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MUSTA: a multi-stage numerical flux. (English) Zbl 1101.65088
Summary: Numerical methods for conservation laws constructed in the framework of finite volume and discontinuous Galerkin finite elements require, as the building block, a monotone numerical flux. In this paper we present some preliminary results on the MUSTA approach [cf. E. F. Toro, Multi-stage predictor-corrector fluxes for hyperbolic equations, Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June (2003)] for constructing upwind numerical fluxes. The scheme may be interpreted as an unconventional approximate Riemann solver that has simplicity and generality as its main features.
When used in its first-order mode we observe that the scheme achieves the accuracy of the Godunov method used in conjunction with the exact Riemann solver, which is the reference first-order method for hyperbolic systems. At least for the scalar model hyperbolic equation, the Godunov scheme is the best of all first-order monotone schemes, it has the smallest truncation error. Extensions of the scheme of this paper are realized in the framework of existing approaches.
Here we present a second-order total variation diminishing (TVD for the scalar case) extension and show numerical results for the two-dimensional Euler equations on non-Cartesian geometries. The schemes find their best justification when solving very complex systems for which the solution of the Riemann problem, in the classical sense, is too complex, too costly or is simply unavailable.

MSC:
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 theory)
Software:
MUSTA
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