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

We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing nite diierence scheme on a xed, uniform spatial grid. We conjecture that certain a priori criteria insure 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 satissed by the hyperbolic system. We support our conjecture with analytical and numerical results for a speciic 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. We also apply our ideas to the numerical solution of stii detonation problems.