Numerical methods for hyperbolic conservation laws with stiff relaxation

We consider the use of a shock capturing finite difference scheme to approximate solutions of hyperbolic systems of conservation laws with stiff, relaxing source terms. We study in particular the problem of obtaining accurate solutions of these systems with a numerical method that uses time and space increments governed solely by the non-stiff part of the equations. We have two main goals in this investigation. The first is to determine when it is possible to obtain first order accurate numerical solutions of these systems without fully resolving the effects of the stiff source terms. The second is to develop a Godunov method for these systems which produces higher order accurate numerical solutions even as the relaxation time tends to zero.; We accomplish the first goal in part I of the thesis. We present criteria which insure that a shock capturing finite difference method does not produce spurious solutions as the relaxation time approaches zero. 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. We consider 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. The analytical and numerical results for this example support our claim that the above criteria are valid. We also generalize our results to determine similar criteria applicable to the numerical solution of stiff detonation problems.; In part II we develop a Godunov method which produces higher order accurate numerical solutions even as the relaxation time tends to zero. We assume that the system of equations satisfies the "subcharacteristic" condition mentioned above. We base our development on a higher order Godunov method developed by Colella for hyperbolic conservation laws with non-stiff source terms. Our method differs from the corresponding method for non-stiff systems in its semi-implicit treatment of the stiff source terms and in the manner in which the characteristic form of the system is used. We apply the method to a system of equations similar to the model presented in part I and to a system of equations for gas flow with heat transfer. We present analytical and numerical results which support the claim that the modifications to the non-stiff method are necessary for obtaining second order accuracy as the relaxation time tends to zero. The numerical results also suggest that certain modifications to the Riemann solver used by the Godunov method would help reduce small numerical oscillations that may be produced by the scheme near discontinuities.