An efficient linear scheme to approximate nonlinear diffusion problems

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. Numér.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 μ in the linear scheme. In this paper, we propose a linear discrete-time scheme replacing the constant μ 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 μ is constant. However, actual errors in numerical computation become significantly smaller if varying μ 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.