engleski

Entropy dissipative approximations of cross-diffusion models

In this talk we present novel structure-preserving numerical schemes for Shigesada-Kawasaki-Teramoto (SKT) cross diffusion model from population dynamics. The main features of the proposed discretizations are the preservation of the nonnegativity and the entropy-dissipation structure. For the analysis we combine linear multistep discretizations and G-stability theory, investigated for ODEs from 1980s on, and entropy dissipation methods, which have been proposed in recent years. Entropy dissipation techniques were intensively used in the mathematical analysis of PDEs for derivation of apriori estimates which represent a cruical tool in proving the existence of solutions and studing their long-time behaviour. Our aim was to translate the continuous entropy estimates to the discrete level with hope to obtain more accurate and stable approximations. It is shown that G-stability of the one-leg scheme is sufficient to derive discrete entropy dissipation estimates. We prove the existence of semi-discrete weak solutions of proposed one-leg multistep time discretizations. Furthermore, under some assumptions on the evolution operator, the second-order convergence of solutions is proved. We note that our results can be applied to some other nonlinear evolution equations as well. In order to underline the theoretical results, few numerical experiments for the population model will be presented.

Linear multistep methods ; entropy dissipation, diffusion equations ; population dynamics ; quantum drift-diffusion equation ; Derrida-Lebowitz-Speer-Spohn equation ; existence of solutions

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