Flux-Explicit IMEX Runge--Kutta Schemes for Hyperbolic to Parabolic Relaxation Problems

We consider the development of Implicit-Explicit (IMEX) Runge--Kutta (R-K) schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. The asymptotic behavior of such systems as the relaxation parameter vanishes is governed by a reduced parabolic system, and it is desirable to have schemes that are able to capture the correct diffusive limit. The hyperbolic part becomes stiff when the system relaxes towards the parabolic equation. For this reason, in previous works part of the hyperbolic terms are treated implicitly [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439], [S. Jin and L. Pareschi, J. Comp. Phys., 161 (2000), pp. 312--330], [G. Naldi and L. Pareschi, SIAM J. Numer. Anal., 37 (2000), pp. 1246--1270]. In particular, in [S. Boscarino, L. Pareschi, and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, submitted] the scheme relaxes to an implicit method for the limit diffusive problem, thus avoiding the $\Delta t \propto \Delta x^2$ restriction. It would be very desirable to have IMEX schemes which treat the hyperbolic part explicitly, because this allows the use of well-tested space discretization with no modification of the original system. However, the development of such methods presents the difficulty that the characteristic speeds diverge in the diffusive limit making the hyperbolic part very stiff. In this paper we show how to overcome these difficulties with the introduction of particular conditions on the coefficients of the IMEX R-K schemes such that we can treat the stiff component on the hyperbolic part explicitly without any manipulation of the original system. Moreover, the schemes proposed in this paper guarantee a CFL condition independent of the diffusive parameter where a CFL hyperbolic condition in the stiff regime is chosen. Such schemes are shown to have the correct diffusion limit and several numerical results confirm the theoretical analysis.