Finite volumes for cross-diffusion systems and discrete entropy methods

par Ansgar Jüngel (TU Vienne)

mardi 9 mars 2021, 14:00 → 15:00 Europe/Paris

Many multi-species applications can be described by cross-diffusion systems,
which are systems of quasilinear parabolic equations. Although the diffusion
matrices are generally neither symmetric and nor positive definite, the
systems often possess an entropy structure. This means that there exists a
change of unknowns such that the transformed diffusion matrix becomes positive
semi-definite and defining a Lyapunov functional (entropy). The resulting
entropy structure yields a priori estimates needed for the existence analysis.
We wish to "translate" this entropy structure to finite-volume discretizations.
The main difficulty is to adapt the nonlinear chain rule to the discrete level.

In this talk, we present two strategies to define a discrete chain rule,
assuming either that the total entropy is the sum of individual entropies
or that the entropy describes volume-filling models. Both strategies use
suitable mean formulas, based on the mean-value theorem and the convexity of
the entropy functional. We prove the existence of finite-volume solutions and
the convergence of the numerical scheme. Our assumptions include the Shigesada-
Kawasaki-Teramoto population model and the vapor-deposition model for solar
panels of Bakhta and Ehrlacher.