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A. Jüngel:
"Discrete entropy-dissipation methods: finite-volume and higher-order in time approximations of nonlinear diffusive equations";
Keynote Lecture: Asymptotic behaviour of systems of PDE arising in physics and biology: theoretical and numerical points of view, Lille (invited); 2013-11-06 - 2013-11-08.

We present numerical discretizations which preserve the entropy structure of the analyzed
nonlinear diffusive equations. More precisely, we develop numerical schemes for which
the discrete entropy is stable or even dissipating. The key idea is to "translate" entropy-
dissipation methods to the discrete case. We consider two situations.
First, an implicit Euler finite-volume approximation of porous-medium or fast-diffusion
equations is investigated. The scheme dissipates all zeroth-order entropies which are
dissipated by the continuous equation. The proof is based on novel discrete generalized
Beckner inequalities. Furthermore, the exponential decay of some first-order entropies is
proved using systematic integration by parts and a convexity property with respect to
the time step parameter.
Second, new one-leg multistep time approximations of general nonlinear evolution
equations are investigated. These schemes preserve both the nonnegativity and the
entropy-dissipation structure of the equations. The key idea is to combine Dahlquist´s G-
stability theory with entropy-dissipation methods. The optimal second-order convergence
rate is proved under a certain monotonicity assumption on the operator. The discretiza-
tion is applied to a cross-diffusion system from population dynamics and a fourth-order
quantum diffusion equation.

Siehe englisches Abstract.

Entropy methods; numerical analysis