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Derivative Riemann solvers for systems of conservation laws and ader methods. (English) Zbl 1087.65590
Summary: We first briefly review the semi-analytical method of E. F. Toro and V. A. Titarev [Proc. Roy. Soc. London, Ser. A, Math. Phys. Eng. Sci. 458, No. 2018, 271–281 (2002; Zbl 1019.35061)] for solving the derivative Riemann problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76N15 Gas dynamics, general
76M20 Finite difference methods applied to problems in fluid mechanics
Full Text: DOI
[1] Ben-Artzi, M.; Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. comput. phys., 55, 1-32, (1984) · Zbl 0535.76070
[2] Bourgeade, A.; LeFloch, P.G.; Raviart, P.-A., An asymptotic expansion for the solution of the generalized Riemann problem. part II: application to the gas dynamics equations, Ann. inst. H. poincare, nonlinear analysis, 6, 437-480, (1989) · Zbl 0703.35106
[3] Casper, J.; Atkins, H., A finite-volume high order ENO scheme for two dimensional hyperbolic systems, J. comput. phys., 106, 62-76, (1993) · Zbl 0774.65066
[4] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[5] M. Dumbser, C.-D. Munz, Building blocks for arbitrary high order discontinuous Galerkin schemes, in press. · Zbl 1115.65100
[6] Godunov, S.K., A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. sbornik, 47, 357-393, (1959)
[7] 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
[8] Kulikovskii, A.G.; Pogorelov, N.V.; Semenov, A.Yu., Mathematical aspects of numerical solution of hyperbolic systems, Monographs and surveys in pure and applied mathematics, vol. 118, (2002), Chapman and Hall
[9] LeFloch, P.G.; Raviart, P.-A., An asymptotic expansion for the solution of the generalized Riemann problem. part I: general theory, Ann. inst. H. Poincaré, nonlinear analysis, 5, 2, 179-207, (1988)
[10] Men’shov, I.S., Increasing the order of approximation of godunov’s scheme using the solution of the generalized Riemann problem, USSR comput. math. math. phys., 30, 5, 54-65, (1990) · Zbl 0739.65079
[11] Men’shov, I.S., Generalized problem of break-up of a single discontinuity, Prikl. matem. mekhan. (J. appl. math. mech.), 55, 1, 86-95, (1991)
[12] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, USSR J. comput. math. phys., 1, 267-279, (1961)
[13] Schwartzkopff, T.; Dumbser, M.; Munz, C.D., Fast high order ADER schemes for linear hyperbolic equations, J. comput. phys., 197, 2, 532-539, (2004) · Zbl 1052.65078
[14] Shi, J.; Hu, C.; Shu, C.-W., A technique for treating negative weights in WENO schemes, J. comput. phys., 175, 108-127, (2002) · Zbl 0992.65094
[15] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. scient. statist. comput., 9, 1073-1084, (1988) · Zbl 0662.65081
[16] Takakura, Y.; Toro, E.F., Arbitrarily accurate non-oscillatory schemes for a non-linear conservation law, Cfd j., 11, 1, 7-18, (2002)
[17] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, J. sci. comput., 17, 609-618, (2002) · Zbl 1024.76028
[18] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional non-linear hyperbolic systems, J. comput. phys., 204, 2, 715-736, (2005) · Zbl 1060.65641
[19] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag · Zbl 0923.76004
[20] E.F. Toro, Riemann solvers with evolved initial conditions, Preprint NI05003-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2005. · Zbl 1108.65091
[21] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high order Godunov schemes, (), 907-940 · Zbl 0989.65094
[22] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten-Lax-Van leer Riemann solver, J. shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[23] Toro, E.F.; Titarev, V.A., Solution of the generalised Riemann problem for advection-reaction equations, Proc. roy. soc. London, 458, 2018, 271-281, (2002) · Zbl 1019.35061
[24] E.F. Toro, V.A. Titarev, MUSTA schemes for systems of conservation laws, Preprint NI04033-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2004. · Zbl 1097.65091
[25] Toro, E.F.; Titarev, V.A., ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions, J. comput. phys., 202, 1, 196-215, (2005) · Zbl 1061.65103
[26] 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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