High–Order Asymptotic–Preserving Methods for Nonlinear Relaxation from Hyperbolic Systems to Convection–Diffusion Equations

In Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761) the authors propose an asymptotic–preserving method based on Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes which are adopted to deal with a class of nonlinear hyperbolic systems containing nonlinear diffusive relaxation. These schemes are able to solve such systems with no stiff nor parabolic restriction on the time step. In the limit when the relaxation parameter vanishes, the proposed scheme relaxes to a semi-implicit scheme for the limit nonlinear diffusion equation, thus overcoming the classical parabolic CFL condition in the time step. In this paper we consider an extension of the numerical treatment of the Kawashima-LeFloch’s model (LeFloch and Kawashima, private communication) proposed in Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761). We show that the same schemes introduced in Boscarino et al. (SIAM J Sci Comput, 6(2), A377–A395, preprint: arxiv.org/pdf/1210.4761) relaxes to an semi-implicit scheme for the limit nonlinear convection-diffusion equation. A numerical example confirms the robustness and the accuracy of the scheme in order to capture the correct behavior of the solution in the hyperbolic–to–parabolic regime.