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Talk Abstract

Towards an Explicit, Uniformly Accurate Godunov Method for Hyperbolic Systems with Relaxation Source Terms

Towards an Explicit, Uniformly Accurate Godunov Method for Hyperbolic Systems with Relaxation Source Terms

**Jeffrey
A.F. Hittinger**

W.M. Keck Foundation CFD Laboratory

Department of Aerospace Engineering

The University of Michigan

jhitt@engin.umich.edu

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).