Implicit-Explicit methods for hyperbolic systems with hyperbolic and parabolic relaxation

In this talk we discuss the problem of constructing effective high order methods for the numerical solution of hyperbolic systems of balance laws, in presence of stiff source. Because of the stiffness, the use of implicit integrators is advisable, so that no restrictions on the time step due to small relaxation time will appear. Two different relaxation systems will be considered, namely hyperbolic and parabolic relaxation. Because of the different nature of the problems, the two cases will be considered separately. A common denominator of both treatments is the choice of space discretization. Most schemes for conservation or balance laws are discretized by finite volume (FV), conservative finite difference (FD), or discontinuous Galerkin (DG). Here we choose conservative finite difference since it is probably the simplest general approach for the construction of high order shock capturing schemes for such problems.