An efficient linear scheme to approximate nonlinear diffusion problems

Abstract

This paper deals with nonlinear diffusion problems including the Stefan problem, the porous medium equation and cross-diffusion systems. A linear discrete-time scheme was proposed by Berger, Brezis and Rogers [RAIRO Anal. Numer. 13 (1979) 297–312] for degenerate parabolic equations and was extended to cross-diffusion systems by Murakawa [Math. Mod. Numer. Anal. 45 (2011) 1141–1161]. There is a constant stability parameter \(\mu \) in the linear scheme. In this paper, we propose a linear discrete-time scheme replacing the constant \(\mu \) with given functions depending on time, space and species. After discretizing the scheme in space, we obtain an easy-to-implement numerical method for the nonlinear diffusion problems. Convergence rates of the proposed discrete-time scheme with respect to the time increment are analyzed theoretically. These rates are the same as in the case where \(\mu \) is constant. However, actual errors in numerical computation become significantly smaller if varying \(\mu \) is employed. Our scheme has many advantages even though it is very easy-to-implement, e.g., the ensuing linear algebraic systems are symmetric, it requires low computational cost, the accuracy is comparable to that of the well-studied nonlinear schemes, the computation is much faster than the nonlinear schemes to obtain the same level of accuracy.