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Riemann solvers with evolved initial conditions. (English) Zbl 1108.65091
Summary: The scope of this paper is three fold. We first formulate upwind and symmetric schemes for hyperbolic equations with non-conservative terms. Then we propose upwind numerical schemes for conservative and non-conservative systems, based on a Riemann solver, the initial conditions of which are evolved nonlinearly in time, prior to a simple linearization that leads to closed-form solutions. The Riemann solver is easily applied to complicated hyperbolic systems. Finally, as an example, we formulate conservative schemes for the three-dimensional Euler equations for general compressible materials and give numerical results for a variety of test problems for ideal gases in one and two space dimensions.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics, general
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