**Balancing Source Terms and Flux Gradients in High-Resolution
Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm **

by Randall J. LeVeque
*J. Comput. Phys.*, 146 (1998) 346-365.

**Abstract.**
Conservation laws with source terms often have steady states in which the
flux gradients are nonzero but exactly balanced by source terms. Many
numerical methods (e.g., fractional step methods) have difficulty preserving
such steady states and cannot accurately calculate small perturbations of
such states. Here a variant of the wave-propagation algorithm is developed
which addresses this problem by introducing a Riemann problem in the center
of each grid cell whose flux difference exactly cancels the source term.
This leads to modified Riemann problems at the cell edges in which the jump
now corresponds to perturbations from the steady state. Computing waves and
limiters based on the solution to these Riemann problems gives
high-resolution results. The 1D and 2D shallow water equations for flow over
arbitrary bottom topography are used as an example, though the ideas apply
to many other systems. The method is easily implemented in the software
package CLAWPACK.

**Note:** The method proposed in this paper is not necessarily the best way to
solve quasi-steady balance laws. Instead I recommend the approach discussed
in Section 7 of the more recent paper
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.

**bibtex entry:**

@Article{rjl:qsteady, author = "R. J. LeVeque", title = "Balancing Source Terms and Flux Gradients in High-Resolution {G}odunov Methods: The Quasi-Steady Wave-Propagation Algorithm", journal = "J. Comput. Phys.", volume = "146", year = "1998", pages = "346--365", url = "http://faculty.washington.edu/rjl/pubs/qsteady/" }