Asymptotically implicit schemes for the hyperbolic heat equation

The main concern of this paper is to derive numerical schemes for the solution of kinetic equations with diffusive scaling which works efficiently for a wide range of the scaling parameter E. We will concentrate on the simple Goldstain-Taylor model from kinetic theory and propose a resolution method based on the reformulation first introduced in [8]. We show how this reformulation corresponds to the use of interpolated fluxes and then we adopt the penalized implicit-explicit Runge-Kutta approach recently introduced in [1] to overcome the parabolic time step restriction in the diffusive regime. The resulting schemes permit to choose a time step Delta t = O(Delta x), independent from epsilon, in all regimes. Some numerical examples show the efficiency and accuracy of the proposed methods.