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

TL;DR

This paper develops sixth-order WENO finite difference methods tailored for reaction-diffusion systems with sharp traveling wave fronts, demonstrating improved accuracy and efficiency over classical methods.

Contribution

The paper adapts WENO methods for parabolic reaction-diffusion equations, providing a high-order, efficient numerical approach for sharp front solutions.

Findings

WENO methods outperform classical finite difference methods in accuracy

WENO methods enable larger time steps and fewer grid points

Numerical analysis confirms high efficiency of WENO for sharp fronts

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.