High order relaxation schemes for non linear diffusion problems

Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems. The present paper focuses onto diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration. Error estimates and convergence analysis are developed for semidiscrete schemes with numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two- and three-dimensional cases.