Analysis of the effect of operator splitting and of the sampling procedure on the accuracy of Glimm's method
- 1 December 1978
- report
- Published by Office of Scientific and Technical Information (OSTI)
Abstract
A study is made of 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. Two fundamental problems are addressed. A highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence is proposed. Error bounds are derived for Glimm's method, with van der Corput sampling, as applied to the inviscid Burgers' equation: for sufficiently small times the error in shock locations, speeds, and strengths is no greater than O(h/sup 1/2/abs. value (log h)), and the error in the continuous part of the solution, away from shocks, is O(h abs. value (log h)). Here h is the spatial increment of the grid, with the estimates holding in the limit of h ..-->.. 0. The improved sampling procedure is tested numerically in the case of inviscid compressible flow in one space dimension; it gives high-resolution results both in the smooth parts of the solution and at discontinuities. The operator splitting procedure by means of which the multidimensional method is constructed is investigated. An O(1) error stemming from the use ofmore » 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, and a method for eliminating this error, by the use of suitable artificial viscosity, is proposed and tested. 33 figures. « lessKeywords
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