On Uniformly Accurate Upwind Methods for Hyperbolic Systems with Relaxation Source Terms
Jeffrey Hittinger, University of Michigan
Many nonequilibrium problems in physics and engineering have natural mathematical formulations as hyperbolic systems with relaxation source terms. Such systems pose interesting challenges for numerical approximation. The source terms cause the system to be dispersive; generally, both the eigenvalues and eigenvectors of the system are modified by the relaxation processes which drive the system towards equilibrium. Another difficult numerical issue arises when the relaxation source terms operate on much smaller time scales than the advection terms; the problem is then said to be stiff. Unfortunately, many problems simultaneously exhibit behavior in both the stiff and non-stiff limits, as well as in between. It is therefore desirable to develop high-resolution numerical algorithms which are uniformly accurate irrespective of the scales of the relaxation source terms. For example, if the data are such that the advection terms permit a time step large enough to bring the flow into local equilibrium, one would wish to see an accurate solution of the equilibrium problem emerge without resolving the details of the transition.
In recent years, there has been much development on such uniformly accurate methods. A variety of splitting schemes, characteristic schemes, and staggered, centered schemes have been proposed. The most promising of these rely on staggered grids to eliminate the difficulty of resolving the Riemann problem, which no longer has a self-similar solution. However, one drawback of centered schemes is that one may lose definition of contact discontinuities, which are likely to be very important in reactive flows. The question still remains as to whether there is a simple, natural extension of the upwind approach which includes the effects of the relaxation source terms and resolves all discontinuities equally well. The close coupling between the advection and relaxation has lead us to explore an upwind scheme based upon an approximate solution to the Riemann initial value problem. Currently, experiments are being performed which apply this method to a model linear system as well as to equations of extended hydrodynamics.
Abstract Author(s): Jeffrey Hittinger