**A wave-propagation method for conservation laws with spatially varying
flux functions**,

by D. S. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith,
*SIAM J. Sci. Comput* 24 (2002), 955-978.

**Abstract.**

A Wave Propagation Method for Conservation Laws
and Balance Laws with Spatially Varying Flux Functions: SIAM Journal on
Scientific Computing Vol. 24, Iss. 3We 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*_{}-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 modify the flux difference before performing the wave
decomposition, and an additional term is derived that must also be included
to obtain full accuracy. This method is particularly useful for
quasi-steady problems close to steady state.
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**bibtex entry:**

@Article{db-rjl-sm-jr:vcflux,

author = "D. Bale and R. J. LeVeque and S. Mitran and J. A. Rossmanith",

title = "A wave-propagation method for conservation laws and

balance laws with spatially varying flux functions",

journal = "SIAM J. Sci. Comput.",

year = "2002",

volume = "24",

pages = "955--978",

DOI = "10.1137/S106482750139738X",

}