  Structure-preserving finite-difference schemes for general nonlinear
fourth-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 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.
