A linear scheme to approximate nonlinear cross-diffusion systems

Abstract

Abstract. 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.Mathematics Subject Classification. 35K55, 35K57, 65M12, 92D25.

Error Estimates for Discrete-Time Approximations of Nonlinear Cross-Diffusion Systems

“…Problem (1.1) will be understood in the sense of the following weak form. The existence of a weak solution of (1.1) under assumptions (H1) and (H2) has been proved by the author [15]. We prove the uniqueness of the weak solution in the following section.…”

“…The following notation is used: and so on. In [15], we imposed the following assumption: There exists a positive constant a such that…”

“…An extended problem of (1.2) and its relaxed system have been studied by Bendahmane et al [4] and Lepoutre, Pierre, and Rolland [13]. The following cooperative cross-diffusion system has been treated in [15]:…”

“…Fully implicit schemes are unfriendly toward mathematicians and students who want to see numerical solutions but do not want to spend a lot of time and effort on programming. The author [15] provided and analyzed a versatile, easy-to-implement, and stable numerical method for the general cross-diffusion system (1.1). The scheme is as follows.…”

“…When you want to consider another nonlinearity, all you have to do is to rewrite the function β. Indeed, we carried out numerical simulations for (1.2), (1.5), and (1.6) (see [15,19]) in which we used the same code and just rewrote the definitions of β and f . The scheme (1.8) can be regarded as an extension of a linear scheme proposed by Berger, Brezis, and Rogers [5] for the degenerate parabolic equations (1.1) with M = 1.…”