A numerical study of WENO approximations to sharp propagating fronts for reaction-diffusion systems

Abstract

Many reaction-diffusion systems exhibit traveling wave solutions that evolve on multiple spatiotemporal scales, where obtaining fast and accurate numerical solutions is challenging. In this work, we employ sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve the reaction-diffusion system for the traveling wave solution with the sharp fronts. It is shown that those WENO methods achieve the expected sixth-order accuracy in the Fisher's, Zeldovich, bistable equations, and the Lotka-Volterra competition-diffusion system. However, we find that the WENO methods converge very slowly in the Newell-Whitehead-Segel equation because of the speed issue, in which one possible way to match the exact speed is to coarsen the spatial grid and decrease the time step simultaneously. It is also seen that the central WENO method could carry the larger time step while preserving the essentially non-oscillatory behavior for the approximations.