. 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 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...

Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.

A class of wave propagation algorithms for three-dimensional conservation laws is developed. This unsplit nite volume method is based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model cross-derivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.

. 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 quasi-steady wave-propagation algorithm was developed and used to compute near-steady shallow water flow over variable topography. In this paper we extend this algorithm to near-steady 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 quasi-steady 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...

... In this paper, we describe how to apply the full multigrid algorithm in this context. In particular, the restriction, interpolation, and coarse grid problem will be described. Numerical results for several model problems are given to demonstrate that good rates can be obtained even when jumps in the coefficients are large and do not align with the grid.

We discuss how to combine a front tracking method with dimensional splitting to solve numerically systems of conservation laws in two space dimensions. In addition we present an adaptive grid refinement strategy. The method is unconditionally stable and allows for moderately high cfl numbers (typically 1-4), and thus it is highly efficient. The method is

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible cfl numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical

Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans

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I present a new algorithm for solving the streamfunction-vorticity equations in irregular, multiply connected regions. To avoid mesh generation difficulties associated with unstructured, body fitted grids, I embed the irregular domain in a uniform Cartesian mesh. The governing partial differential equations are discretized using standard finite volume and finite difference methods away from the irregular boundary. Near the irregular boundary, special discretizations are used to impose boundary conditions. I solve the vorticity transport equation using the high-resolution algorithms in the clawpack package and capacity form differencing to handle the irregular geometry. By modifying the capacity function for small cells cut by the boundary, I can avoid the small cell instability problem associ...