Relaxed schemes for nonlinear evolutionary PDEs

In this work we present a class of relaxed schemes for nonlinear convection diffusion problems, which can tackle also the cases of degenerate diffusion and of convection dominated regimes. These schemes can achieve any order of accuracy and they give non-oscillatory solutions even in the presence of singularities. "Relaxation approximations to non-linear PDE's are based on the replacement of the original PDE with a semi-linear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ϵ. When ϵ→0, the system reduces to the original PDE. A consistent discretization of the relaxation system for ϵ=0 yields a consistent discretization of the original PDE. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term, still enjoying the robustness of upwind discretizations. We also present a numerical test for a strongly degenerate convection diffusion equation.