zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI
[1] Chen, G.Q.; Toro, E.F., Centred schemes for non-linear hyperbolic equations, Journal of hyperbolic differential equations, 1, 1, 531-566, (2004) · Zbl 1063.65076
[2] Cockburn, B.; Shu, C.W., TVB runge – kutta local projection discontinuous Galerkin method for conservation laws II: general framework, Math. comp., 52, 411, (1989) · Zbl 0662.65083
[3] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. sci. statist. comput., 6, 104-117, (1985) · Zbl 0562.76072
[4] 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
[5] Glimm, J., Solution in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[6] Godlewski, E.; Raviart, P.A., Numerical approximation of hyperbolic systems of conservation laws, (1996), Springer Berlin · Zbl 0860.65075
[7] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271-306, (1959) · Zbl 0171.46204
[8] Godunov, S.K.; Zabrodin, A.V.; Prokopov, G.P., A difference scheme for two-dimensional unsteady aerodynamics, J. comput. math. math. phys. USSR, 2, 6, 1020-1050, (1961) · Zbl 0146.23004
[9] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357-393, (1983) · Zbl 0565.65050
[10] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high order accuracy essentially non-oscillatory schemes III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067
[11] Harten, A.; Osher, S., Uniformly high-order accurate nonoscillatory schemes I, SIAM J. numer. anal., 24, 2, 279-309, (1987) · Zbl 0627.65102
[12] Laney, C.B., Computational gasdynamics, (1998), Cambridge University Press Cambridge · Zbl 0947.76001
[13] Lax, P.D.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802
[14] LeVeque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040
[15] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Mathematical modelling and numerical analysis, 33, 547-571, (1999) · Zbl 0938.65110
[16] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408-463, (1990) · Zbl 0697.65068
[17] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. comp., 38, 158, 339-374, (1982) · Zbl 0483.65055
[18] Kolgan, V.P., Application of the principle of minimum derivatives to the construction of difference schemes for computing discontinuous solutions of gas dynamics, Uch. zap. tsagi, Russia, 3, 6, 68-77, (1972), (in Russian)
[19] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[20] Schwartzkopff, T.; Munz, C.D.; Toro, E.F., ADER: high-order approach for linear hyperbolic systems in 2D, J. sci. comput., 17, 231-240, (2002) · Zbl 1022.76034
[21] C.W. Shu, Essentially non-oscillatory and weighted non-oscillatory schemes for hyperbolic conservation laws, Technical Report ICASE Report No. 97-65, NASA, 1997 · Zbl 0927.65111
[22] Strang, G., On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 3, 506-517, (1968) · Zbl 0184.38503
[23] Takakura, Y.; Toro, E.F., Arbitrarily accurate non-oscillatory schemes for a non-linear conservation law, J. comput. fluid dynamics, 11, 1, 7-18, (2002)
[24] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, J. sci. comput., 17, 609-618, (2002) · Zbl 1024.76028
[25] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional hyperbolic systems, J. comput. phys., 204, 715-736, (2005) · Zbl 1060.65641
[26] 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
[27] Toro, E.F., A weighted average flux method for hyperbolic conservation laws, Proc. roy. soc. London A, 423, 401-418, (1989) · Zbl 0674.76060
[28] E.F. Toro, On Glimm-related schemes for conservation laws. Technical Report MMU-9602, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1996
[29] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004
[30] Toro, E.F.; Billett, S.J., Centred TVD schemes for hyperbolic conservation laws, IMA J. numer. anal., 20, 47-79, (2000) · Zbl 0943.65100
[31] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high-order Godunov schemes, (), 905-937 · Zbl 0989.65094
[32] Toro, E.F.; Siviglia, A., PRICE: primitive centred schemes for hyperbolic systems, Int. J. numer. methods fluids, 42, 1263-1291, (2003) · Zbl 1078.76566
[33] 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
[34] Toro, E.F.; Titarev, V.A., Solution of the generalised Riemann problem for advection – reaction equations, Proc. roy. soc. London A, 458, 271-281, (2002) · Zbl 1019.35061
[35] Toro, E.F.; Titarev, V.A., TVD fluxes for high-order ADER schemes, J. scientific computing, 24, 3, 285-309, (2005) · Zbl 1096.76029
[36] Toro, E.F.; Titarev, V.A., ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions, J. comput. phys., 202, 1, 196-215, (2005) · Zbl 1061.65103
[37] ()
[38] van der Vegt, J.J.W.; van der Ven, H.; Boelens, O.J., Discontinuous Galerkin methods for partial differential equations, () · Zbl 0989.65115
[39] van Leer, B., Towards the ultimate conservative difference scheme I. the quest for monotonicity, Lecture notes in physics, 18, 163-168, (1973)
[40] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, engquist – osher and roe, SIAM J. sci. statist. comput., 5, 1, 1-20, (1984) · Zbl 0547.65065
[41] 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
[42] Yee, H.C., Construction of explicit and implicit symmetric TVD schemes and their applications, J. comput. phys., 68, 151-179, (1987) · Zbl 0621.76026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.