A Second-Order Unsplit Godunov Scheme for Two- and Three-Dimensional Euler Equations

Wenlong Dai, Paul R. Woodward

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Abstract

A second-order finite difference scheme is presented in this paper for two- and three-dimensional Euler equations. The scheme is based on nondirectionally split and single-step Eulerian formulations of Godunov approach. A new approach is proposed for constructing effective left and right states of Riemann problems arising from interfaces of one-, two-, and three-dimensional numerical grids. The Riemann problems are solved through an approximate solver in order to calculate a set of time-averaged fluxes needed in the scheme. The scheme is tested through numerical examples involving strong shocks. It is shown that the scheme offers the principle advantages of a second order Godunov scheme: robust operation in the presence of strong waves and thin shock fronts.

Original languageEnglish (US)
Pages (from-to)261-281
Number of pages21
JournalJournal of Computational Physics
Volume134
Issue number2
DOIs
StatePublished - Jul 1 1997

Bibliographical note

Funding Information:
The authors thank Drs. J. Grieger and E. Kevin Edgar for their useful discussion. This work was supported by the Department of Energy under Grant DE-FG02-87ER25035, the National Science Foundation under FIG. 20. The solution of a test problem with four constant states when reflecting boundary conditions are used. The upper image is obtained from the scheme presented in this paper, and the lower image is obtained from PPM in which a steepener is used for contact discontinuities.

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