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Structure-preserving finite-difference schemes for general nonlinearfourth-order 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 associatedto the logarithmic entropy also preserves the positivity. The idea of thederivation is to reformulate the equations in such a way that the chain rule isavoided. A central finite-difference discretization is then applied to thereformulation. In this way, the same dissipation rates as in the continuouscase are recovered. The strategy can be extended to a multi-dimensionalthin-film equation. Numerical examples in one and two space dimensionsillustrate the dissipation properties.

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations