Geodesic
A linear scheme to approximate nonlinear cross-diffusion systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1141-1161

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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.

DOI : 10.1051/m2an/2011010
Classification : 35K55, 35K57, 65M12, 92D25
Keywords: cross-diffusion systems, nonlinear diffusion, discrete-time schemes, numerical schemes, reaction-diffusion system approximations 
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     author = {Murakawa, Hideki},
     title = {A linear scheme to approximate nonlinear cross-diffusion systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1141--1161},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {6},
     year = {2011},
     doi = {10.1051/m2an/2011010},
     mrnumber = {2833176},
     zbl = {1269.65090},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011010/}
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Murakawa, Hideki. A linear scheme to approximate nonlinear cross-diffusion systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1141-1161. doi: 10.1051/m2an/2011010

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