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cross-diffusion systems

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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A linear scheme to approximate nonlinear cross-diffusion systems *
Hideki Murakawa
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 45 / Issue 6 / November 2011

Published online by Cambridge University Press:

04 July 2011, pp. 1141-1161

Print publication:

November 2011
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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.