Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher's equation

Abstract

In this paper, a discrete singular convolution (DSC) algorithm is introduced to solve Fisher's equation, to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem. Two novel numerical treatments, a moving frame scheme and a new asymptotic scheme, are designed to overcome the subtle difficulties involved in the numerical solution of Fisher's equation. The moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem. The new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long-time integrations. The resulting DSC algorithm is able to correctly predict long-time traveling wave behavior. The performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches, i.e., methods of the Fourier pseudospectral, the accurate spatial derivatives, and the Crank-Nicolson schemes. The spatial and temporal accuracies of all four methods are analyzed and numerically verified.