				# Get the physical vertices of the considered edge
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				Nei  = self.Mesh.Neighborhoods[i][f]
+				Edge = Ele[f:f+2]
+				
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				Flags = Solution.FluidFlag[DEdge]
+
+				# Get the average state of the neighboring element
+				Dele2 = np.array(self.Mesh.ElementsBoundaryDofs[Nei])
+				Var2  = np.mean(Solution.Sol[:,Dele2], axis=1)
+
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = np.array(2*[0.5*(np.sum(self.Mesh.CartesianPoints[Edge, :], axis=0))])
+				MidFluxes    = fflx[i,f,:,:]
+				sprctrad     = self.Problem.GoverningEquations.SpectralRadius(np.array([Var1, Var2]).T, Flags, np.array(2*[n]), EdgeMidPoint)
+
+				# Compute the pondered contribution
+				MidFluxes[:,:,1] = -MidFluxes[:,:,1]
+				avrg  = 0.5*np.sum(MidFluxes, axis=2)
+				FluxN = np.dot(avrg.T,n.T)
+				
+				# Add the contribution
+				res[:] += L*(FluxN - np.max(sprctrad)*(Var1[:]-Var2[:]))
+
+			# Compute the average of du over the cell and compute its associated 
+			# contribution over the element
+			dubar = (np.sum(du[:,dele], axis=1)/len(ele))*self.Mesh.InnerElementsVolume[i]
+			
+			# Distribute it equally to each submit
+			#Res[:,dele] += (dt*res[:].T+dubar)*self.Mesh.DualAreas[i][:].T/self.Mesh.InnerElementsVolume[i]
+			Res[:,dele] += (dt*res[:].T+dubar)/3
+			
+		# Updating each nodal value (here representing the average) by the residual
+		# pondered by the control volume's area
+		Res[:, :] = Res[:, :]
+		
+		return(Res)
+
+	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
+	def ReconstructSolutionAtVertices(self, Solution):
+		""" Routine that maps the values from the Dofs to the Physical vertices of the mesh 
+		    of the FluidSpotter, solution and flags.
+		    
+			Input:  SolverSpecs,  string,  the name of the setting file
+			Output: None, fills direclty the Parameters data structure
+			
+		Note: This is only for later plotting purposes.
+		----------------------------------------------------------------------------------- """
+		
+		# Stupid reconstruction at the closest value for Dofs #TODO better
+		Solution.RLSValues  = np.zeros(np.shape(Solution.LSValues))
+		Solution.RSol       = np.zeros(np.shape(Solution.Sol))
+		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
+		
+		for i in range(self.Mesh.NbMeshPoints):
+			xy = self.Mesh.CartesianPoints[i,:]
+			xyDofs = self.Mesh.DofsXY[:,:]
+			
+			norm = np.linalg.norm(xy-xyDofs,2, axis=1)
+			index = np.argsort(norm)[0]
+			
+			Solution.RLSValues[:,i] = Solution.LSValues[:,index]
+			Solution.RFluidFlag[i]  = Solution.FluidFlag[index]
+			Solution.RSol[:,i]      = Solution.Sol[:,index]
+
+	
diff --git a/SourceCode/_Solver/SpatialSchemes/CellCentredFV_Natural.py b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_Natural.py
new file mode 100644
index 0000000..ca9f354
--- /dev/null
+++ b/SourceCode/_Solver/SpatialSchemes/CellCentredFV_Natural.py
@@ -0,0 +1,191 @@
+"""
+################################################################################
+#	Module that defines a Vertex centred finite volume scheme                  #
+################################################################################
+
+Contains (in scrolling order):
+	-> User defined data reader tools
+
+	-> Structure definition
+
+	-> Initialisation routines
+
+Usage:
+
+Note:
+-----------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+# ==============================================================================
+#	Parameter (and reader) class of scheme properties and runtime informations
+# ==============================================================================
+# Name of the module to be externally accessible
+SchemeName = "Finite Volumes -- Vertex centred with natural flux"
+
+# Mapping to the mesh that should be loaded
+class AssociatedMeshType():
+	""" Class that only gives out the mesh type that should be loaded when tackling the problem
+	with this scheme, given the wished variational order. Access the associated mesh properties
+	through the inner variables
+
+	MeshType,   string,  the mesh type according to fenics's classification
+	MeshOrder,  integer, the mesh order
+
+	Note: This is only a product-type class: no method is available
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, VariationalOrder = 0):
+		""" Automatic initialisation routine defining the type of mesh
+        to be loaded for this solver."""
+
+		self.MeshType = "DG"
+		self.MeshOrder = 0	# Or dependent on VariationalOrder
+		self.ControlVolumes = "Primal"
+
+
+# ==============================================================================
+#	Class furnishing all the scheme tools
+# ==============================================================================
+class Scheme():
+	""" Class furnishing all the schemes tools and registers the scheme's wished
+	parameters. The main iteration is accessible with the method "Iteration".
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, Problem, Mesh, Params = []):
+		""" Automatic initialisation routine. Just registers the user defined scheme parameters
+		within the instance"""
+		self.Problem = Problem
+		self.Params  = Params
+		self.Mesh    = Mesh
+
+    #### Emulates a flux computation as done in the iteration routine ####
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		""" Emulates a flux computation as done in the iteration routine in order
+			to be used in the DeC subtimestep evaluation.
+			
+			Input: 
+				FluidFlag,   numpy float array,  buffer vector containing the current fluid flags at each dof (NbDofs)
+				U,           numpy float array,  buffer containing the current state values at each dof (NbVars   x NbDofs)
+				LSValues,    numpy float array,  buffer containing the current FluidSpotters values at each dof(NbFluids x NbDofs)
+		
+			Output:
+				fluxes,     numpy float (multiD)array,   fluxes in the format requires by the Iteration routine
+		---------------------------------------------------- """
+	
+        # --------------------------  Initialisations -------------------------------------------------
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables,1))
+		
+		# ------------- Flux computation at the control's volume interface ----------------------------
+		for i in range(self.Mesh.NbInnerElements):
+			# Retrieving the information on the local degree of freedom
+			dele  = self.Mesh.ElementsInnerDofs[i][0]
+			ele   = self.Mesh.InnerElements[i]
+			Eele  = self.Mesh.InnerElements[i] + [self.Mesh.InnerElements[i][0]]
+			
+			# Looping on each edge
+			for f in range(len(ele)):
+				
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Flags = np.array([FluidFlag[dele]])
+				Var   = np.array([U[:,dele]])
+				
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = np.array([0.5*np.sum(self.Mesh.CartesianPoints[Eele[f:f+2],:], axis=0)])
+				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Var, EdgeMidPoint, Flags)
+				
+		
+		# Returning the full array
+		return(fluxes)
+
+    #### Main routine, definining the scheme ####
+	def Iteration(self,  Solution, fluxes, du=0, dt=1):
+		""" Main iteration of the scheme, implementing the most stupid scheme
+		you can think of.
+		
+		Args:
+			Solution,   solution structure,   structure containing the current solution's values to iterate
+			fluxes,     numpy (multiD)array,  pre-computed fluxes at the points of interest. For this scheme, 
+											  access with fluxes[element, face, coordinate(fx or fy), variable, pointindex]
+			du,         float numpy array,    (optional) when using DeC, the difference in the time iteration 
+			dt,         float,                (optional) when using DeC, the full time step
+			
+		Returns:
+			Resu,       float numpy array,    the computed residuals (NbVars x NbDofs)
+		---------------------------------------------------------------------"""
+
+		# -------- Initialising the mollification vector (residuals) -------------
+		Resu = np.zeros(np.shape(Solution.Sol))
+		
+		# ------- Getting the residuals for each element -------------------------
+		for i in range(self.Mesh.NbInnerElements):
+			# Retrieving the information on the local degree of freedom
+			dele  = self.Mesh.ElementsInnerDofs[i][0]
+			ele   = self.Mesh.InnerElements[i]
+			
+			# Looping on each edge
+			for f in range(len(ele)):
+				# Get the physical properties of the considered edge
+				L = self.Mesh.ElementsBoundaryWiseWidth[i][f]
+				n = self.Mesh.ElementsBoundaryWiseNormals[i][f,:]
+				
+				# Retrieving the flux computed through DeC
+				MidFluxes = fluxes[i,f,:,:,0]
+				
+				# Compute the pondered contribution
+				FluxN = np.dot(MidFluxes.T,n)
+				quant = L*FluxN
+
+				# Distribute it equally to each submit
+				Resu[:,dele] +=  dt*quant
+				
+		
+		#----- Modifying the residual on each Dof by addin the DeC increment --------
+		# Adapting the residual to Dec
+		Resu += du[:,:]/self.Mesh.Lumping
+		
+		# Returning the result
+		return(Resu)
+
+	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
+	def ReconstructSolutionAtVertices(self, Solution):
+		""" Routine that maps the values from the Dofs to the Physical vertices of the mesh 
+		    of the FluidSpotter, solution and flags.
+		    
+			Input:  SolverSpecs,  string,  the name of the setting file
+			Output: None, fills direclty the Parameters data structure
+			
+		Note: This is only for later plotting purposes.
+		----------------------------------------------------------------------------------- """
+		
+		# -------------- Initialising the reconstructed values ----------------------------
+		Solution.RFluidFlag = np.zeros(self.Mesh.NbMeshPoints)
+		Solution.RLSValues  = np.zeros((np.shape(Solution.LSValues)[0], self.Mesh.NbMeshPoints))
+		Solution.RSol       = np.zeros((np.shape(Solution.Sol)[0], self.Mesh.NbMeshPoints))
+		
+		# ------------- Reconstructing the values -----------------------------------------
+		# Extract the cartesian coordinates of the Dofs
+		xyDofs = self.Mesh.DofsXY[:,:]
+		
+		# Spanning all the mesh points where to fill the values
+		for i in range(self.Mesh.NbMeshPoints):
+            
+            # Retrieve the catesian coordinates of the considered point
+			xy = self.Mesh.CartesianPoints[i,:]
+					
+			# Stupid reconstruction at the closest value for Dofs #TODO better
+			norm = np.linalg.norm(xy-xyDofs,2, axis=1)
+			index = np.argsort(norm)[0]
+			
+			# Fill the reconstructed solution upon the found index
+			Solution.RLSValues[:,i] = Solution.LSValues[:,index]
+			Solution.RFluidFlag[i]  = Solution.FluidFlag[index]
+			Solution.RSol[:,i]      = Solution.Sol[:,index]
+			
+			
+
diff --git a/SourceCode/_Solver/SpatialSchemes/ContinuousGalerkin.py b/SourceCode/_Solver/SpatialSchemes/ContinuousGalerkin.py
new file mode 100644
index 0000000..83259cc
--- /dev/null
+++ b/SourceCode/_Solver/SpatialSchemes/ContinuousGalerkin.py
@@ -0,0 +1,300 @@
+"""
+################################################################################
+#	Module that defines a Vertex centred finite volume scheme                  #
+################################################################################
+
+Contains (in scrolling order):
+	-> User defined data reader tools
+
+	-> Structure definition
+
+	-> Initialisation routines
+
+Usage:
+
+Note:
+-----------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+import BarycentricTools as BT
+import BarycentricBasisFunctions as BF
+import _Solver.QuadratureRules.HandBased as QR
+import copy
+import matplotlib.pyplot as plt
+# ==============================================================================
+#	Parameter (and reader) class of scheme properties and runtime informations
+# ==============================================================================
+# Name of the module to be externally accessible
+SchemeName = "Continuous Galerkin"
+
+# Mapping to the mesh that should be loaded
+class AssociatedMeshType():
+	""" Class that only gives out the mesh type that should be loaded when tackling the problem
+	with this scheme, given the wished variational order. Access the associated mesh properties
+	through the inner variables
+
+	MeshType,   string,  the mesh type according to fenics's classification
+	MeshOrder,  integer, the mesh order
+
+	Note: This is only a product-type class: no method is available
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, SchemeParams = [1]):
+		""" Automatic initialisation routine defining the type of mesh
+        to be loaded for this solver."""
+		
+		# Extracting the schemes properties that are required to determine the Mesh
+		VariationalOrder = int(SchemeParams[0])
+		
+		# Selecting the corresponding mesh type
+		self.MeshType  = "CG"
+		self.MeshOrder = VariationalOrder
+		self.ControlVolumes = "Dual"
+
+
+# ==============================================================================
+#	Class furnishing all the scheme tools
+# ==============================================================================
+class Scheme():
+	""" Class furnishing all the schemes tools and registers the scheme's wished
+	parameters. The main iteration is accessible with the method "Iteration".
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, Problem, Mesh, Params = []):
+		""" Automatic initialisation routine. Just registers the user defined scheme parameters
+		within the instance"""
+		self.Problem = Problem
+		self.Params  = Params
+		self.Mesh    = Mesh
+
+	#### Emulates a flux computation as done in the iteration routine ####
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		""" Emulates a flux computation as done in the iteration routine in order
+			to be used in the DeC subtimestep evaluation.
+			
+			Input: 
+				FluidFlag,   numpy float array,  buffer vector containing the current fluid flags at each dof (NbDofs)
+				U,           numpy float array,  buffer containing the current state values at each dof (NbVars   x NbDofs)
+				LSValues,    numpy float array,  buffer containing the current FluidSpotters values at each dof(NbFluids x NbDofs)
+		
+			Output:
+				fluxes,     numpy float (multiD)array,   fluxes in the format requires by the Iteration routine
+		---------------------------------------------------- """
+	
+        # --------------------------  Initialisations -------------------------------------------------
+		# Defining the output fluxes (inner and boundary ones)
+		# The very first coordinate corresponds to the inner flux. Only Flux[0, :, 0, :, :, :] is filled
+		# The second coordinate corresponds to the inner flux. Only Flux[1, :, :, :, :, 0:3] is filled
+		Flux   = np.zeros((2, self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables, 12))
+		
+		# Retrieving the scheme order 
+		Order = int(self.Params[0])
+				
+		# ------------- Flux computation at the control's volume interface ----------------------------
+		# Spanning on each element
+		for i in range(self.Mesh.NbInnerElements):
+			# ---- Getting the element and state informations
+			# Retrieving the physical information on the spanned element's vertices 
+			ele = self.Mesh.InnerElements[i]
+			vol = self.Mesh.InnerElementsVolume[i]
+			n   = self.Mesh.ElementsBoundaryWiseNormals[i]
+			
+			# Retrieving the information on the local degree of freedom
+			dele = np.array(self.Mesh.ElementsDofs[i])
+			LS   = LSValues[:, dele]
+			UU   = U[:, dele]
+			
+			# Get the coordinates of the vertices in the physical order
+			elevert = self.Mesh.CartesianPoints[ele,:]
+			
+			# ---- Reordering the normal's according to the edge's order given by the Dofs
+			# Get the coordinates of the vertices in the variational order
+			reordering = np.array(self.Mesh.Dofs2Vertex)[dele[self.Mesh.VertexLocation[i]]].tolist()
+			eleverts = self.Mesh.CartesianPoints[reordering,:]
+			
+			# Getting the local indexing from the Dof to the vertex
+			LocIdx = self.Mesh.TabulatedVertex[i]
+			
+			# Reordering the normal according to the Dofs order (to be used in the definition of the gradients)
+			if (LocIdx[1]==LocIdx[0]+1 or (LocIdx[0] == len(ele)-1 and LocIdx[1]==0)):	NIdx = LocIdx
+			else: NIdx = np.roll(LocIdx,-1)
+			
+			# ---- Computing the plain integral 
+			# Retrieving the quadrature points as barycentric coordinates and convert them to cartesian 
+			# in the Dof's vertex ordering referential
+			bqp, weights  = QR.InnerQuadrature()
+			qp = BT.GetCartesianCoordinates(bqp, eleverts)
+			
+			# Evaluate the basis function in the order of the Dofs
+			EvaluatedBase      = BF.SimplicialBasis("B", Order, bqp)
+			EvaluatedGradients = BF.SimplicialGradients("B", Order, bqp, n[NIdx,:], vol)	# reorder the n since we are in the referential of the dofs
+			
+			# Reconstructing the solution and its gradient at the quadrature point
+			Uloc  = np.dot(UU,  EvaluatedBase.T)
+			LSloc = np.dot(LS, EvaluatedBase.T)
+			
+			# Compute the flux according to the fluid's flag
+			Flags = np.argmax(LSloc, axis=0)
+			Flux[0, i, 0, :, :]  = self.Problem.GoverningEquations.Flux(Uloc, qp, Flags)
+			
+			for face in range(len(ele)):
+				# Get the coordinates of the quadrature points on the edge, flipping the 
+				# points to match the Dof's order with the physical vertices
+				qbbp, weights  = QR.BoundaryQuadrature(face)
+				bqp = BT.GetCartesianCoordinates(qbbp[:,LocIdx], elevert)
+				
+				# Compute each basis function value at those points
+				# (get the order of the basis in the order of the physical vertices:
+				# the dele have to be ordered in the same output as the evaluated base:
+				# check also for higher order the order of the dele given in the mesh and 
+				# the order of the basis function given here)
+				EvaluatedBase = BF.SimplicialBasis("B", Order, qbbp[:,LocIdx])	# To move as qbbp are not tabulated at the right dof location,, tabultaed on physical and not dof ordered
+				
+				# Reconstruct the solution's value at this point
+				Uloc  = np.dot(UU,  EvaluatedBase.T)
+				LSloc = np.dot(LS, EvaluatedBase.T)
+				
+				# Evaluate the flux there
+				Flags = np.argmax(LSloc, axis=0)
+				Flux[1,i,face,:,:,0:3] = self.Problem.GoverningEquations.Flux(Uloc, bqp, Flags)
+		
+		# Returning the full array
+		return(Flux)
+
+	#### Main routine, definining the scheme ####
+	def Iteration(self,  Solution, fluxes, du=0, dt=1):
+		""" Main iteration of the scheme, implementing the most stupid scheme
+		you can think of.
+		
+		Args:
+			Solution,   solution structure,   structure containing the current solution's values to iterate
+			fluxes,     numpy (multiD)array,  pre-computed fluxes at the points of interest. For this scheme, 
+											  access with fluxes[element, face, coordinate(fx or fy), variable, pointindex]
+			du,         float numpy array,    (optional) when using DeC, the difference in the time iteration 
+			dt,         float,                (optional) when using DeC, the full time step
+			
+		Returns:
+			Resu,       float numpy array,    the computed residuals (NbVars x NbDofs)
+		---------------------------------------------------------------------"""
+
+		# -------- Initialisation ------------------------------------------------
+		# Retrieving the scheme order 
+		Order = int(self.Params[0])
+		# Initialising the mollification vector (residuals)
+		Resu  = np.zeros(np.shape(Solution.Sol))
+		
+		# Retrieving the inner and boundary fluxes
+		Flx = fluxes[0, :, 0, :, :]
+		Fln = fluxes[1,:,:,:,:,0:3]
+		
+		# ------- Getting the residuals for each element -------------------------
+		# Spanning on each element
+		for i in range(self.Mesh.NbInnerElements):
+			# ---- Getting the element and state informations
+			# Retrieving the physical information on the spanned element's vertices 
+			ele = self.Mesh.InnerElements[i]
+			vol = self.Mesh.InnerElementsVolume[i]
+			n   = self.Mesh.ElementsBoundaryWiseNormals[i]
+			
+			# Retrieving the information on the local degree of freedom
+			dele = np.array(self.Mesh.ElementsDofs[i])
+			dU   = du[:, dele]
+			
+			# ---- Reordering the normal's according to the edge's order given by the Dofs
+			# Getting the local indexing from the Dof to the vertex
+			LocIdx = self.Mesh.TabulatedVertex[i]
+			
+			# Reordering the normal according to the Dofs order (to be used in the definition of the gradients)
+			if (LocIdx[1]==LocIdx[0]+1 or (LocIdx[0] == len(ele)-1 and LocIdx[1]==0)):	NIdx = LocIdx
+			else: NIdx = np.roll(LocIdx,-1)
+			
+			# ---- Computing the plain integral 
+			# Retrieving the quadrature points as barycentric coordinates and convert them to cartesian 
+			# in the Dof's vertex ordering referential
+			bqp, weights  = QR.InnerQuadrature()
+			
+			# Evaluate the basis function in the order of the Dofs
+			EvaluatedBase      = BF.SimplicialBasis("B", Order, bqp)
+			EvaluatedGradients = BF.SimplicialGradients("B", Order, bqp, n[NIdx,:], vol)	# reorder the n since we are in the referential of the dofs
+			
+			# Reconstructing the solution and its gradient at the quadrature point
+			dUloc = np.dot(dU, EvaluatedBase.T)
+			
+			# Compute the flux according to the fluid's flag
+			Flux = Flx[i]
+			
+			# Complete the resodual at each Dof by the relevant quadrature value
+			for dof in range(len(dele)):
+				# Evaluating the flux at the quadrature point for the reconstructed flux variable
+				qt  = np.sum([weights[q]*(EvaluatedGradients[dof,q,0]*Flux[0,:,q]+EvaluatedGradients[dof,q,1]*Flux[1,:,q]) for q in range(len(weights))], axis=0)
+				DEC = np.sum(EvaluatedBase[:,dof]*dUloc, axis=1)
+				
+				# Updating the residual at the spanned Dof by the local contribution
+				Resu[:,dele[dof]] += (DEC - dt*qt)*vol
+			
+			# ---- Computing the boundary integral 
+			# Spanning each edge in the physical element			
+			for face in range(len(ele)):
+				# Get the coordinates of the quadrature points on the edge, flipping the 
+				# points to match the Dof's order with the physical vertices
+				qbbp, weights  = QR.BoundaryQuadrature(face)
+				
+				# Compute each basis function value at those points
+				# (get the order of the basis in the order of the physical vertices:
+				# the dele have to be ordered in the same output as the evaluated base:
+				# check also for higher order the order of the dele given in the mesh and 
+				# the order of the basis function given here)
+				EvaluatedBase = BF.SimplicialBasis("B", Order, qbbp[:,LocIdx])	# To move as qbbp are not tabulated at the right dof location,, tabultaed on physical and not dof ordered
+				
+				# Evaluate the flux there
+				Flux = Fln[i,face]
+				
+				# Compute the flux's normal contribution to each dof
+				Qt = (Flux[0,:,:]*n[face,0]+Flux[1,:,:]*n[face,1])
+				WeigthedBaseWiseQt = np.array([weights[pt]*np.array([Qt[:,pt]]).T*np.array([EvaluatedBase[pt,:]]) for pt in range(len(weights))])
+				PonderedQt = np.sum(WeigthedBaseWiseQt, axis=0)
+				
+				# Adjust the residuals
+				Resu[:,dele] += dt*PonderedQt
+		
+		# Returning the result
+		return(Resu)
+
+	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
+	def ReconstructSolutionAtVertices(self, Solution):
+		""" Routine that maps the values from the Dofs to the Physical vertices of the mesh 
+		    of the FluidSpotter, solution and flags.
+		    
+			Input:  SolverSpecs,  string,  the name of the setting file
+			Output: None, fills direclty the Parameters data structure
+			
+		Note: This is only for later plotting purposes.
+		----------------------------------------------------------------------------------- """
+		
+		# -------------- Initialising the reconstructed values ----------------------------
+		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
+		Solution.RLSValues  = np.zeros(np.shape(Solution.LSValues))
+		Solution.RSol       = np.zeros(np.shape(Solution.Sol))
+		
+		# ------------- Reconstructing the values -----------------------------------------
+		# Extract the cartesian coordinates of the Dofs
+		xyDofs = self.Mesh.DofsXY[:,:]
+		
+		# Spanning all the mesh points where to fill the values
+		for i in range(self.Mesh.NbMeshPoints):
+            
+			# Averaging all the Dofs values located at the corresponding physical vertex 
+			index = self.Mesh.Vertex2Dofs[i]
+			RS  = np.mean(Solution.LSValues[:,index], axis=1)
+			RSO = np.mean(Solution.Sol[:,index],      axis=1)
+			FF  = int(np.mean(Solution.FluidFlag[index]))
+			
+			# Fill the reconstructed solution upon the found index
+			Solution.RLSValues[:,i] = RS
+			Solution.RFluidFlag[i]  = FF
+			Solution.RSol[:,i]      = RSO
diff --git a/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_LFX.py b/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_LFX.py
index af58c9c..48c799e 100644
--- a/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_LFX.py
+++ b/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_LFX.py
@@ -24,7 +24,7 @@ import numpy as np
 #	Parameter (and reader) class of scheme properties and runtime informations
 # ==============================================================================
 # Name of the module to be externally accessible
-SchemeName = "Finite Volumes -- Vertex centred -- LFX flux"
+SchemeName = "Finite Volumes -- Vertex centred with natural flux"
 
 # Mapping to the mesh that should be loaded
 class AssociatedMeshType():
@@ -40,10 +40,12 @@ class AssociatedMeshType():
 
 	#### Automatic initialisation routine ####
 	def __init__(self, VariationalOrder = 0):
-		""" Automatic initialisation routine."""
+		""" Automatic initialisation routine defining the type of mesh
+        to be loaded for this solver."""
 
 		self.MeshType = "CG"
 		self.MeshOrder = 1	# Or dependent on VariationalOrder
+		self.ControlVolumes = "Dual"
 
 
 # ==============================================================================
@@ -62,85 +64,111 @@ class Scheme():
 		self.Params  = Params
 		self.Mesh    = Mesh
 
-	def ComputeFlux(self, U):
-		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.NbVars, 2))
-		for i in range(self.Mesh.NbInnerElements):
-			ala = self.Mesh.InnerElements[i]
-				
-			# Looping on each edge
-			for f in range(len(ala)):
-				edge = np.array(self.Mesh.ElementsBoundaryWiseDofs[i][f])
-				Vars  = U[:, edge]
-				Flags = FluidFlag[edge]
-
-				Coords = np.array(len(Vars[0,:])*[0.5*(self.Mesh.DualCenters[i][f] + self.Mesh.DualCenters[i][-1])])
+    #### Emulates a flux computation as done in the iteration routine ####
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		""" Emulates a flux computation as done in the iteration routine in order
+			to be used in the DeC subtimestep evaluation.
 			
-				
-				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Vars, Coords, Flags)
-		return(fluxes)
+			Input: 
+				FluidFlag,   numpy float array,  buffer vector containing the current fluid flags at each dof (NbDofs)
+				U,           numpy float array,  buffer containing the current state values at each dof (NbVars   x NbDofs)
+				LSValues,    numpy float array,  buffer containing the current FluidSpotters values at each dof(NbFluids x NbDofs)
+		
+			Output:
+				fluxes,     numpy float (multiD)array,   fluxes in the format requires by the Iteration routine
+		---------------------------------------------------- """
+	
+        # --------------------------  Initialisations -------------------------------------------------
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables,2))
+		
+		# ------------- Flux computation at the control's volume interface ----------------------------
+		# Spanning the elements
 		for i in range(self.Mesh.NbInnerElements):
-			ala = self.Mesh.InnerElements[i]
-				
-			# Looping on each edge
-			for f in range(len(ala)):
-				edge = np.array(self.Mesh.ElementsBoundaryWiseDofs[i][f])
-				Coords = np.array(len(Vars[0,:])*[0.5*(self.Mesh.DualCenters[i][f] + self.Mesh.DualCenters[i][-1])])
+			# Retrieving the variational information
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
 			
-				Vars  = Solution.Sol[:, edge]
-				Flags = Solution.FluidFlag[edge]
-
-				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Vars, Coords, Flags)
+			# Looping on each edge
+			for f in range(len(ele)):
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				Flags = FluidFlag[DEdge]
+
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Var1 = U[:,DEdge[0]]
+				Var2 = U[:,DEdge[1]]
+				Var = np.array([Var1, Var2])				# To approximate properly wrt basis functions
+				
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = 0.5*np.array(2*[self.Mesh.DualCenters[i][-1,:]+self.Mesh.DualCenters[i][f]])
+				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Var, EdgeMidPoint, Flags)
+		
+		# Returning the full array
 		return(fluxes)
 
-	#### Main routine, defininin the scheme ####
-	def Iteration(self,  Solution):
-		""" Main iteration of the scheme
-		Input:  SolverSpecs,  string,  the name of the setting file
-		Output: None, fills direclty the Parameters data structure
+    #### Main routine, definining the scheme ####
+	def Iteration(self,  Solution, fluxes, du=0, dt=1):
+		""" Main iteration of the scheme, implementing the most stupid scheme
+		you can think of.
+		
+		Args:
+			Solution,   solution structure,   structure containing the current solution's values to iterate
+			fluxes,     numpy (multiD)array,  pre-computed fluxes at the points of interest. For this scheme, 
+											  access with fluxes[element, face, coordinate(fx or fy), variable, pointindex]
+			du,         float numpy array,    (optional) when using DeC, the difference in the time iteration 
+			dt,         float,                (optional) when using DeC, the full time step
+			
+		Returns:
+			Resu,       float numpy array,    the computed residuals (NbVars x NbDofs)
 		---------------------------------------------------------------------"""
 
-		# Initialising the mollification vector (residuals)
-		Res = np.zeros(np.shape(Solution.Sol))
-		Sol = Solution.Sol[:,:]
-
-		# Looping on each element
+		# -------- Initialising the mollification vector (residuals) -------------
+		Resu = np.zeros(np.shape(Solution.Sol))
+		
+		# ------- Getting the residuals for each element -------------------------
 		for i in range(self.Mesh.NbInnerElements):
-			ala = self.Mesh.InnerElements[i]
+            # Retrieving the information on the local degree of freedom
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
 			
 			# Looping on each edge
-			for f in range(len(ala)):
-				# Getting the geometrical properties
-				edge = np.array(self.Mesh.ElementsBoundaryWiseDofs[i][f])
-				L    = self.Mesh.DualEdgesWidths[i][f,:]
-				n    = self.Mesh.DualNormals[i][f]
-
-				# Getting the variational quantitites
-				Vars  = Solution.Sol[:, edge]
-				Flags = Solution.FluidFlag[edge]
-
-				# Creating the compatibility quantities to call the spectrum and flux functions
-				Coords = np.array(len(Vars[0,:])*[0.5*(self.Mesh.DualCenters[i][f] + self.Mesh.DualCenters[i][-1])])
+			for f in range(len(ele)):
+                # Getting the Dofs of this edge
+				DEdge = Dele[f:f+2]
+				Flags = Solution.FluidFlag[DEdge]
+
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Var1 = Solution.Sol[:,DEdge[0]]
+				Var2 = Solution.Sol[:,DEdge[1]]
+				Vars = np.array([Var1, Var2]).T				# To approximate properly wrt basis functions
+				
+				# Get the physical properties of the considered edge
+				L = self.Mesh.DualEdgesWidths[i][f]
+				n = self.Mesh.DualNormals[i][f,:]
+				Coords = 0.5*np.array(2*[self.Mesh.DualCenters[i][-1,:]+self.Mesh.DualCenters[i][f]])
 				nn     = np.array(len(Vars[0,:])*[n])
 				
-				# Computing the fluxes and the spectral radiuses
-				fluxes = self.Problem.GoverningEquations.Flux(Vars, Coords, Flags)
-				spr    = self.Problem.GoverningEquations.SpectralRadius(Vars, Flags, nn, Coords)
-
-				# Creating the LFX flux between the two control volumes
-				fluxes[:,:,1] = - fluxes[:,:,1]
-				avrg  = 0.5*np.sum(fluxes, axis=2)
-				Fluxn = np.dot(avrg.T,n)
-
-				# Distibuting the residuals to the associated Dof
-				Quantity = L*(Fluxn + np.max(spr)*(Vars[:,0]-Vars[:,1]))
-				Res[:,edge[0]]  += Quantity
-				Res[:,edge[-1]] -= Quantity
+				# Retrieving the flux computed through DeC
+				MidFluxes = fluxes[i,f,:,:,:]
+				MidFluxes[:,:,1] = -MidFluxes[:,:,1]
+				spr = self.Problem.GoverningEquations.SpectralRadius(Vars, Flags, nn, Coords)
+
+				# Compute the pondered contribution
+				FluxN = np.dot(0.5*np.sum(MidFluxes, axis=2).T,n)
+				quant = L*FluxN + L*np.max(spr)*(Vars[:,0]-Vars[:,1])
+
+				# Distribute it equally to each submit
+				Resu[:,DEdge[0]] +=  dt*quant 
+				Resu[:,DEdge[1]] += -dt*quant
 		
-		# Updating each nodal value (here representing the average) by the residual
-		# pondered by the control volume's area
-		Res[:, :] = Res[:, :]/self.Mesh.DualAreasTot[:].T
+		#----- Modifying the residual on each Dof by addin the DeC increment --------
+		# Adapting the residual to Dec
+		Resu += du[:,:]/self.Mesh.Lumping
 		
-		return(Res)
+		# Returning the result
+		return(Resu)
 
 	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
 	def ReconstructSolutionAtVertices(self, Solution):
@@ -153,20 +181,27 @@ class Scheme():
 		Note: This is only for later plotting purposes.
 		----------------------------------------------------------------------------------- """
 		
-		# Stupid reconstruction at the closest value for Dofs #TODO better
+		# -------------- Initialising the reconstructed values ----------------------------
+		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
 		Solution.RLSValues  = np.zeros(np.shape(Solution.LSValues))
 		Solution.RSol       = np.zeros(np.shape(Solution.Sol))
-		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
 		
+		# ------------- Reconstructing the values -----------------------------------------
+		# Extract the cartesian coordinates of the Dofs
+		xyDofs = self.Mesh.DofsXY[:,:]
+		
+		# Spanning all the mesh points where to fill the values
 		for i in range(self.Mesh.NbMeshPoints):
+            
+            # Retrieve the catesian coordinates of the considered point
 			xy = self.Mesh.CartesianPoints[i,:]
-			xyDofs = self.Mesh.DofsXY[:,:]
-			
-			norm = np.linalg.norm(xy-xyDofs,2, axis=1)
+					
+			# Stupid reconstruction at the closest value for Dofs #TODO better
+			norm = np.linalg.norm(xy-xyDofs, 2, axis=1)
 			index = np.argsort(norm)[0]
 			
+			# Fill the reconstructed solution upon the found index
 			Solution.RLSValues[:,i] = Solution.LSValues[:,index]
 			Solution.RFluidFlag[i]  = Solution.FluidFlag[index]
 			Solution.RSol[:,i]      = Solution.Sol[:,index]
 
-	
diff --git a/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_Natural.py b/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_Natural.py
new file mode 100644
index 0000000..fab5b4d
--- /dev/null
+++ b/SourceCode/_Solver/SpatialSchemes/VertexCentredFV_Natural.py
@@ -0,0 +1,204 @@
+"""
+################################################################################
+#	Module that defines a Vertex centred finite volume scheme                  #
+################################################################################
+
+Contains (in scrolling order):
+	-> User defined data reader tools
+
+	-> Structure definition
+
+	-> Initialisation routines
+
+Usage:
+
+Note:
+-----------------------------------------------------------------------------"""
+
+# ==============================================================================
+#	Preliminary imports
+# ==============================================================================
+import numpy as np
+
+# ==============================================================================
+#	Parameter (and reader) class of scheme properties and runtime informations
+# ==============================================================================
+# Name of the module to be externally accessible
+SchemeName = "Finite Volumes -- Vertex centred with natural flux"
+
+# Mapping to the mesh that should be loaded
+class AssociatedMeshType():
+	""" Class that only gives out the mesh type that should be loaded when tackling the problem
+	with this scheme, given the wished variational order. Access the associated mesh properties
+	through the inner variables
+
+	MeshType,   string,  the mesh type according to fenics's classification
+	MeshOrder,  integer, the mesh order
+
+	Note: This is only a product-type class: no method is available
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, VariationalOrder = 0):
+		""" Automatic initialisation routine defining the type of mesh
+        to be loaded for this solver."""
+
+		self.MeshType = "CG"
+		self.MeshOrder = 1	# Or dependent on VariationalOrder
+		self.ControlVolumes = "Dual"
+
+
+# ==============================================================================
+#	Class furnishing all the scheme tools
+# ==============================================================================
+class Scheme():
+	""" Class furnishing all the schemes tools and registers the scheme's wished
+	parameters. The main iteration is accessible with the method "Iteration".
+	--------------------------------------------------------------------------------------"""
+
+	#### Automatic initialisation routine ####
+	def __init__(self, Problem, Mesh, Params = []):
+		""" Automatic initialisation routine. Just registers the user defined scheme parameters
+		within the instance"""
+		self.Problem = Problem
+		self.Params  = Params
+		self.Mesh    = Mesh
+
+    #### Emulates a flux computation as done in the iteration routine ####
+	def ComputeFlux(self, FluidFlag, U, LSValues):
+		""" Emulates a flux computation as done in the iteration routine in order
+			to be used in the DeC subtimestep evaluation.
+			
+			Input: 
+				FluidFlag,   numpy float array,  buffer vector containing the current fluid flags at each dof (NbDofs)
+				U,           numpy float array,  buffer containing the current state values at each dof (NbVars   x NbDofs)
+				LSValues,    numpy float array,  buffer containing the current FluidSpotters values at each dof(NbFluids x NbDofs)
+		
+			Output:
+				fluxes,     numpy float (multiD)array,   fluxes in the format requires by the Iteration routine
+		---------------------------------------------------- """
+	
+        # --------------------------  Initialisations -------------------------------------------------
+		fluxes = np.zeros((self.Mesh.NbInnerElements, 3, 2, self.Problem.GoverningEquations.NbVariables,1))
+		
+		# ------------- Flux computation at the control's volume interface ----------------------------
+		# Spanning the elements
+		for i in range(self.Mesh.NbInnerElements):
+			# Retrieving the variational information
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			
+			# Looping on each edge
+			for f in range(len(ele)):
+				# Get the variational information at each Dof lying on the edge
+				DEdge = Dele[f:f+2]
+				Flags = FluidFlag[np.argmax(LSValues[:,DEdge], axis=1)]
+
+				# Reconstructing the solution's value at the considered point (usually quadrature point)
+				Var1 = U[:,DEdge[0]]
+				Var2 = U[:,DEdge[1]]
+				Var = np.array([0.5*(Var1+Var2)])				# To approximate properly wrt basis functions
+				
+				# Compute the input exchange from the neighboring element at the edge midpoint
+				EdgeMidPoint = 0.5*np.array([self.Mesh.DualCenters[i][-1,:]+self.Mesh.DualCenters[i][f]])
+				fluxes[i,f,:,:,:] = self.Problem.GoverningEquations.Flux(Var, EdgeMidPoint, Flags)
+		
+		# Returning the full array
+		return(fluxes)
+
+    #### Main routine, definining the scheme ####
+	def Iteration(self,  Solution, fluxes, du=0, dt=1):
+		""" Main iteration of the scheme, implementing the most stupid scheme
+		you can think of.
+		
+		Args:
+			Solution,   solution structure,   structure containing the current solution's values to iterate
+			fluxes,     numpy (multiD)array,  pre-computed fluxes at the points of interest. For this scheme, 
+											  access with fluxes[element, face, coordinate(fx or fy), variable, pointindex]
+			du,         float numpy array,    (optional) when using DeC, the difference in the time iteration 
+			dt,         float,                (optional) when using DeC, the full time step
+			
+		Returns:
+			Resu,       float numpy array,    the computed residuals (NbVars x NbDofs)
+		---------------------------------------------------------------------"""
+
+		# -------- Initialising the mollification vector (residuals) -------------
+		Resu = np.zeros(np.shape(Solution.Sol))
+		
+		# ------- Getting the residuals for each element -------------------------
+		for i in range(self.Mesh.NbInnerElements):
+            # Retrieving the information on the local degree of freedom
+			dele = self.Mesh.ElementsBoundaryDofs[i]
+			ele  = self.Mesh.InnerElements[i]
+			Dele = np.array(dele + [dele[0]], dtype=int)
+			n = self.Mesh.DualNormals[i]
+			
+			vf = []
+			for f in range(len(ele)):
+
+				vf += [np.where([self.Mesh.ElementsBoundaryWiseDofs[i][p] == Dele[f:f+2].tolist() or self.Mesh.ElementsBoundaryWiseDofs[i][p] == Dele[f:f+2][::-1].tolist() for p in range(3)])[0][0]]
+			
+			n = n[vf,:]
+			# Looping on each edge
+			for f in range(len(ele)):
+                # Getting the Dofs of this edge
+				DEdge = Dele[f:f+2]
+				
+				# Get the physical properties of the considered edge
+				L = self.Mesh.DualEdgesWidths[i][f]
+				
+				
+				# Retrieving the flux computed through DeC
+				MidFluxes = fluxes[i,f,:,:,0]
+				
+				# Compute the pondered contribution
+				FluxN = np.dot(MidFluxes.T,n[f,:].T)
+				quant = L*FluxN
+
+				# Distribute it equally to each submit
+				Resu[:,DEdge[0]] +=  dt*quant
+				Resu[:,DEdge[1]] += -dt*quant
+		
+		#----- Modifying the residual on each Dof by addin the DeC increment --------
+		# Adapting the residual to Dec
+		Resu += du[:,:]/self.Mesh.Lumping
+		
+		# Returning the result
+		return(Resu)
+
+	#### Routine that maps the values from the Dofs to the Physical vertices of the mesh ####
+	def ReconstructSolutionAtVertices(self, Solution):
+		""" Routine that maps the values from the Dofs to the Physical vertices of the mesh 
+		    of the FluidSpotter, solution and flags.
+		    
+			Input:  SolverSpecs,  string,  the name of the setting file
+			Output: None, fills direclty the Parameters data structure
+			
+		Note: This is only for later plotting purposes.
+		----------------------------------------------------------------------------------- """
+		
+		# -------------- Initialising the reconstructed values ----------------------------
+		Solution.RFluidFlag = np.zeros(np.shape(Solution.FluidFlag))
+		Solution.RLSValues  = np.zeros(np.shape(Solution.LSValues))
+		Solution.RSol       = np.zeros(np.shape(Solution.Sol))
+		
+		# ------------- Reconstructing the values -----------------------------------------
+		# Extract the cartesian coordinates of the Dofs
+		xyDofs = self.Mesh.DofsXY[:,:]
+		
+		# Spanning all the mesh points where to fill the values
+		for i in range(self.Mesh.NbMeshPoints):
+            
+            # Retrieve the catesian coordinates of the considered point
+			xy = self.Mesh.CartesianPoints[i,:]
+					
+			# Stupid reconstruction at the closest value for Dofs #TODO better
+			norm = np.linalg.norm(xy-xyDofs,2, axis=1)
+			index = np.argsort(norm)[0]
+			
+			# Fill the reconstructed solution upon the found index
+			Solution.RLSValues[:,i] = Solution.LSValues[:,index]
+			Solution.RFluidFlag[i]  = Solution.FluidFlag[index]
+			Solution.RSol[:,i]      = Solution.Sol[:,index]
+
diff --git a/SourceCode/_Solver/SpatialSchemes/__pycache__/CellCentredFV_LFX.cpython-38.pyc b/SourceCode/_Solver/SpatialSchemes/__pycache__/CellCentredFV_LFX.cpython-38.pyc
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diff --git a/SourceCode/_Solver/TemporalSchemes/DeC.py b/SourceCode/_Solver/TemporalSchemes/DeC.py
index c9e5dae..e021d45 100644
--- a/SourceCode/_Solver/TemporalSchemes/DeC.py
+++ b/SourceCode/_Solver/TemporalSchemes/DeC.py
@@ -208,12 +208,12 @@ def DeC(schemes, dt, Mesh, Sol, M, K, type = "Equispaced"):
 			[upres, lpres] = SPL.RD(Theta, m, M, dt, schemes, Mesh, SolBuff, up, lp)
 			
 			# Retrieving a DeC-adapted increment on the residuals
-			ua[:,:,m] = ua[:,:,0] + upres
-			la[:,:,m] = la[:,:,0] + lpres
+			ua[:,:,m] = ua[:,:,m] - upres
+			la[:,:,m] = la[:,:,m] - lpres
 
 	# Updating the subtime step
 	U  = ua[:,:,M]
 	LS = la[:,:,M]
-
+	
 	# Returning the full time*step
 	return([U, LS])
diff --git a/SourceCode/_Solver/TemporalSchemes/RK.py b/SourceCode/_Solver/TemporalSchemes/RK.py
deleted file mode 100644
index 25f1672..0000000
--- a/SourceCode/_Solver/TemporalSchemes/RK.py
+++ /dev/null
@@ -1,2 +0,0 @@
-def Iteration(params, Var):
-	return(0)
diff --git a/SourceCode/_Solver/TemporalSchemes/__pycache__/DeC.cpython-38.pyc b/SourceCode/_Solver/TemporalSchemes/__pycache__/DeC.cpython-38.pyc
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diff --git a/SourceCode/_Solver/__pycache__/Solver_Spatial.cpython-38.pyc b/SourceCode/_Solver/__pycache__/Solver_Spatial.cpython-38.pyc
index b5a204db9819a4506cd45a02187f6969b8ffa798..d6f933d59ff3b54f58e809f1789e0326d3abe91c 100644
GIT binary patch
delta 1058
zcmcgqL2J}N6rRZ>+059@)^<^pA_E06hiVb^P)a>mq(Zx>wU-55;?C?QHQB6_EY+H_
z7iDu@_7B*jUIh<c1b;^G4|wU(vv0DX{s8g8eDA%P_r5o8<`vCJ(><uy83K9q=JT^N
zcj~T;Id}rKk-)-;aNDRfX%j(&`4+WJVeBK%^1ot)*y=wPwo1p8-VY<mv%t^zP$k`w
zk|_@dnN&T$3uaFxF`P=Cb^{qhU{ZP8@*bs}^Ht!W^eB)6K1!wFM`0F(1MYL63B|~d
zxCqlsh0jM>m<)i;kL6LKUO-ZnKK}Pju?O)mf~!Igg7>7K>6SJaUE8V_J4pS$1fR8%
zOpZ4gTl;VFL;roNOSPIy11*A~HwdhBX&N6>V?rk8#Om48v)!G%p3{z9z#Lpc?`t_d
z#TbD{z;kk%FXSvIrzT7deg_dJZtmvRA#H*s$VmgdMqf{m^XVu4MuKxtWLXz_2+&74
zdb)Fh3!LME^dKFG5fGyS9pb{8lg=^6Jt$e|@q|Ku@12q+ni?-nC}Jp>Qq=N{_H4oC
z7eCmgv!6uO&zIJdKxTevM8nx#`xxJzeX(C{dbp)rNKqz$N0fdCrSGWf8mgm)w)7+2
zftBh&8yU!BvyaZT*>~si&D!oL9)^RyxA-T&#CxS7GPMBVrMV~5KsjK}Y1!-sy*eHZ
zVJzBKl!BU9s_6(_t~4HRuYgW30HE{CT*vF=7QRkx?2;9m875Iz;KZ}aTBjqDZl_aP
o1GvR&)r4)eq(KMFd8;;l_fk5WaUw=bk=)X+mD&L2vV)!4FY;g6DgXcg

delta 1014
zcmZ`%&1%~~5Z)ijmh4#mZJ>>5Ljxr`w1kr0LN7Ief@4aZ^dNA9@ybquV!JCjq+x?E
zDb_XU1>$25g+ec-$3BMW3-m2IyK>@~Gz&YU)$GoE-#0U<gcWNqm&+hn%g?`mJT)(^
z>YFN@7-HB%4pyExgb_wBkV6@6B3t__A2d3qbhA|gZ@{Lo{fp=oIZM-tHqpC!^pn(=
z19C(Ia-~i1giK}uW<<ZpO4(YBbtyCD|5je)g}I*+Ik4$aW8@*aGK4wB7{Q6c$rPqo
z5*b0K8p8~Kfg9(A+06{ZERkJ^6-7Z5djuF{e$?r2x+{s7CEzVEJtR<T^Bjj*;E;47
zSx6azGAKkx7}BZ|i50jD0H&mZF3Bei5)zIfMZ+d>bdp{z+j?v?2i_<+9Q0#-_rML>
z(YN#&ejKeEr{k}<!E<n(>tJlkrH));;Y-q}kR`wJV4lGCH83eeP)Am|UIadF7}VjQ
z7}S6Py2Ic$`Y}o1hd@ZGb}uDapkZS!dzxU?71o$|!vt?uM*ai(Dg3_|bi|C6{X}3n
zUkDH=o1nUulow_0J2J^ROKLC5w066qosIq&>n)?+{LuDBt{+oqhGT4d?Otur^<(pG
z;PSR?j$`c&SvzpIXE*KAyc;pzi3tn1BpGWvu7AMggeQDU(L3|m*gNcZxTBvWZZzbs
z@93TnP9e8b$qUusIr6z`vdb}fUrS5$kHDZphHl^u`VzlTJB3r&qE%d_d=;X$7VG`d
l@$fT$04KgFhgSs>p_C*yy2QrT@qmpy_qCiF)eZw2`aguyy37Co

diff --git a/SourceCode/_Solver/__pycache__/Solver_Temporal.cpython-38.pyc b/SourceCode/_Solver/__pycache__/Solver_Temporal.cpython-38.pyc
index c593fdeea1704c9f3faf5580dfa9623779b0228a..c459cf0672908c1d744e46778e55dde957e35bd8 100644
GIT binary patch
delta 997
zcmZuv&ubGw6rP!#O*We!rm?NYBCZ$PqO_$4741Q3ixzCD#`aLqu+2`Iq?_Gzc2aGb
zrC{;mNmxYi)PNVkiwJ@Uo_ZF~^Ec?tlLztbrUj7>yxDKQ_ujWN@4X+T??Z)^eBMN`
z-s)c-e=dF~+$4J>J|8U#Uju9wEO-x_#4QvOfe(m8Ev<(os!~ZDyh-?)(1A<s<2m$b
zD%1c|u(Z_5NJnJ^*od*n9JBj)1YJWKnn7Eayd_W}pmMNuqKtW>Ov3&n<9de3I@&&d
zh8<c#<GTGcIwUVft)o@(-EGFN#HT*AnqDAmuf`HCvb3QQ+jT9`U{;XCjs{l8=60J2
z#)D`<d$XtLT-SD0YMD3Nw!6DH&$-WUSS#M9=Rff*vD0CeU$<($C%Erg+rHavvk7aq
z!D^e<j^DL{T7$Kj<*<6wYl^1tO<1?reV(AD@ti8m_ET4r%s>&Zqx4<I{XnIb)}ue#
z$}Kf8?}COWf(6fGe9o^2%3QVGE=z7RP8D5bK|`hbuT&b`O5x@^#0Q~=zt^J;I$E5r
zE;pRXS>Iv7<h{y+x#%^$d$t6Alu40-Zr}o*qWzr@M?dNLy>H-ia7YD`q=l14e+qTL
zMCxS}rT}RPN(&^Ygi%OhUWSHDgHMlr(kMgILl}!zN~R$z11Rl4^Pdx$t`lHPh=q(k
zmoQn8IWXk{X=RTkHgFGzIhe-WL0+0NBXcs_qhTHb3zF6`NSume;arsY9(J@dXav0?
zkVUtqKOX9X<l+F&z=L=^|0<sp^0=zJ5IgeGuk=-{N0*KJI2VORd8`<R8c@`3+%2U$
z%oR329kUv9nW8{2h3nB*<HBBC>ZIO$C=STpVd7!4%_@R*t^*OAgAJKfD-aW#T9K5n
ziA!1u4}*<qd=k9lx=L58j$f-*mF{)hogJ<N&4=SH4(RLu_a|ef?K|CQEi-^G`~{@V
B|Dpf@

delta 644
zcmYLHyKWOf6rDS>kJ<I^+Ok53aS)*ayCJDUVJj2?0R;$Tpa4M>X~u~yKgjN+o0SU-
zDyT;3?E(op5I?{VAW^T7_yG!PM8Ta65m&lr&YXMZbv574{j4XC<2WQ3lg*DWzDA#t
z4xNFW{|kyE$K1uto+2o+08NH;iMYp<gWR3NP4avzw>rc0G&G1y8U(p-x#DN)9Gc`h
zVWdlr)<o%VqHB){+9PeYzm7+5m<#N+`NK}413V7n*Nyw_z<?`O+O%-)$?ULz!3UDY
zSo{6VymPMx4-UoV?o`Oke0T38_P)eVAc1R;R!!HtIQxQ%fZS0;(;<}ieV@C?sdgJA
z_Y_e~d-K!Eo7~5WV=LOnD%Wi2YK|;G<_Rk3E2?>)q7tIh*Zfo~m5Snv0M+np{Q)dH
zkVhJZl|`gO9q33`mQEg{G1g8W(41H^&Z|0Jf?(%Klbm3SbLXh_`Le%4*K4VVPe^@)
zBECDB$n@4oj^#vUX=t7V+d$3R;2{L&N6=_htc&-iqhmFgP6}7Zk=joqt1>x~1@sFC
zHEA}&OFOoWWiQjj8Y;J?dNdfzo{}%GppdO0kkDZX4OxvQ5JH{RVHLB*(w3Qps~!7E
g(yE1)h5tL2tlXA&$5U}Uk~{WuW%eyjjdLFV1y*~9F8}}l

diff --git a/SourceCode/__pycache__/Mappers.cpython-38.pyc b/SourceCode/__pycache__/Mappers.cpython-38.pyc
index b5770bd567a6b250b10f1868b930e884bcd8e9a1..b6b948892c366c7b2490b4cbcfca4fd58d8a725f 100644
GIT binary patch
delta 466
zcmbQD_FJ7Vl$V!_0SMNnZHap;wUO@uCu82^_nh)!H4G`t*-S-RB@9{2!3;ni3y^04
z<gtKxtU#U{kjDz<u>pA@Kpq>I#}4Er0eS4f3?&>X9L<bPj47PK44Pb3VUrDcBq!T&
zXiSdh5|d{H+FQf~BA7u03y^pTCRjl%w#n68cO*G)F$FtE2|4HIm1O3X=9d<`C+4IU
zWoPD1&gT|&jS_WE&B<|2%_}KNO>qm0_e(4(ElSLZ5)Vr)DoL%tln_K!@8cE`B|f>3
zTUa{3C^H@1G@vX<c=8;6>B;lCg;nH$&SGHUU|?g!ftiYAC+}jH-F%<>0V8Ai<O96o
zjPjE&a{6pm;akWk+5yyE3=V9V8O%kBn{V+eF*44a{8vDlQDw8Fpgkj_%;ZQR1xC5a
nRYE!{3P3$Y+#o^-L`Z-LSr8!)A{0S{%H&f*uB@Uwf*gVXev@+Z

delta 267
zcmeyZK1Gc$l$V!_0SNAWUKaOJawFdZ&dGQ9^clk@^YBYfuH~|uoW-9!*_^vxg5?%d
zuyd54b81eGb822mQEG}?SiFy0#N_kbqLZ5iq$d~g2n&FXV&Pz5V+3NRBB{v-1w<yF
zW9Qs_jOPKPWD(GwVm^><KqV|dC44|E!c-(P`68$6<~w{B7)6^wO3+M^+k8+!iBWbk
zNJ<E#n~4Lc52T-kiHjLT7AbBLvS(zJoa`X1z$iUAQCLSs7O1X>8$^J#6iI*xDG(t8
TBIH1X;^d9OuB-w)f*b+>^dL2)

diff --git a/SourceCode/__pycache__/PostProcess.cpython-38.pyc b/SourceCode/__pycache__/PostProcess.cpython-38.pyc
index e64e72d6d8bb475a6967f80c44e9cc2614458f3c..39a2bc46b43912df6b76bcd69833ef769c977512 100644
GIT binary patch
delta 20
acmezD{Mng1l$V!_0SGRg+`N(dg(3h)-3J5!

delta 20
acmezD{Mng1l$V!_0SHb#T(*(>g(3h)^alq3

diff --git a/SourceCode/__pycache__/Solve.cpython-38.pyc b/SourceCode/__pycache__/Solve.cpython-38.pyc
index e885fdaf0696d3645458e8e481e8191b579d4f98..91308aac5e0ffd2bb217894876f757952cc6f224 100644
GIT binary patch
delta 305
zcmeyMwm_XXl$V!_0SF#+Z;P8Zkyn<nYod0UFk=c+j##dElmsI~3Udle3qzFT#ub7r
zjB6(EV)>ECnj+B5$jDH_p2e|1xQ3~QWg$~7YYA(LAY4?WhP8&JhIt`#EgMi=2re!P
z5(lbg2a585MOjmXp|WBN#DRL)7BVq1OcrFD$UCEkBZWnrp_T(^kJ#k(Z0YPQ*-R4{
zi*8LWWK)=I%Pz`zYjZHW8QbK|yakgxd9^kx^9eFdjunuZY$~9!xk8|raq>q&vB``=
z=8RI4Erq@++!D-9EGfvzFUiSF(krM0v2L+JY372QlF53)8Zr{VuwaoAU=d+tW99%t
Q9%dd!9!3sE4kjKJ09qVR3jhEB

delta 310
zcmZ3W{y~j5l$V!_0SFolHpI0~<dtQto2XqT#GJy?!Vo2v!W7J)$(s0#nSo()AY;&G
zLB=i2jH@OeWBC!!o+Z3Mq=sc7BO^lzYYI;@gcg9(f?%3Gi(`Rk4HHlmQ!Oh{l@MH1
ztcJCQrG|MSb1fTCTo^7c4iX2dW}ht2Hj#H)4Mz%#I72N*3DCC5d)d<2nX{QDFc#gI
zT+gO3*_&OI@y6yvb~Cog?0j02r}GwUHsljznj9;jDSeAQpeR2nCpGsLTX23(S!&TO
z4j^4xl9`{kxlN#$QIMH~k%y6kg@ciUkq?Lk7$<WH*)WPvb{G0OIYwARN(>kzEK&k2
TB8+Uz96-p!%mdQE$iW5x1Wigd

-- 
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