Hyperbolic systems with relaxation source terms can be used to describe many non-equilibrium flows. The numerical simulation of this type of system can be challenging, particularly if regions exist where the characteristic time scales of the relaxation processes are much smaller than those of the wave propagation across the local computational cells. In this case, the source terms are stiff. Typically, the coupling of the two processes is such that the wave speeds and strengths change significantly as the relaxation drives the solution to equilibrium. High-resolution approaches based upon split physics, therefore, often lose accuracy in the stiff limit; the coupling must somehow be accounted for in any successful numerical scheme.

For a properly linearized hyperbolic system with relaxation, an exact transformation exists from which a finite volume update strategy can be devised, automatically capturing details of the advection/relaxation coupling. The implementation of this scheme on both staggered and non-staggered computational grids will be considered. Of particular difficulty in the non-staggered case is the solution of the Riemann problem at cell interfaces. Using asymptotic information about the wave structures of the Riemann problem, strategies will be discussed for the design of upwind numerical flux functions which sufficiently account for the relaxation effects.

Joint work with Philip Roe (The University of Michigan).

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