AN EFFICIENT LINEAR NUMERICAL SCHEME FOR THE STEFAN PROBLEM, THE POROUS MEDIUM EQUATION AND NON-LINEAR CROSS-DIFFUSION SYSTEMS



Proceedings of EQUADIFF 2017
pp. 305–314

MOTLATSI MOLATI∗ AND HIDEKI MURAKAWA†

∗Department of Mathematics and Computer Science, National University of Lesotho, P.O. Roma, 180, Lesotho (m.molati@nul.ls).

†Faculty of Mathematics, Kyushu University, 744 Motooka, Nishiku, Fukuoka 819-0395,
Japan(murakawa@math.kyushu-u.ac.jp) (Corresponding author).

Abstract

This paper deals with nonlinear diffusion problems which include the Stefan problem,
the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages. Namely, the implementation is very easy and the ensuing linear algebraic systems are symmetric, which show low computational cost. Moreover, this scheme has the accuracy comparable to that of the well-studied nonlinear schemes and make it possible to realize the much faster computation rather than the nonlinear schemes with the same level of accuracy. In this paper, numerical experiments are carried out to demonstrate efficiency of the proposed scheme.