The effects of slope limiting on asymptotic-preserving numerical methods for hyperbolic conservation laws

Many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates. An asymptotic-preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit. We explore the consequences of applying a slope limiter to the discontinuous Galerkin (DG) method, with linear elements, for hyperbolic systems with stiff relaxation terms. Without a limiter, the DG method gives an accurate discretization of the Chapman–Enskog approximation of the system when the relaxation length scale is not resolved. It is well known that the first-order upwind (or “step”) method fails to obtain the proper asymptotic limit. We show that using the minmod slope limiter also fails, but that using double minmod gives the proper asymptotic limit. Despite its effectiveness in the asymptotic limit, the double minmod limiter allows artificial extrema at cell interfaces, referred to as “sawteeth”. We present a limiter that eliminates the sawteeth, but maintains the proper asymptotic limit. The systems that we analyze are the hyperbolic heat equation and the Pn thermal radiation equations. Numerical examples are used to verify our analysis.