Preprint 2002-006

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

Abstract: We study a general approach to solving conservation laws of the form $q_t+f(q,x)_x=0$ where the flux function $f(q,x)$ has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function $f_i(q)$ over the $i$th grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences $f_i(Q_i)-f_{i-1}(Q_{i-1})$ into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws $q_t+f(q,x)_x=\psi(q,x)$ are also considered, in which case the source term is used to modifiy the flux differene before performing the wave decomposition. An additional term is derived that must also be included to obtain full accuracy.

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Author(s):

Derek S. Bale, <dbale@amath.washington.edu>

Randall J. LeVeque, <rjl@amath.washington.edu>

Sorin Mitran, <mitran@amath.washington.edu>

James A. Rossmanith, <jrossman@amath.washington.edu>

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Submitted by:

<rjl@amath.washington.edu> January 27 2002.