A Numerical Study of WENO Approximations to Sharp Propagating Fronts for Reaction-Diffusion Systems

TL;DR

This paper tackles accurate numerical propagation of sharp traveling fronts in one-dimensional reaction-diffusion systems by applying sixth-order WENO schemes designed for parabolic terms (including CWENO, MWENO, and WENO-LSZ) and comparing them to centered finite differences across models such as Fisher, Zeldovich, Newell–Whitehead–Segel, bistable, and Lotka–Volterra. The central finding is that CWENO provides stable and accurate traveling-front solutions on coarser grids and with larger time steps, while speed accuracy exhibits a nontrivial dependence on the grid size $N$ and CFL number, with an observed optimal pairing for matching the front speed $c$ in the Newell–Whitehead–Segel case. The work identifies a theoretical speed floor $c_{\min}=2\sqrt{\rho D}$ and demonstrates that, for certain nonlinearities, reducing the CFL number does not guarantee speed convergence, highlighting a practical consideration for RD simulations with sharp fronts. Overall, CWENO is recommended for sharp-front reaction-diffusion problems, and the study points to future work on systematically locating optimal time-step choices to reproduce correct wave speeds.

Abstract

Many reaction-diffusion systems in various applications exhibit traveling wave solutions that evolve on multiple spatio-temporal scales. These traveling wave solutions are crucial for understanding the underlying dynamics of the system. In this work, we present sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve reaction-diffusion systems. The WENO method allows us to use fewer grid points and larger time steps compared to classical finite difference methods. Our focus is on solving the reaction-diffusion system for the traveling wave solution with the sharp front. Although the WENO method is popular for hyperbolic conservation laws, especially for problems with discontinuity, it can be adapted for the equations of parabolic type, such as reaction-diffusion systems, to effectively handle sharp wave fronts. Thus, we employed the WENO methods specifically developed for equations of parabolic type. We considered various reaction-diffusion equations, including Fisher's, Zeldovich, Newell-Whitehead-Segel, bistable equations, and the Lotka-Volterra competition-diffusion system, all of which yield traveling wave solutions with sharp wave fronts. Numerical examples in this work demonstrate that the central WENO method is highly more accurate and efficient than the commonly used finite difference method. We also provide an analysis related to the numerical speed of the sharp propagating front in the Newell-Whitehead-Segel equation. The overall results confirm that the central WENO method is highly efficient and is recommended for solving reaction-diffusion equations with sharp wave fronts.

Figures (18)

• Figure 1: Exact solutions to the Fisher's equation (top left), Zeldovich equation (top right), Newell–Whitehead–Segel equation with $\alpha=2$ (bottom left) and bistable equation with $\beta=0.2$ (bottom right) at $t=0$ for different values of $\rho$.
• Figure 2: Exact solutions \ref{['eq:nws_exact']} to the Newell–Whitehead–Segel equation ($\rho=50$) at $t=0$ for different values of $\alpha$.
• Figure 3: Exact solutions \ref{['eq:bistable_exact']} to the bistable equation ($\rho=50$) at $t=0$ for different values of $\beta$.
• Figure 4: Exact solutions $u$ (left) and $v$ (right) to Lotka-Volterra competition-diffusion system \ref{['eq:lotka_volterra']} at $t=0$ for different values of $\rho$.
• Figure 5: The numerical flux $\hat{g}_{i+1/2}$ is constructed on the stencil $S^6 = \{ x_{i-2}, \cdots, x_{i+3} \}$ with six uniform points, as well as three $4$-point substencils $S_0, S_1, S_2$.

Theorems & Definitions (5)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5