Relaxation schemes for partial differential equations and applications to degenerate diffusion problems

In the recent years several relaxation approximations to partial differential equations of various types have been proposed, from the classical kinetic schemes for gas dynamics to the relaxation schemes for conservation laws and Hamilton-Jacobi equations and the diffusive relaxation schemes for convection-diffusion problems. In the present paper we review some basic results on numerical methods derived from relaxation approximations of differential equations and present some recent developments and open problems related to their extension to second- and fourth-order degenerate diffusion equations. We present a class of numerical schemes for the approximation of second-order degenerate diffusion (or convection-diffusion) equations. The schemes are based on a suitable relaxation approximation that permits to reduce the second-order diffusion equations to first-order semilinear hyperbolic systems with stiff terms. In a similar way we show how fourth-order diffusion equations can be approximated by second- or first-order relaxation systems with stiff terms. The numerical passage from the relaxation system to the nonlinear diffusion equation is realized by suitable semi-implicit or fully implicit time discretizations combined with upwind and central diffences in space. Applications to porous-media equations and thin-film equations in one and two space dimensions are presented.