A linear scheme to approximate nonlinear cross-diffusion systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, p. 1141-1161

This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer. 13 (1979) 297-312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile, easy to implement and efficient numerical scheme for the cross-diffusion systems. Numerical experiments are carried out to demonstrate the effectiveness of the proposed scheme.

DOI : https://doi.org/10.1051/m2an/2011010
Classification:  35K55,  35K57,  65M12,  92D25
Keywords: cross-diffusion systems, nonlinear diffusion, discrete-time schemes, numerical schemes, reaction-diffusion system approximations
@article{M2AN_2011__45_6_1141_0,
     author = {Murakawa, Hideki},
     title = {A linear scheme to approximate nonlinear cross-diffusion systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {6},
     year = {2011},
     pages = {1141-1161},
     doi = {10.1051/m2an/2011010},
     zbl = {1269.65090},
     mrnumber = {2833176},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_6_1141_0}
}
Murakawa, Hideki. A linear scheme to approximate nonlinear cross-diffusion systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 6, pp. 1141-1161. doi : 10.1051/m2an/2011010. http://www.numdam.org/item/M2AN_2011__45_6_1141_0/

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