A unified Imex Runge-Kutta approach for hyperbolic systems with multiscale relaxation

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior, which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation lose their efficiency, and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable of capturing the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis.