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MUSTA fluxes for systems of conservation laws. (English) Zbl 1097.65091
Summary: 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. 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.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
76N15 Gas dynamics, general
76W05 Magnetohydrodynamics and electrohydrodynamics
76M12 Finite volume methods applied to problems in fluid mechanics
Software:
GFORCE; HE-E1GODF; HLLE
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[1] Barmin, A.A.; Kulikovskii, A.G.; Pogorelov, N.V., Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics, J. comput. phys., 126, 77-90, (1996) · Zbl 0863.76039
[2] Billett, S.J.; Toro, E.F., WAF-type schemes for multidimensional hyperbolic conservation laws, J. comput. phys., 130, 1-24, (1997) · Zbl 0873.65088
[3] Brio, M.; Wu, C.C., An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. comput. phys., 75, 400-422, (1988) · Zbl 0637.76125
[4] Chen, G.Q.; Toro, E.F., Centred schemes for non-linear hyperbolic equations, J. hyperbolic diff. eq., 1, 1, 531-566, (2004) · Zbl 1063.65076
[5] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171-200, (1990) · Zbl 0694.65041
[6] Colella, P.; Glaz, H.H., Efficient solution algorithms for the Riemann problem for real gases, J. comput. phys., 59, 264-289, (1985) · Zbl 0581.76079
[7] Courant, R.; Isaacson, E.; Rees, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. pure appl. math., 5, 243-255, (1952) · Zbl 0047.11704
[8] Einfeldt, B.; Munz, C.D.; Roe, P.L.; Sjögreen, B., On Godunov-type methods near low densities, J. comput. phys., 92, 273-295, (1991) · Zbl 0709.76102
[9] Glaister, P., An approximate linearised Riemann solver for the three-dimensional Euler equations for real gases using operator splitting, J. comput. phys., 77, 361-383, (1988) · Zbl 0644.76088
[10] Glimm, J., Solution in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[11] Godlewski, E.; Raviart, P.A., Numerical approximation of hyperbolic systems of conservation laws, (1996), Springer Germany · Zbl 0860.65075
[12] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Mater. sb., 47, 271-306, (1959) · Zbl 0171.46204
[13] Godunov, S.K.; Zabrodin, A.V.; Prokopov, G.P., A difference scheme for two-dimensional unsteady aerodynamics, USSR J. comp. math. math. phys. USSR, 2, 6, 1020-1050, (1961) · Zbl 0146.23004
[14] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 1, 35-61, (1983) · Zbl 0565.65051
[15] Jiang, G.S.; Tadmor, E., Non-oscillatory central schemes for multi-dimensional hyperbolic conservation laws, SIAM J. sci. comput., 19, 6, 1892-1917, (1998) · Zbl 0914.65095
[16] Jiang, G.S.; Wu, C.C., A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics, J. comput. phys., 150, 561-594, (1999) · Zbl 0937.76051
[17] Kulikovskii, A.G.; Pogorelov, N.V.; Semenov, A.Y., Mathematical aspects of numerical solutions of hyperbolic systems, Monographs and surveys in pure and applied mathematics, (2002), Chapman & Hall London
[18] Kurganov, A.; Tadmor, E., New high-resolution central schemes for non-linear conservation laws, J. comput. phys., 160, 241-282, (2000) · Zbl 0987.65085
[19] Lax, P.D.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802
[20] LeVeque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040
[21] Levy, D.; Puppo, G.; Russo, G., A fourth order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. sci. comput., 24, 2, 480-506, (2002) · Zbl 1014.65079
[22] Liska, R.; Wendroff, B., Composite schemes for conservation laws, SIAM J. numer. anal., 35, 6, 2250-2271, (1998) · Zbl 0920.65054
[23] Love, E.H.; Pidduck, F.B., Lagrange’s ballistic problem, Phil. trans. roy. soc. lond., 222, 167-228, (1922) · JFM 48.0912.01
[24] Menikoff, R.; Plohr, B.J., The Riemann problem for fluid flow of real materials, Rev. modern phys., 61, 75-130, (1989) · Zbl 1129.35439
[25] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408-463, (1990) · Zbl 0697.65068
[26] Kolgan, V.P., Application of the principle of minimum derivatives to the construction of difference schemes for computing discontinuous solutions of gas dynamics (in Russian), Uch. zap. tsagi, Russia, 3, 6, 68-77, (1972)
[27] Quartapelle, L.; Castelletti, L.; Guardone, A.; Quaranta, G., Solution of the Riemann problem of classical gas dynamics, J. comput. phys., 190, 1, 118-140, (2003) · Zbl 1236.76054
[28] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[29] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, J. comput. math. phys. USSR, 1, 267-279, (1961)
[30] Shi, C.; Hu, J.; Shu, C.W., A technique for treating negative weights in WENO schemes, J. comput. phys., 175, 1, 108-127, (2002) · Zbl 0992.65094
[31] Titarev, V.A.; Toro, E.F., Finite volume WENO schemes for three-dimensional conservation laws, J. comput. phys., 201, 1, 238-260, (2004) · Zbl 1059.65078
[32] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional hyperbolic systems, J. comput. phys., 204, 715-736, (2005) · Zbl 1060.65641
[33] Titarev, V.A.; Toro, E.F., MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements, Int. J. numer. meth. fluid, 49, 117-147, (2005) · Zbl 1073.65553
[34] 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.
[35] Toro, E.F., A fast Riemann solver with constant covolume applied to the random choice method, Int. J. numer. meth. fluid, 9, 1145-1164, (1989) · Zbl 0673.76090
[36] Toro, E.F., A weighted average flux method for hyperbolic conservation laws, Proc. roy. soc. lond., A423, 401-418, (1989) · Zbl 0674.76060
[37] E.F. Toro, Riemann-problem based techniques for computing reactive two-phase flows, in: Dervieux, Larrouturrou (Eds.), Proceedings of the 3rd International Conference on Numerical Combustion, Lecture Notes in Physics, vol. 351, Antibes, France, May, 1989, pp. 472-481.
[38] E.F. Toro, On Glimm-Related Schemes for Conservation Laws. Technical Report MMU-9602, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1996.
[39] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag Germany · Zbl 0923.76004
[40] E.F. Toro, Riemann Solvers with Evolved Initial Conditions. Int. J. Numer. Meth. Fluids, 2006 (to appear). · Zbl 1108.65091
[41] Toro, E.F.; Billett, S.J., Centred TVD schemes for hyperbolic conservation laws, IMA J. numer. anal., 20, 47-79, (2000) · Zbl 0943.65100
[42] Toro, E.F.; Chakraborty, A., Development of an approximate Riemann solver for the steady supersonic Euler equations, Aeronaut. J., 98, 325-339, (1994)
[43] Toro, E.F.; Hu, W., Unsplit centred finite volume schemes: preliminary results, () · Zbl 0990.65095
[44] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high-order Godunov schemes, (), 905-937 · Zbl 0989.65094
[45] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[46] van Leer, B., Towards the ultimate conservative difference scheme I. the quest for monotonicity, Lect. notes phys., 18, 163-168, (1973)
[47] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
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