A review of some numerical methods for reaction-diffusion equations

Abstract

Some fixed-node finite-difference schemes and a finite element method are applied to a reaction-diffusion equation which has an exact traveling wave solution. The accuracy of the methods is assessed in terms of the computed steady state wave speed which is compared with the exact speed. The finite element method uses a semi-discrete Galerkin approximation. The finite-difference schemes discussed in this review include two explicit algorithms, three methods of lines, two implicit procedures, two majorant operator-splitting techniques, four time-linearization schemes and the Crank-Nicolson method. The effects of the truncation errors and linearization on the computed wave speed are determined. The application of these techniques to reaction-diffusion equations appearing in combustion theory is also discussed. The review is limited to fixed-node techniques and does not include moving or adaptive finite-difference and adaptive finite element methods.