1D single-species reaction diffusion#

This problem focuses on the following 1D diffusion reaction PDE:

\[\frac{\partial \phi}{\partial t} = D \frac{\partial^2 \phi}{\partial x^2} + k \phi^2 + u(x, t)\]

  • The problem is adapted from this paper

  • D, k, u(x, t) can be provided to the problem constructor (more below)

  • Initial condition: \(\phi(x, 0) = 0\)

  • Default settings:

    • D = 0.01

    • k = 0.01

    • \(u(x, t) = 4 x^2\sin(\pi x) \cos(4 \pi x)\)

  • Domain is \([0,1]\) with homogeneous Dirichlet BC

Mesh#

python3 pressio-demoapps/meshing_scripts/create_full_mesh_for.py \
       --problem diffreac1d -n <N> --outDir <destination-path>

where:

  • N is the number of cells you want

  • <destination-path> is where you want the mesh files to be generated. The script creates the directory if it does not exist.

Notes:#

Important

For this problem, only viscous schemes are applicable. Note also that the stencil cannot be (yet) set because the implementation currently only supports a thee-point stencil that allows a first order viscous flux reconstruction, yielding a second-order scheme. We might relax this in the future if additional scheme are added.

C++ synopsis#

#include "pressiodemoapps/diffusion_reaction1d.hpp"

int main(){
  namespace pda      = pressiodemoapps;
  const auto meshObj = pda::load_cellcentered_uniform_mesh_eigen("path-to-mesh");

  // A. constructor for problem using default values
  {
    const auto probId  = pda::DiffusionReaction1d::ProblemA;
    auto problem = pda::create_problem_eigen(meshObj, probId);
  }

  // B. setting custom coefficients
  {
    using scalar_type = typename decltype(meshObj)::scalar_t;
    const auto diffCoeff = static_cast<scalar_type(0.5);
    const auto reacCoeff = static_cast<scalar_type(0.2);
    auto problem = pda::create_diffusion_reaction_1d_problem_A_eigen(meshObj, diffCoeff, reacCoeff);
  }

  // C. setting custom coefficients and custom source function
  {
    using scalar_type = typename decltype(meshObj)::scalar_t;
    const auto diffCoeff = static_cast<scalar_type(0.5);
    const auto reacCoeff = static_cast<scalar_type(0.2);

    auto mySource = [](const scalar_type & x,
                       const scalar_type & t,
                       scalar_type & result)
    {
      // x, t are the location and time where the source must be evaluated
      result = /* do whatever you want */;
    };

    auto problem = pda::create_diffusion_reaction_1d_problem_A_eigen(meshObj, mySource,
                                                                     diffCoeff, reacCoeff);
  }
}

Python synopsis#

import pressiodemoapps as pda

meshObj = pda.load_cellcentered_uniform_mesh("path-to-mesh")

# A. constructor for problem using default values
probId  = pda.DiffusionReaction1d.ProblemA
problem = pda.create_problem(meshObj, probId)

# B. setting custom coefficients
myD, myK = 0.2, 0.001
problem = pda.create_diffusion_reaction_1d_problem_A(meshObj, myD, myK)

# C. setting custom coefficients and custom source function
myD, myK = 0.55, 0.002
mysource = lambda x, time : np.sin(math.pi*x) *x*x * 4.*np.cos(4.*math.pi*x)
problem = pda.create_diffusion_reaction_1d_problem_A(meshObj, probId, \
                                                     mysource, myD, myK)