IMEX Runge-Kutta schemes for hyperbolic systems with diffusive relaxation

The purpose of the talk is to give a review on effective methods for the numerical solution of hyperbolic systems with parabolic relaxation. We consider mathematical models described by a set of hyperbolic equations with relaxation. As the relaxation parameter vanishes, the characteristic speeds of the system diverge, and the system reduces to a parabolic-type equation (typically a convection-diffusion equation). Several numerical schemes have been proposed in the literature, that are able to capture the limiting behavior, in spite of the divergence of the characteristic speeds. Such methods are usually based on an implicit treatment of the relaxation term and of the divergent part of the flux derivative. In general the approach leads to an explicit scheme for the underlying limit parabolic equation, thus suffering from the standard parabolic CFL restriction t / x2. We present two approaches to treat the problem. The first one, that we call partitioned, is very close to what is already available in the literature. We show that implicit-explicit Runge-Kutta methods allow the construction of high order schemes for the numerical solution of such problems. With this approach, classical implementation of IMEX-RK would lead to an explicit scheme for the limit equation. In order to overcome this difficulty, a penalization method is proposed, which consists in adding two opposite terms, and to treat one of them explicitly and one implicitly. A second approach, that we call additive is based on an explicit treatment of the flux. Because of the divergence of the characteristic speeds, an additional condition has to be satisfied by the IMEX-RK to reach consistency in the relaxation limit. The parabolic restriction can be overcome also in this case with the same penalization technique. A numerical example is presented that shows the effectiveness of both approaches.