Higher Order Godunov Methods for General Systems of Hyperbolic Conservation Laws

We describe an extension of higher order Godunov methods to general systems of hyper-bolic conservation laws. This extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave tields. The method constructs an approximation of the Riemann problem from local wave information. A generalization of the Engquist-Osher flux for systems is then used to compute a numerical flux based on this approximation. This numerical flux replaces the Godunov numerical flux in the algorithm, thereby eliminating the need for a global Riemann problem solution. The additional modifications to the Godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail. The method is applied to some simple model problems for which the global analytic structure is known. The method is also applied to the black-oil model for multiphase flow in petroleum reservoirs. Over the last 15 years, there has been an extensive effort in the development of conservative finite difference methods for computing discontinuous solutions to hyperbolic systems of conservation laws. The goal of this effort has been to develop viable numerical algorithms to solve problems in several specific applications, in particular, compressible fluid flow, plasma physics, and combustion. For reviews of various aspects of this work, see [l-3]. As a result of this effort, a set of heuristics for the design of " high resolution " methods for hyperbolic conservation laws has emerged. They can be summarized as follows. (1) The numerical flux should consist of a hybridization of a flux for a first order method, and a flux for a higher order method. The first order method should be sufficiently dissipative so that, if it were used by itself, the numerical solution would converge to a weak solution satisfying appropriate entropy conditions. The rule by which the two fluxes are hybridized, known as the limiter, should include a sufficient amount of low-order flux at discontinuities and underresolved gradients