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Solvers for the high-order Riemann problem for hyperbolic balance laws. (English) Zbl 1148.65066
The authors study three methods for solving the Cauchy problem for a system of nonlinear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin, a high-order Riemann problem. Two of the methods in the paper are new; one of the them results from a re-interpretation of the high-order numerical methods of A. Harten, B. Engquist, S. Osher, and S. Chakravarthy [J. Comput. Phys. 71, 231–303 (1987; Zbl 0652.65067)], and the other is a modification of the solver of E. F. Toro and V. A. Titarev [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2018, 271–281 (2002; Zbl 1019.35061)].
Some interesting results are presented. In particular, schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations and the shallow water equations with source terms are constructed. The authors also address the question of balance between flux gradients and source terms, for steady flow. It is found that the arbitrary accuracy derivative (ADER) schemes may be termed asymptotically well-balanced, in the sense that the well-balanced property is attained as the order of the method increases, and this without introducing any ad-hoc fixes to the schemes or the equations.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76N15 Gas dynamics (general theory)
76M20 Finite difference methods applied to problems in fluid mechanics
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[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] Ben-Artzi, M.; Falcovitz, J., Generalized Riemann problems in computational fluid dynamics, (2003), Cambridge University Press · Zbl 1017.76001
[3] Ben-Artzi, M.; Li, J.; Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. comput. phys., 218, 19-43, (2006) · Zbl 1158.76375
[4] Ben-Artzi, M.; Li, J., Hyperbolic balance laws: Riemann invariants and hyperbolic balance laws, Numerische Mathematik, 106, 369-425, (2007) · Zbl 1123.65082
[5] Bourgeade, A.; LeFloch, P.; Raviart, P.A., An asymptotic expansion for the solution of the generalized Riemann problem. part 2: application to the Euler equations of gas dynamics, Ann. inst. Henri Poincaré. analyse non lineáre, 6, 6, 437-480, (1989) · Zbl 0703.35106
[6] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. sci. stat. comput., 6, 104-117, (1985) · Zbl 0562.76072
[7] M. Dumbser, Arbitrary high order schemes for the solution of hyperbolic conservation laws in complex domains. PhD thesis, Institut für Aero- un Gasdynamik, Universität Stuttgart, Germany, 2005.
[8] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. comput. phys., 221, 693-723, (2007) · Zbl 1110.65077
[9] Dumbser, M.; Munz, C.D., ADER discontinuous Galerkin schemes for aeroacoustics, Comptes rendus Mécanique, 333, 683-687, (2005) · Zbl 1107.76044
[10] Dumbser, M.; Schwartzkopff, T.; Munz, C.D., Arbitrary high order finite volume schemes for linear wave propagation, Computational science and high performance computing II: notes on numerical fluid mechanics and multidisciplinary design, vol. 91, (2006), Springer, 129-144
[11] Dumbser, M.; Käser, M.; Titarev, V.A.; Toro, E.F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. comput. phys., 226, 204-243, (2007) · Zbl 1124.65074
[12] S.K. Fok, Extension of Glimm’s method to the problem of gas flow in a duct of variable cross-section. PhD Thesis, Department of Mathematics, University of California, Berkeley, 1981.
[13] Glimm, J., Solution in the large for nonlinear hyperbolic systems of equations, Commun. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[14] Glimm, J.; Marshall, G.; Plohr, B., A generalized Riemann problem for quasi one-dimensional gas flows, Advances in applied mathematics, 5, 1-30, (1984) · Zbl 0566.76056
[15] 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
[16] Harabetian, E., A convergent series expansion for hyperbolic systems of conservation laws, Trans am. math. soc., 294, 383-424, (1986) · Zbl 0599.35103
[17] 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
[18] Harten, A.; Osher, S., Uniformly high – order accurate nonoscillatory schemes I, SIAM J. numer. anal., 24, 2, 279-309, (1987) · Zbl 0627.65102
[19] Hu, C.; Shu, C.W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090
[20] Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 130, 202-228, (1996) · Zbl 0877.65065
[21] M. Käser, Adaptive Methods for the Numerical Simulation of Transport Processes. PhD thesis, Institute of Numerical Mathematics and Scientific Computing, University of Munich, Germany, 2003.
[22] Käser, M.; Iske, A., ADER schemes for the solution of conservation laws on adaptive triangulations, Mathematical methods and modelling in hydrocarbon exploration and production. mathematics in industry, vol. 7, (2005), Springer-Verlag, pp. 323-385
[23] Käser, M.; Iske, A., Adaptive ADER schemes for the solution of scalar non-linear hyperbolic problems, J. comput. phys., 205, 486-508, (2005) · Zbl 1072.65116
[24] L. Tatsien, Y. Wenci. Boundary-value problems for quasi-linear hyperbolic systems, Duke University Mathematics Series, 1985. · Zbl 0582.35078
[25] Le Floch, P.; Raviart, P.A., An asymptotic expansion for the solution of the generalized Riemann problem. part 1: general theory, Ann. inst. Henri Poincaré. analyse non lineáre, 5, 2, 179-207, (1988) · Zbl 0679.35064
[26] Le Floch, P.; Tatsien, L., A global asymptotic expansion for the solution of the generalized Riemann problem, Ann. inst. Henri Poincaré. analyse non lineáre, 3, 321-340, (1991) · Zbl 0731.35006
[27] Liu, T.P., Quasilinear hyperbolic systems, Commun. math. phys., 68, 141-172, (1979) · Zbl 0435.35054
[28] Men’shov, I.S., Increasing the order of approximation of godunov’s scheme using the generalized Riemann problem, USSR comput. math. phys., 30, 5, 54-65, (1990) · Zbl 0739.65079
[29] 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)
[30] H. Schardin, in: Proceedings of VII International Congress on High Speed Photg. Darmstadt, O. Helwich Verlag, 1965.
[31] T. Schwartzkopff, Finite-Volumen Verfahren hoher Ordnung und heterogene Gebietszerlegung ü die numerische Aeroakustik. PhD thesis, Institut für Aero- un Gasdynamik, Universität Stuttgart, Germany, 2005.
[32] 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
[33] Schwartzkopff, T.; Dumbser, M.; Munz, C.D., Fast high-order ADER schemes or linear hyperbolic equations, J. comput. phys., 197, 532-539, (2004) · Zbl 1052.65078
[34] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. comput. phys., 83, 32-78, (1988) · Zbl 0674.65061
[35] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[36] Takakura, Y.; Toro, E.F., Arbitrarily accurate non-oscillatory schemes for a non-linear conservation law, J. comput. fluid dyn., 11, 1, 7-18, (2002)
[37] Titarev, V.A.; Toro, E.F., ADER: arbitrary high order Godunov approach, J. sci. comput., 17, 609-618, (2002) · Zbl 1024.76028
[38] Titarev, V.A.; Toro, E.F., ADER schemes for three-dimensional hyperbolic systems, J. comput. phys., 204, 715-736, (2005) · Zbl 1060.65641
[39] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer-Verlag · Zbl 0888.76001
[40] Toro, E.F., Primitive, conservative and adaptive schemes for hyperbolic conservation laws, (), 323-385 · Zbl 0958.76062
[41] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag · Zbl 0923.76004
[42] Toro, E.F.; Millington, R.C.; Nejad, L.A.M., Towards very high-order Godunov schemes, (), 905-937, (Edited Review) · Zbl 0989.65094
[43] Toro, E.F., Shock-capturing methods for free-surface shallow flows, (2001), Wiley and Sons Ltd. · Zbl 0996.76003
[44] 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
[45] 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
[46] Toro, E.F.; Titarev, V.A., Derivative Riemann solvers for systems of conservation laws and ADER methods, J. comput phys., 212, 1, 150-165, (2006) · Zbl 1087.65590
[47] van Leer, B., Towards the ultimate conservative difference scheme I. the quest for monotonicity, Lecture notes in physics, 18, 163-168, (1973)
[48] van Leer, B., On the relationship between the upwind-differencing schemes of Godunov, engquist – osher and roe, SIAM J. sci. stat. comput., 5, 1, 1-20, (1985) · Zbl 0547.65065
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