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Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . 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 suc ..."
Abstract

Cited by 54 (5 self)
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. 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 wavepropagation 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 highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
Wave Propagation Methods for Conservation Laws with Source Terms
 In preparation
, 1998
"... . An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm ..."
Abstract

Cited by 12 (3 self)
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. An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm was developed and used to compute nearsteady shallow water flow over variable topography. In this paper we extend this algorithm to nearsteady flow of an ideal gas subject to a static gravitational field. The method is implemented in the software package clawpack. The ability of this method to capture perturbed quasisteady solutions is demonstrated with numerical examples. 1. Introduction We consider the Euler equations in conservation form @ t q +r \Delta f (q) = / (q) (1) where q 2 R m is a vector of conserved quantities, f : R m ! R m is the flux, and / is a source term due to a static gravitational field. It is well known that if f is a nonlinear function of q as for the Eule...