A Class of upwind methods for Conservation Laws

Abstract

Various new methods for the solution of hyperbolic systems of conservation laws in one, two and three space dimensions are developed. All are explicit, conservative timemarching methods that are second order accurate in space and time in regions of smooth flow and make use of local Riemann problems at intercell boundaries. In one space dimension, the Weighted Average Flux (w af ) approach of Toro is extended to generate a scheme that is stable with timesteps twice as large as those allowed by the stability conditions of the original scheme. A Riemann problem based extension of the Warming-Beam scheme is considered. Total Variation Diminishing (t v d ) conditions are enforced for both schemes. Numerical results for the Euler Equations of Gas Dynamics are presented. In two and three space dimensions, finite volume versions of the waf scheme on Cartesian grids are derived for the linear advection equation. Two two dimensional schemes are found that are second order accurate in space and time. One of these is extended for the solution of nonlinear systems of hyperbolic conservation laws in two separate ways. The resulting schemes are tested on the Shallow Water equations. The equivalent three dimensional schemes are also discussed. The two dimensional schemes are then extended for use on structured, body-fitted grids of quadrilaterals and one of these extensions is used to demonstrate the phenomena of Mach reflection of shallow water bores.