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

1993; Society for Industrial and Applied Mathematics; Volume: 14; Issue: 4 Linguagem: Inglês

10.1137/0914052

1095-7197

Richard B. Pember,

Fluid Dynamics and Turbulent Flows

A higher-order Godunov method is presented for hyperbolic systems of conservation laws with stiff, relaxing source terms. The goal is to develop a Godunov method that produces higher-order accurate solutions using time and space increments governed solely by the nonstiff part of the system, i.e., without fully resolving the effect of the stiff source terms. It is assumed that the system satisfies a certain "subcharacteristic" condition. The method is a semi-implicit form of a method developed by Colella for hyperbolic conservation laws with nonstiff source terms. In addition to being semi-implicit, our method differs from the method for nonstiff systems in its treatment of the characteristic form of the equations. The method is applied to a model system of equations and to a system of equations for gas flow with heat transfer. Our analytical and numerical results show that the modifications to the nonstiff method are necessary for obtaining second-order accuracy as the relaxation time tends to zero. Our numerical results also suggest that certain modifications to the Riemann solver used by the Godunov method would help reduce numerical oscillations produced by the scheme near discontinuities. The development of a modified Riemann solver is a topic of future work.