@@ -9,7 +9,7 @@ We provide some *Julia* codes and Jupyter notebooks to reproduce the simulations

### Julia codes

In folder [simulations](simulations/) there are:

* the implementations of all the Patankar and modified Patankar schemes ([Patankar Runge--Kutta](simulations/mPRK.jl) and [modified Patankar Deferred Correction](simulations/mPDeC.jl)) and some auxiliary functions ([Lagrange polynomials file](simulations/Lagrange_polynomials.jl))

* the study of the general **linear $2\times 2$ system** searching for the maximum $\Delta t$ bound for which we do not observe oscillations for all the schemes is in the [CFL notebook](simulations/SearchingCFLconditions.ipynb)

* the study of the general **linear $`2\times 2`$ system** searching for the maximum $`\Delta t`$ bound for which we do not observe oscillations for all the schemes is in the [CFL notebook](simulations/SearchingCFLconditions.ipynb)

* the study of the spurious steady state for all the methods is in the [Inconsistency notebook](simulations/Inconsistency.ipynb)

* the test of the stability bound previously found on **scalar nonlinear problem** is in [Nonlinear Problem notebook](simulations/Nonlinear_Problems.ipynb)

* the test of the inconsistency on **Robertson problem** is in [Robertson notebook](simulations/Robertson_Problem.ipynb)

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@@ -17,11 +17,11 @@ In folder [simulations](simulations/) there are:

### Mathematica notebooks

In folder [symbolic_computations](simbolic_computations/) contains:

* the computations of the $\Delta t$ bound on the general **linear $2\times 2$ system** for the **MPRK(2,2,1)** scheme is in the [MPRK(2,2,1) notebook](simbolic_computations/MPRK_2_2_1_generalSystem.nb)

* the computations of the $\Delta t$ bound and the Taylor expansion to understand which scheme has inconsistency on the **symmetric linear $2\times 2$ system** for some of the **MPRK(2,2,$\alpha$)** schemes is in the [MPRK(2,2,$\alpha$) notebook](simbolic_computations/MPRK_2_2_alpha.nb)

* the Taylor expansion to understand if **MPRK(3,2)** has inconsistency on the **symmetric linear $2\times 2$ system** is in the [MPRK(3,2) notebook](simbolic_computations/MPRK_3_2.nb)

* the Taylor expansion to understand if **mPDeC2** and **mPDeC3** have inconsistencies on the **symmetric linear $2\times 2$ system** is in the [MPDeC notebook](simbolic_computations/MPDeC.nb)

In folder [symbolic_computations](symbolic_computations/) contains:

* the computations of the $`\Delta t`$ bound on the general **linear $`2\times 2`$ system** for the **MPRK(2,2,1)** scheme is in the [MPRK(2,2,1) notebook](symbolic_computations/MPRK_2_2_1_generalSystem.nb)

* the computations of the $`\Delta t`$ bound and the Taylor expansion to understand which scheme has inconsistency on the **symmetric linear $`2\times 2`$ system** for some of the **MPRK(2,2,$`\alpha`$)** schemes is in the [MPRK(2,2,$\alpha$) notebook](symbolic_computations/MPRK_2_2_alpha.nb)

* the Taylor expansion to understand if **MPRK(3,2)** has inconsistency on the **symmetric linear $`2\times 2`$ system** is in the [MPRK(3,2) notebook](symbolic_computations/MPRK_3_2.nb)

* the Taylor expansion to understand if **mPDeC2** and **mPDeC3** have inconsistencies on the **symmetric linear $`2\times 2`$ system** is in the [MPDeC notebook](symbolic_computations/MPDeC.nb)

### Supplementary material

In folder [supplementary_material](supplementary_material/) there is the [file](supplementary_material/supplementary_material.pdf) with the supplementary results which are only described in the report