2D Gray Scott reaction-diffusion#

This problem focuses on the following 2D reaction-diffusion system of PDE:

\[ \begin{align}\begin{aligned}\frac{\partial A}{\partial t} &= D_a \left(\frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2}\right) - AB^2 + F(1-A)\\\frac{\partial B}{\partial t} &= D_b \left(\frac{\partial^2 B}{\partial x^2} + \frac{\partial^2 B}{\partial y^2}\right) + AB^2 - (F+K)B\end{aligned}\end{align} \]

  • \(D_a, D_b, F, K\) can be provided to the constructor (more below)

  • Initial conditions:

    \(\left\{\begin{matrix}A=1/2, B=1/4; & \text{if }|x| < 1/10 \text{ and } |y|< 1/10 \\ A=1, B=0; & \text{otherwise} \end{matrix}\right.\)

  • Default settings:

    • \(D_a = 2\cdot 10^{-4}, D_b = D_a/4\)

    • \(F=0.042, K=0.062\)

  • Domain is \([-5/4,5/4]^2\) with periodic BCs

  • For more details on the problem, see the following references: link1, link2.

Mesh#

python3 pressio-demoapps/meshing_scripts/create_full_mesh_for.py \
        --problem grayscott2d -n Nx Ny --outDir <destination-path>

where Nx, Ny is the number of cells you want along \(x\) and \(y\) respectively, and <destination-path> is where you want the mesh files to be generated.

C++ synopsis#

#include "pressiodemoapps/diffusion_reaction2d.hpp"

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

  // A. constructor for problem using default values
  {
    const auto probId  = pda::DiffusionReaction2d::GrayScott;
    auto problem = pda::create_problem_eigen(meshObj, probId, scheme);
  }

  // B. setting custom coefficients
  {
    using scalar_type = typename decltype(meshObj)::scalar_t;
    const scalar_type Da  = /*some_value*/;
    const scalar_type Db  = /*some_value*/;
    const scalar_type F   = /*some_value*/;
    const scalar_type K   = /*some_value*/;
    auto problem      = pda::create_gray_scott_2d_problem(meshObj, scheme, Da, Db, F, K);
  }
}

Python synopsis#

import pressiodemoapps as pda

scheme  = pda.ViscousFluxReconstruction.FirstOrder

# A. constructor for problem using default values
probId  = pda.DiffusionReaction2d.GrayScott
problem = pda.create_problem(meshObj, probId, scheme)

# B. setting custom coefficients
Da, Db, F, K = ...
problem = pda.create_gray_scott_2d_problem(meshObj, scheme, Da, Db, F, K)

Notes:#

Important

Note that this problem does not have advection, so inviscid schemes are not applicable but only viscous schemes are. Currently, we only support a first order viscous flux reconstruction, which leads to a second-order scheme.

Sample Plot#

Representative plots at selected time \(t=1000\) obtained using time step \(dt=0.5\), Runge-Kutta4 integration, a mesh of 160x160 and default values for \(D_a, D_b, F, K\).

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