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A. Jüngel, S. Schuchnigg:
"Entropy- dissipating semi-discrete Runge-Kutta schemes for non-linear diffusion schemes";
in: "ASC Report 26/2015", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-08-7, 1 - 29.

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equa-tion. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

Entropy-dissipative numerical schemes, Runge-Kutta schemes, entropy method,

http://www.asc.tuwien.ac.at/preprint/2015/asc26x2015.pdf