Abstract

this paper, we consider a simple and efficient finite volume method for the Euler equation for compressible flows. The 1D system of the Euler equation is given by # t U + # x F(U ) = 0, U = 2 4 # m E 3 5 , F = 2 4 m #u 2 + p u(E + p) 3 5 , (1.1) where #,u,m=#u, and E are density, velocity, momentum, and total energy, respectively, and the pressure is obtained from the equation of state, p = (# -1)(E-#u 2 /2). The equation 1 Current address: Courant Institute, New York, NY 10012. 2 Current address: Institute for Physical Sciences and Technology, and Department of Mathematics, University of Maryland, College Park, MD 20742. Research was supported in part by NSF Grant DMS-9505275. 237 0021-9991/98 $25.00 Copyright c # 1998 by Academic Press All rights of reproduction in any form reserved. 238 CHOI AND LIU in multidimensional space is given similarly. Most modern shock capturing schemes for the solution of Eq. (1.1) are of Godunov-type, which reconstructs the solution after fieldby -field decomposition and solves a Riemann problem for time evolution. The well-known Godunov-type schemes are MUSCL [30], PPM [32], and ENO schemes [7, 26]. We explore an efficient implementation by using a flux splitting for the time evolution for dynamic and steady state computations. In the flux splitting, F = F + + F - , the Jacobians of the split fluxes F have only positive or negative eigenvalues. That is, each split flux always keeps only one wind direction. For this reason, they are sometimes called upwind fluxes. The high order accuracy of the scheme is achieved by directly reconstructing each component of upwind fluxes by a piecewise polynomial. Then the numerical flux in the finite volume method is determined by evaluating the reconstructed upwind ...

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