A fast solver for systems of reaction-diffusion equations

TL;DR

The paper develops a fast, stable solver for reaction-diffusion systems with stiff chemistry by combining the method of characteristics for advection with a postprocessing, Fourier-based filter to stabilize the explicit diffusion step. The main approach uses backward Euler time stepping for diffusion, a low-frequency shift and the $2\pi$-extension, and an eighth-order Gottlieb-type filter with a tunable stretch factor $\kappa$, enabling larger time steps without forming full Jacobians. Key contributions include a practical, parallelizable stabilization technique, extension to 1D and 2D domains including domain decomposition, and demonstration on heat and predator–prey models with robust performance under periodic boundary forcing. The method offers a simple, scalable option for large-scale environmental RD models where fast, explicit diffusion is desirable.

Abstract

In this paper we present a fast algorithm for the numerical solution of systems of reaction-diffusion equations, $\partial_t u + a \cdot \nabla u = Δu + F (x, t, u)$, $x \in Ω\subset \mathbf{R}^3$, $t > 0$. Here, $u$ is a vector-valued function, $u \equiv u(x, t) \in \mathbf{R}^m$, $m$ is large, and the corresponding system of ODEs, $\partial_t u = F(x, t, u)$, is stiff. Typical examples arise in air pollution studies, where $a$ is the given wind field and the nonlinear function $F$ models the atmospheric chemistry.

Figures (3)

• Figure 1: Accuracy of the stabilized explicit scheme for the heat equation. The horizontal coordinate is $3\Delta t/h^2$. * : first-order shift; o : third-order shift.
• Figure 2: Accuracy of the stabilized explicit scheme for the predator-prey system. The horizontal coordinate is $3 \Delta t/h^2$. * : first-order shift; o : third-order shift.
• Figure 3: Accuracy of the stabilized explicit scheme applied to the heat equation with overlapping subdomains.