An upwind differencing scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
On Godunov-type methods near low densities
Journal of Computational Physics
Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics
Journal of Computational Physics
On WAF-type schemes for multidimensional hyperbolic conservation laws
Journal of Computational Physics
Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Composite Schemes for Conservation Laws
SIAM Journal on Numerical Analysis
A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Journal of Computational Physics
A technique of treating negative weights in WENO schemes
Journal of Computational Physics
A Fourth-Order Central WENO Scheme for Multidimensional Hyperbolic Systems of Conservation Laws
SIAM Journal on Scientific Computing
Solution of the Riemann problem of classical gasdynamics
Journal of Computational Physics
Finite-volume WENO schemes for three-dimensional conservation laws
Journal of Computational Physics
ADER schemes for three-dimensional non-linear hyperbolic systems
Journal of Computational Physics
High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems
Journal of Scientific Computing
Journal of Computational Physics
Journal of Scientific Computing
Conservative Models and Numerical Methods for Compressible Two-Phase Flow
Journal of Scientific Computing
A new TVD flux-limiter method for solving nonlinear hyperbolic equations
Journal of Computational and Applied Mathematics
Modeling and numerical approximation of a 2.5D set of equations for mesoscale atmospheric processes
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.46 |
This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach [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], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov's method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.