IMEX Runge-Kutta schemes and hyperbolic systems of conservation laws with stiff diffusive relaxation

Hyperbolic system of conservation laws often have relaxation terms that, under a suitable scaling, lead to a reduced system of parabolic or hyperbolic type. The development of numerical methods to solve systems of this form his an active area of research. These systems in addition to the stiff relaxation term have the convection term stiff too. In this paper we will mainly concentrate on the study of the stiff regime. In fact in this stiff regime most of the popular methods for the solution of these system fail to capture the correct behavior of the relaxation limit unless the small relaxation rate is numericaly resolved. We will show how to overcome this difficulties and how to construct numerical schemes with the correct asymnptotic limit, i.e., the correct zero-relaxation limit should be preserved at a discrete level.