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

This paper is concerned with nonlinear and linear discrete-time algorithms for cross-diffusion systems. The nonlinear scheme corresponds to backward differences in time, and the linear algorithm is a very-easy-to-implement scheme proposed by the author [ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 1141--1161]. The main purpose of this paper is to derive convergence rates of the discrete-time schemes. We obtain the same orders for both the nonlinear and the linear scheme. Moreover, these orders are optimal. We also establish uniqueness and regularity results of weak solutions of the cross-diffusion systems.