Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems in the Diffusion Limit

Several mathematical models are described by hyperbolic systems with stiff relaxation, defined by a small relaxation time epsilon. In the so-called diffusive relaxation, as the epsilon vanishes, the system relaxes to a parabolic equation or convection-diffusion equation. From a numerical point of view, to solve hyperbolic systems with diffusive relaxation is much complicated because the characteristic speeds of the hyperbolic part depend on epsilon and diverge as epsilon -> 0. There are methods that allow to overcome such stiffness, and that allow the construction of asymptotic preserving schemes that, in the limit of infinite stiffness, reduce to a consistent explicit scheme for the underlying diffusion equation, [5, 6]. Here we consider IMEX Runge-Kutta (RK) schemes for hyperbolic systems of conservation laws and we present two techniques for the construction of such schemes which capture the diffusive limit without the classical stability restriction on the time step Delta t = O(Delta x(2)). The first one [7], is based on an implicit treatment of some hyperbolic terms while the second one, which treats the hyperbolic terms explicitly, is obtained by applying additional condition on the RK coefficients. Several numerical tests will be presented that illustrate the robustness and generality of the methods.