Glimm's Method for Gas Dynamics

We investigate Glimm's method, a method for constructing approximate solutions to systems of hyperbolic conservation laws in one space variable by sampling explicit wave solutions. It is extended to several space variables by operator splitting. We consider two problems. 1) We propose a highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence. We test the improved sampling procedure numerically in the case of inviscid compressible flow in one space dimension and find that it gives high resolution results both in the smooth parts of the solution, as well as at discontinuities. 2) We investigate the operator splitting procedure by means of which the multidimensional method is constructed. An $O(1)$ error stemming from the use of this procedure near shocks oblique to the spatial grid is analyzed numerically in the case of the equations for inviscid compressible flow in two space dimensions. We present a hybrid method which eliminates this error, consisting of Glimm's method, used in continuous parts of the flow, and the nonlinear Godunov method, used in regions where large pressure jumps are generated. The resulting method is seen to be a substantial improvement over either of the component methods for multidimensional calculations.

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