Foundations for the generalization of the Godunov method to hyperbolic systems with stiff relaxation source terms.

Abstract

Hyperbolic systems of partial differential equations with relaxation source terms arise in the modeling of many physical problems where internal processes return non-equilibrium disturbances to equilibrium. A challenge in numerically approximating such systems is that the relaxation may take place on time scales much shorter than the time scales of the flow evolution. In such cases, it is desirable for numerical methods to accurately approximate the solution even if the relaxation scales are underresolved. High-resolution Godunov methods are very successful shock-capturing algorithms for the solution of hyperbolic systems of conservation laws. It is desirable to extend this methodology to properly preserve the asymptotic behavior of hyperbolic-relaxation systems such that underresolved solutions can be accurately approximated. Godunov schemes solve or approximate Riemann problems at cell interfaces to estimate numerical fluxes that respect the physics, but, due to coupling between relaxation and wave propagation in hyperbolic-relaxation systems, the Riemann problem becomes much more complicated and its exact solution is no longer feasible. Evidence presented here suggests that to obtain robust, non-oscillatory, upwind discretizations that accurately compute underresolved solutions, aspects of this physical coupling must be included in the numerical flux calculations. A simple model system is extensively analyzed using Fourier and asymptotic analysis on both the system and its integral solution for both smooth and discontinuous initial conditions. Specifically, the early- and late-time asymptotic behaviors of the Riemann problem are determined, and the results are generalized to <italic>m</italic> x <italic>m</italic> constant-coefficient systems. A nonlinear physical example, a set of eleven macroscopic transport equations for a diatomic gas, is constructed from the Boltzmann equation and is investigated to verify the applicability of the linear analysis. Current numerical methods are reviewed, and possibilities are proposed for new procedures that include the effects of the coupling between relaxation and wave propagation in the numerical discretization, specifically in a finite-volume update strategy and in the numerical flux evaluated at cell interfaces. Promising preliminary results are presented for the model system.