A time discretization of the abstract Cauchy parabolic problem with diffusion operator defined on some Sobolev space is investigated. The authors are interested in entropy-dissipating in time semidiscrete Runge-Kutta schemes from the stability point of view. They extend previous results for the schemes of order \(p\geq1\) for general diffusion equations. First they determine an abstract condition under which the discrete entropy-dissipation inequality holds. The specification of the obtained abstract conditions is derived for a quasilinear diffusion equation, porous media or fast diffusion equations, a linear system and the fourth-order Derrida-Lebowitz-Speer-Spohn equation.
Moreover, the concavity property which is a consequence of the concavity of the difference of the entropies at two time consecutive steps is related to the Bakry-Emery approach and the geodesic convexity of the entropy. Finally, numerical experiments for the one-dimensional porous medium equation using various Runge-Kutta finite difference discretizations show that the entropy-dissipation property is global.