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On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

  • Michael Dumbser (a1) and Eleuterio F. Toro (a1)

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gas-dynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in [9].

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[1]Balsara, D., Total variation diminishing scheme for relativistic magnetohydrodynamics, Astrophys. J. Supp. Ser., 132 (2001), 83–101.
[2]Barton, P. T., Drikakis, D., Romenski, E. and Titarev, V. A., Exact and approximate solutions of riemann problems in non-linear elasticity, J. Comput. Phys., 228 (2009), 7046–7068.
[3]Canestrelli, A., Dumbser, M., Siviglia, A. and Toro, E. F., Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water. Res., 33 (2010), 291–303.
[4]Castro, C. E. and Toro, E. F., Solvers for the high-order Riemann problem for hyperbolic balance laws, J. Comput. Phys., 227 (2008), 2481–2513.
[5]Castro, M. J., Gallardo, J. M. and Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products, applications to shallow-water systems, Math. Comput., 75 (2006), 1103–1134.
[6]Dedner, A., Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T. and Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175 (2002), 645–673.
[7]Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids., 39 (2010), 60–76.
[8]Dumbser, M. and Balsara, D. S., High-order unstructured one-step PNPM schemes for the viscous and resistive MHD equations, CMES-Comput. Model. Eng. Sci., 54 (2009), 301–333.
[9]Dumbser, M., Balsara, D. S., Toro, E. F. and Munz, C. D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, J. Comput. Phys., 227 (2008), 8209–8253.
[10]Dumbser, M., Castro, M., Parés, C. and Toro, E. F., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Comput. Fluids., 38 (2009), 1731–2748.
[11]Dumbser, M., Enaux, C. and Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227 (2008), 3971–4001.
[12]Dumbser, M., Hidalgo, A., Castro, M., Parés, C. and Toro, E. F., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods. Appl. Mech. Eng., 199 (2010), 625–647.
[13]Dumbser, M. and Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221 (2007), 693–723.
[14]Dumbser, M., Käser, M., Titarev, V. A and Toro, E. F., Quadrature-freenon-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226 (2007), 204–243.
[15]Dumbser, M. and Munz, C. D., Building blocks for arbitrary high order discontinuous Galerkin schemes, J. Sci. Comput., 27 (2006), 215–230.
[16]Dumbser, M. and Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J. Comput. Phys., 228 (2009), 6991–7006.
[17]Einfeldt, B., Munz, C. D., Roe, P. L. and Sjögreen, B., On Godunov-type methods near low densities. J. Comput. Phys., 92 (1991), 273–295.
[18]Falle, S.A.E.G., On the inadmissibility of non-evolutionary shocks, J. Plasma. Phys., 65 (2001), 29–58.
[19]Giacomazzo, B. and Rezzolla, L., The exact solution of the Riemann problem in relativistic magnetohydrodynamics, J. Fluid. Mech., 562 (2006), 223–259.
[20]Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), 271–306.
[21]Godunov, S. K. and Romenski, E. I., Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates, J. Appl. Mech. Tech. Phys., 13 (1972), 868–885.
[22]Godunov, S. K. and Romenski, E. I., Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media, in Computational Fluid Dynamics Review, pages 19–31, John Wiley, NY, 1995.
[23]Godunov, S. K. and Romenski, E. I., Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic/Plenum Publishers, 2003.
[24]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), 231–303.
[25]Harten, A., Lax, P. D. and Leer, B. van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25(1) (1983), 35–61.
[26]Honkkila, V. and Janhunen, P., HLLC solver for ideal relativistic MHD, J. Comput. Phys., 223 (2007), 643–656.
[27]Hu, C. and Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150 (1999), 97–127.
[28]Jiang, G. and Shu, C. W., On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 62 (1994), 531–538.
[29]Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical approximation, Commun. Pure. Appl. Math., 7 (1954), 159–193.
[30]Osher, S., Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal., 21 (1984), 217–235.
[31]Osher, S. and Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. Comput., 38 (1982), 339–374.
[32]Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal., 44 (2006), 300–321.
[33]Qiu, J., Dumbser, M. and Shu, C. W., The discontinuous Galerkin method with Lax-Wendroff type time discretizations, Comput. Methods. Appl. Mech. Eng., 194 (2005), 4528–4543.
[34]Rezzolla, L. and Zanotti, O., An improved exact Riemann solver for relativistic hydrodynamics, J. Fluid. Mech., 449 (2001), 395–411.
[35]Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), 357–372.
[36]Roe, P. L. and Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. Appl. Math., 56 (1996), 57–67.
[37]Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR., 1 (1961), 267–279.
[38]Stroud, A. H., Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1971.
[39]Titarev, V. A., Romenski, E. I. and Toro, E. F., MUSTA-type upwind fluxes for non-linear elasticity, Int. J. Numer. Methods. Eng., 73 (2008), 897–926.
[40]Titarev, V. A. and Toro, E. F., ADER: Arbitrary high order Godunov approach, J. Sci. Comput., 17 (2002), 609–618.
[41]Titarev, V. A. and Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204 (2005), 715–736.
[42]Toro, E. F., Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, 2001.
[43]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, third edition, 2009.
[44]Toro, E. F. and Billet, S. J., Centered TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20 (2000), 44–79.
[45]Toro, E. F., Hidalgo, A. and Dumbser, M., FORCE schemes on unstructured meshes I: conservative hyperbolic systems, J. Comput. Phys., 228 (2009), 3368–3389.
[46]Toro, E. F., Spruce, M. and Speares, W., Restoration of the contact surface in the Harten-Lax-van Leer Riemann solver, J. Shock. Waves., 4 (1994), 25–34.
[47]Torrilhon, M., Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics, J. Comput. Phys., 192 (2003), 73–94.
[48]Toumi, I., A weak formulation of Roe’s approximate Riemann solver, J. Comput. Phys., 102 (1992), 360–373.
[49]Waals, J. D. van der, Over de Continuiteit van den Gas-en Vloei Stof Toestand, PhD thesis, University of Leiden, 1873.
[50]Leer, B. van, Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), 101–136.
[51]Del Zanna, L., Bucciantini, N. and Londrillo, P., An efficient shock-capturing central-type scheme for multidimensional relativistic flows II: magnetohydrodynamics, Astr. Astrophys., 400 (2003), 397–413.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
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