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A linear scheme to approximate nonlinear cross-diffusion systems*

Published online by Cambridge University Press:  04 July 2011

Hideki Murakawa
Hideki Murakawa*
Affiliation:
Faculty of Mathematics, Kyushu University, 744 Motooka, Nishiku, Fukuoka, 819-0395 Japan. murakawa@math.kyushu-u.ac.jp

Abstract

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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.

Keywords

Cross-diffusion systemsnonlinear diffusiondiscrete-time schemesnumerical schemesReaction-diffusion system approximations

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 6 , November 2011 , pp. 1141 - 1161
Copyright
© EDP Sciences, SMAI, 2011

References

Barrett, J.W. and Blowey, J.F., Finite element approximation of a nonlinear cross-diffusion population model. Numer. Math. 98 (2004) 195–221. CrossRef
Beckett, G., Mackenzie, J.A. and Robertson, M.L., A moving mesh finite element method for the solution of two-dimensional Stefan problems. J. Comp. Phys. 168 (2001) 500–518. CrossRef
Berger, A.E., Brezis, H. and Rogers, J.C.W., A numerical method for solving the problem ut-Δf(u) = 0. RAIRO Anal. Numer. 13 (1979) 297–312. CrossRef
H. Brézis, Analyse Fonctionnelle. Masson (1983).
Chen, L. and Jüngel, A., Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal. 36 (2004) 301–322. CrossRef
Chen, L. and Jüngel, A., Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differ. Equ. 224 (2006) 39–59. CrossRef
Galiano, G., Garzón, M.L. and Jüngel, A., Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. Rev. R. Acad. Cien. Ser. A Mat. 95 (2001) 281–295.
Galiano, G., Garzón, M.L. and Jüngel, A., Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model. Numer. Math. 93 (2003) 655–673. CrossRef
M.E. Gurtin, Some mathematical models for population dynamics that lead to segregation. Quart. Appl. Math. 32 (1974) 1–9.
Jäger, W. and Kačur, J., Solution of porous medium type systems by linear approximation schemes. Numer. Math. 60 (1991) 407–427. CrossRef
Kačur, J., Handlovičová, A. and Kačurová, M., Solution of nonlinear diffusion problems by linear approximation schemes. SIAM J. Numer. Anal. 30 (1993) 1703–1722.
T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms. J. Math. Anal. Appl. 323 (2006) 1387–1401.
E.H. Kerner, Further considerations on the statistical mechanics of biological associations. Bull. Math. Biophys. 21 (1959) 217–255.
Magenes, E., Nochetto, R.H. and Verdi, C., Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. Math. Mod. Numer. Anal. 21 (1987) 655–678. CrossRef
Mimura, M. and Kawasaki, K., Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9 (1980) 49–64. CrossRef
Murakawa, H., Reaction-diffusion system approximation to degenerate parabolic systems. Nonlinearity 20 (2007) 2319–2332. CrossRef
Murakawa, H., A relation between cross-diffusion and reaction-diffusion. Discrete Contin. Dyn. Syst. S 5 (2012) 147–158. CrossRef
Nochetto, R.H. and Verdi, C., An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation. Math. Comput. 51 (1988) 27–53. CrossRef
Nochetto, R.H. and Verdi, C., The combined use of a nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems. Numer. Funct. Anal. Optim. 9 (1988) 1177–1192. CrossRef
Nochetto, R.H., Paolini, M. and Verdi, C., An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part I: stability and error estimates. Math. Comput. 57 (1991) 73–108.
Nochetto, R.H., Paolini, M. and Verdi, C., A fully discrete adaptive nonlinear Chernoff formula. SIAM J. Numer. Anal. 30 (1993) 991–1014. CrossRef
Nochetto, R.H., Schmidt, A. and Verdi, C., A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69 (1999) 1–24. CrossRef
Pang, P.Y.H. and Wang, M.X., Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200 (2004) 245–273. CrossRef
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species. J. Theor. Biol. 79 (1979) 83–99.
R. Temam, Navier-Stokes equation theory and numerical analysis. AMS Chelsea Publishing, Providence, RI (2001).
Verdi, C., Numerical aspects of parabolic free boundary and hysteresis problems. Lecture Notes in Mathematics 1584 (1994) 213–284.