Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation II. Higher-Order Godunov Methods

We present a higher order Godunov method for hyperbolic systems of conservation laws with stii, relaxing source terms. Our goal is to develop a Godunov method which produces higher order accurate solutions using time and space increments governed solely by the non-stii part of the system, i.e., without fully resolving the eeect of the stii source terms. We assume that the system satisses a certain \subcharacteristic" condition. The method is a semi-implicit form of a method developed by Colella for hyperbolic conservation laws with non-stii source terms. In addition to being semi-implicit, our method diiers from the method for non-stii systems in its treatment of the characteristic form of the equations. We apply the method to a model system of equations and to a system of equations for gas ow with heat transfer. Our analytical and numerical results show that the modiications to the non-stii method are necessary for obtaining second order accuracy as the relaxation time tends to zero. Our numerical results also suggest that certain modiications to the Riemann solver used by the Godunov method would help reduce numerical oscillations produced by the scheme near discontinuities. The development of a modiied Riemann solver is a topic of future work.