Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation

Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve, and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.