MUSTA fluxes for systems of conservation laws 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 [cf. {it 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 nonlinear 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 nonlinear 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 nonlinear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

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  1. Mignone, A.; Del Zanna, L.: Systematic construction of upwind constrained transport schemes for MHD (2021)
  2. Nogueira Sondermann, Carina; Viggiano, Raphael; de Freitas Rachid, Felipe Bastos; Bodstein, Gustavo C. R.: A suitability analysis of transient one-dimensional two-fluid numerical models for simulating two-phase gas-liquid flows based on benchmark problems (2021)
  3. Dumbser, Michael (ed.); Titarev, Vladimir (ed.); Saurel, Richard (ed.): Preface to the special issue “Recent progress in numerical methods for nonlinear time-dependent flow & transport problems” (2018)
  4. Porter-Sobieraj, Joanna; Słodkowski, Marcin; Kikoła, Daniel; Sikorski, Jan; Aszklar, Paweł: A MUSTA-FORCE algorithm for solving partial differential equations of relativistic hydrodynamics (2018)
  5. Wu, Jie; He, Yuan-yuan; Ding, Guo-hao; Han, Yi-yu: Hybrid optimized low-dissipation and adaptive MUSCL reconstruction technique for hyperbolic conservation laws (2018)
  6. Schmidtmann, Birte; Torrilhon, Manuel: A hybrid Riemann solver for large hyperbolic systems of conservation laws (2017)
  7. Fraysse, F.; Redondo, C.; Rubio, G.; Valero, E.: Upwind methods for the Baer-Nunziato equations and higher-order reconstruction using artificial viscosity (2016)
  8. Romenski, Evgeniy; Belozerov, Alexander A.; Peshkov, Ilya M.: Conservative formulation for compressible multiphase flows (2016)
  9. Knaus, R.; Pantano, C.: A computational approach to flame hole dynamics using an embedded manifold approach (2015)
  10. Castro Díaz, M. J.; Fernández-Nieto, E. D.; Narbona-Reina, G.; de la Asunción, M.: A second order PVM flux limiter method. Application to magnetohydrodynamics and shallow stratified flows (2014)
  11. Castro, Manuel J.; Gallardo, José M.; Marquina, Antonio: A class of incomplete Riemann solvers based on uniform rational approximations to the absolute value function (2014)
  12. Morales de Luna, Tomás; Castro Díaz, Manuel J.; Parés, Carlos: Relation between PVM schemes and simple Riemann solvers (2014)
  13. Thanh, Mai Duc: Building fast well-balanced two-stage numerical schemes for a model of two-phase flows (2014)
  14. Jerez, Silvia: Monotone first-order weighted schemes for scalar conservation laws (2013)
  15. Morales de Luna, T.; Castro Díaz, M. J.; Parés, C.: Reliability of first order numerical schemes for solving shallow water system over abrupt topography (2013)
  16. Castro Díaz, M. J.; Fernández-Nieto, E.: A class of computationally fast first order finite volume solvers: PVM methods (2012)
  17. Kalise, Dante; Lie, Ivar: Modeling and numerical approximation of a 2.5D set of equations for mesoscale atmospheric processes (2012)
  18. Castro, Manuel J.; Pardo, Alberto; Parés, Carlos; Toro, E. F.: On some fast well-balanced first order solvers for nonconservative systems (2010)
  19. Jerez Galiano, Silvia; Uh Zapata, Miguel: A new TVD flux-limiter method for solving nonlinear hyperbolic equations (2010)
  20. Bouchut, F.; Morales de Luna, T.: Semi-discrete entropy satisfying approximate Riemann solvers. The case of the Suliciu relaxation approximation (2009)

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