This page describes the current functionality of the reaction-diffusion module. The source code can be found in src/rd/mod_rd_phys.t.

Physics

Systems of \( s \) reaction-diffusion equations are of the form

\[ \frac{\partial u_i}{\partial t} = D_i \nabla^2 u_i + f_i(u_1, \ldots, u_s) \qquad i \in \{1,\ldots, s\} \]

where \( D_i \) are the diffusion coefficients and \( f_i \) the reaction functions. Several of these systems are available in MPI-AMRVAC, the systems of interest are given below. The prefixes relate to the names given to the systems in the code, to be supplied in the .par file or prefixing the variables. Some systems have a good starting resource linked to them.

lg = logistic: Fisher-KPP or the diffusive logistic equation

\[ u_t = D \nabla^2 u + \lambda u(1-u) \]
gs = gray-scott: Gray-Scott

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u - uv^2 + F(1-u) \\ v_t &=& D_2 \nabla^2 v + uv^2 - (F+k)v \end{eqnarray*}
br = brusselator: Brusselator

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u + u^2 v + A - (B+1) u \\ v_t &=& D_2 \nabla^2 v - u^2 v + Bu \end{eqnarray*}
sb = schnakenberg: Schnakenberg

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u + \gamma (\alpha + u^2 v - u) \\ v_t &=& D_2 \nabla^2 v + \gamma (\beta - u^2 v) \end{eqnarray*}
ebr = ext-brusselator: Extended Brusselator (where \( f(u,v) \) and \( g(u,v) \) are the reaction functions in the Brusselator model (see above))

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u + f(u,v) - Cu + Dw \\ v_t &=& D_2 \nabla^2 v + g(u,v) \\ w_t &=& D_3 \nabla^2 w + Cu - Dw \end{eqnarray*}
bzfn = belousov_fieldnoyes: Oregonator (Field-Noyes model of the Belousov-Zhabotinski reaction)

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u + \frac{1}{\epsilon} (\lambda u - u w + u - u^2) \\ v_t &=& D_2 \nabla^2 v + u - v \\ w_t &=& D_3 \nabla^2 w + \frac{1}{\delta} (-\lambda w - u w + \mu v)) \end{eqnarray*}
lor = lorenz: The Lorenz system

\begin{eqnarray*} u_t &=& D_1 \nabla^2 u + \sigma (v-u) \\ v_t &=& D_2 \nabla^2 v + u(r - w) - v \\ w_t &=& D_3 \nabla^2 w + uv - bw \end{eqnarray*}

Numerics

The reaction-diffusion module doesn't use advection, hence none of the flux schemes or slope limiters are applicable to this module. The spatially discretized diffusion and the local reactions are added as source terms. For the moment, diffusion is discretized with a second order centered finite difference scheme. There is a possibility to use IMEX schemes, where the diffusion part is handled implicitly and reaction explicitly. The implicit system is then solved using the AMR-compatible multigrid solver coupled to MPI-AMRVAC.

<!-- J. Teunissen, R. Keppens,
A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC,
Computer Physics Communications,
Volume 245,
2019,
106866,
ISSN 0010-4655,
https://doi.org/10.1016/j.cpc.2019.106866. -->

Practical usage

A number of examples can be found in tests/rd/. The namelist for the reaction-diffusion module is as follows:

&rd_list
    D1, D2, D3,                  ! Diffusion coefficients
    sb_alpha, sb_beta, sb_kappa, ! Parameters for Schnakenberg
    gs_F, gs_k,                  ! Parameters for Gray-Scott
    br_A, br_B,                  ! Parameters for Brusselator
    br_C, br_D,                  ! Parameters for extended Brusselator
    lg_lambda,                   ! Parameter for Fisher-KPP equation
    bzfn_epsilon, bzfn_delta, bzfn_lambda, bzfn_mu, ! Parameters for Oregonator model
    lor_r, lor_sigma, lor_b,     ! Parameters for Lorenz system
    equation_name,               ! Name of the system to simulate
    dtreacpar,                   ! Timestep restriction parameter
    rd_particles, rd_source_split 
/