Numerical Methods for Hyperbolic Conservation Laws With Stiff Relaxation I. Spurious Solutions

The author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock-capturing finite difference scheme on a fixed, uniform spatial grid. It is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain “subcharacteristic”condition be satisfied by the hyperbolic system. This conjecture is supported with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov’s method is coupled with the backward Euler method. Similar ideas are applied to the numerical solution of stiff detonation problems.