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M. Braukhoff, A. Jüngel:
"Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations";
in: "ASC Report 1/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 22.

Structure-preserving finite-difference schemes for general nonlinear fourthorder parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida{Lebowitz{Speer{Spohn equations. The schemes conserve the
mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a
way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation.
Numerical examples in one and two space dimensions illustrate the dissipation properties.

Entropy, finitee differences, thin-film equation, DLSS equation, discrete chain rule, denoing

http://www.asc.tuwien.ac.at/preprint/2020/asc01x2020.pdf