×

zbMATH — the first resource for mathematics

Multidimensional Riemann problem with self-similar internal structure. I: Application to hyperbolic conservation laws on structured meshes. (English) Zbl 1349.76303
Summary: Multidimensional Riemann solvers have been formulated recently by the author [ibid. 229, No. 6, 1970–1993 (2010; Zbl 1303.76140); ibid. 231, No. 22, 7476–7503 (2012; Zbl 1284.76261)]. They operate at the vertices of a two-dimensional mesh, taking input from all the neighboring states and yielding the resolved state and fluxes as output. The multidimensional Riemann problem produces a self-similar strongly interacting state which is the result of several one-dimensional Riemann problems interacting with each other. The prior work was restricted to the use of one-dimensional HLLC Riemann solvers as building blocks. In this paper, we formulate the problem in similarity variables. As a result, any self-similar one-dimensional Riemann solver can be used as a building block for the multidimensional Riemann solver. This paper focuses on the structure of the strongly-interacting state. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)In this work the strongly-interacting state is expanded in a set of basis functions that depend on the similarity variables. Consequently, the resolved state and the fluxes can be endowed with considerably richer sub-structure compared to prior work. Unlike the multidimensional HLLC Riemann solver, the need to independently specify a direction for the evolution of the contact discontinuity is eliminated. The richer sub-structure in the strongly-interacting state naturally accommodates waves that may be moving in any direction relative to the mesh, thereby minimizing mesh-imprinting. Two formulations are presented. The first formulation does not linearize the problem around a favorable state. Its derivation takes a few cues from the derivation of the multidimensional HLL Riemann solver. The second formulation identifies such a state and carries out a linearization of the fluxes about that state. This paper is the very first time that a series solution of the multidimensional Riemann problem has been presented. Explicit formulae are presented for up to quartic variation in the self-similar variables. While linear variations are sufficient for numerical work, the higher order terms in the series solution could prove useful for analytical studies of the multidimensional Riemann problem. The formulation presented here is general enough to accommodate any hyperbolic conservation law. It can also accommodate any one-dimensional Riemann solver and yields a multidimensional version of the same. It has been incorporated in the author’s RIEMANN code. As examples of the different types of hyperbolic conservation laws, we use Euler flow, Magnetohydrodynamics (MHD) and relativistic MHD. As examples of different types of Riemann solvers, we show multidimensional formulations of HLL, HLLC and HLLD Riemann solvers for MHD all working fluently within this formulation. Several stringent test problems are presented.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
Software:
HLLE; RIEMANN
PDF BibTeX Cite
Full Text: DOI
References:
[1] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des équations d’Euler par une méthode de type roe, I: la linéarisation, C. R. Acad. Sci., Ser. 1 Math., 319, 499, (1994) · Zbl 0813.76074
[2] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des équations d’Euler par une méthode de type roe, II: solution du problème de Riemann approché, C. R. Acad. Sci., Ser. 1 Math., 319, 625, (1994) · Zbl 0813.76075
[3] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, 45-58, (1994) · Zbl 0822.65062
[4] Anton, L.; Miralles, J. A.; Marti, J. M.; Ibanez, J. M.; Aloy, M. A.; Mimica, P., A full wave decomposition Riemann solver in RMHD, Astrophys. J. Suppl. Ser., 187, 1, (2010)
[5] Balsara, D. S., Total variation diminishing scheme for relativistic magnetohydrodynamics, Astrophys. J. Suppl. Ser., 132, 83-101, (2001)
[6] Balsara, D. S., Multidimensional HLLE Riemann solver; application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[7] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503, (2012) · Zbl 1284.76261
[8] Balsara, D. S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 119, (1998)
[9] Balsara, D. S.; Spicer, D. S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149, 270-292, (1999) · Zbl 0936.76051
[10] Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452, (2000) · Zbl 0961.65078
[11] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648, (2001) · Zbl 1157.76369
[12] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (2004)
[13] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228, 5040-5056, (2009) · Zbl 1280.76030
[14] Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480, (2009) · Zbl 1275.76169
[15] Balsara, D. S., Self-adjusting positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231, 7504-7517, (2012)
[16] Balsara, D. S.; Dumbser, M.; Meyer, C.; Du, H.; Xu, Z., Efficient implementation of ADER schemes for Euler and magnetohydrodynamic flow on structured meshes - comparison with Runge-Kutta methods, J. Comput. Phys., 235, 934-969, (2013) · Zbl 1291.76237
[17] Balsara, D. S.; Dumbser, M.; Abgrall, R., Multidimensional HLL and HLLC Riemann solvers for unstructured meshes - with application to Euler and MHD flows, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426
[18] T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 28th Aerospace Sciences Meeting, January 1990.
[19] Batten, P.; Clarke, N.; Lambert, C.; Causon, D. M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18, 1553-1570, (1997) · Zbl 0992.65088
[20] Billett, S. J.; Toro, E. F., On WAF-type schemes for multidimensional hyperbolic conservation laws, J. Comput. Phys., 130, 1-24, (1997) · Zbl 0873.65088
[21] Boscheri, W.; Dumbser, M.; Balsara, D. S., High order Lagrangian ADER-WENO schemes on unstructured meshes - application of several node solvers to hydrodynamics and magnetohydrodynamics, Int. J. Numer. Methods Fluids, (2014), to appear
[22] Boscheri, W.; Balsara, D. S.; Dumbser, M., Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers, J. Comput. Phys., 267, 112-138, (2014) · Zbl 1349.76309
[23] Brio, M.; Zakharian, A. R.; Webb, G. M., Two-dimensional Riemann solver for Euler equations of gas dynamics, J. Comput. Phys., 167, 177-195, (2001) · Zbl 1043.76042
[24] Cargo, P.; Gallice, G., Roe matrices for ideal MHD and systematic construction of roe matrices for systems of conservation laws, J. Comput. Phys., 136, 446, (1997) · Zbl 0919.76053
[25] Chakraborty, A.; Toro, E. F., Development of an approximate Riemann solver for the steady supersonic Euler equations, Aeronaut. J., 98, 325-339, (1994)
[26] Chorin, A. J., Random choice solutions of hyperbolic systems, J. Comput. Phys., 22, 517, (1976) · Zbl 0354.65047
[27] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[28] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comput., 6, 104, (1985) · Zbl 0562.76072
[29] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171, (1990) · Zbl 0694.65041
[30] Colella, P.; Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 174-201, (1984) · Zbl 0531.76082
[31] Dedner, A.; Kemm, F.; Kröener, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for MHD equations, J. Comput. Phys., 175, 645-673, (2002) · Zbl 1059.76040
[32] Del Zanna, L.; Bucciantini, N.; Londrillo, P., An efficient shock-capturing central-type scheme for multidimensional relativistic flows, Astron. Astrophys., 400, 397-414, (2003) · Zbl 1222.76122
[33] Dumbser, M.; Käser, 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
[34] Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C.-D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[35] Dumbser, M.; Toro, E. F., On universal osher-type schemes for general non-linear hyperbolic conservation laws, Commun. Comput. Phys., 10, 635-671, (2011) · Zbl 1373.76125
[36] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 3, 294-318, (1988) · Zbl 0642.76088
[37] Einfeldt, B.; Munz, C.-D.; Roe, P. L.; Sjogreen, B., On Godunov-type methods near low densities, J. Comput. Phys., 92, 273-295, (1991) · Zbl 0709.76102
[38] Fey, M., Multidimensional upwinding 1. the method of transport for solving the Euler equations, J. Comput. Phys., 143, 159, (1998) · Zbl 0932.76050
[39] Fey, M., Multidimensional upwinding 2. decomposition of the Euler equation into advection equation, J. Comput. Phys., 143, 181, (1998) · Zbl 0932.76051
[40] Gardiner, T.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys., 205, 509, (2005) · Zbl 1087.76536
[41] Gardiner, T.; Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. Comput. Phys., 227, 4123, (2008) · Zbl 1317.76057
[42] Gilquin, H.; Laurens, J.; Rosier, C., Multidimensional Riemann problems for linear hyperbolic systems, Notes Numer. Fluid Mech., 43, 284, (1993) · Zbl 0921.35090
[43] Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. USSR Sb., 47, 271-306, (1959) · Zbl 0171.46204
[44] Godunov, S. K., Numerical solution of multi-dimensional problems in gas dynamics, (1976), Nauka Press Moscow
[45] Gurski, K. F., An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics, SIAM J. Sci. Comput., 25, 2165, (2004) · Zbl 1133.76358
[46] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 289-315, (1983) · Zbl 0565.65051
[47] Honkkila, V.; Janhunen, P., HLLC solver for relativistic MHD, J. Comput. Phys., 223, 643-656, (2007) · Zbl 1111.76036
[48] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065
[49] Kim, J. H.; Balsara, D. S., A stable HLLC Riemann solver for relativistic magnetohydrodynamics, J. Comput. Phys., 270, 634-639, (2014) · Zbl 1349.76618
[50] Komissarov, S. S., A Godunov-type scheme for relativistic MHD, Mon. Not. R. Astron. Soc., 303, 343, (1999)
[51] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 1, 241-282, (2000) · Zbl 0987.65085
[52] Kurganov, A.; Noelle, S.; Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23, 3, 707-740, (2001) · Zbl 0998.65091
[53] LeVeque, R. J., Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys., 131, 327, (1997) · Zbl 0872.76075
[54] Li, S.-T., An HLLC Riemann solver for magnetohydrodynamics, J. Comput. Phys., 203, 344, (2005) · Zbl 1299.76302
[55] Lukacsova-Medvidova, M.; Morton, K. W.; Warnecke, G., Finite volume evolution Galerkin methods for Euler equations of gas dynamics, Int. J. Numer. Methods Fluids, 40, 425, (2002) · Zbl 1023.76026
[56] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 4, 1781-1824, (2007) · Zbl 1251.76028
[57] Mignone, A.; Bodo, G., An HLLC Riemann solver for relativistic flows II - magnetohydrodynamics, Mon. Not. R. Astron. Soc., 368, 1040, (2006)
[58] Mignone, A.; Ugliano, M.; Bodo, G., A five-wave HLL Riemann solver for relativistic MHD, Mon. Not. R. Astron. Soc., 393, 1141, (2009)
[59] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315-344, (2005) · Zbl 1114.76378
[60] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation-laws, J. Comput. Phys., 87, 2, 408-463, (1990) · Zbl 0697.65068
[61] Orszag, S. A.; Tang, C. M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 129, (1979)
[62] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 158, 339, (1982) · Zbl 0483.65055
[63] Roe, P. L., Approximate Riemann solver, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[64] Roe, P. L., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. Comput. Phys., 63, 458, (1986) · Zbl 0587.76126
[65] Roe, P. L.; Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. Appl. Math., 56, 57, (1996) · Zbl 0845.35092
[66] Rumsey, C. B.; van Leer, B.; Roe, P. L., A multidimensional flux function with application to the Euler and Navier-Stokes equations, J. Comput. Phys., 105, 306, (1993) · Zbl 0767.76039
[67] Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, USSR Comput. Math. Math. Phys., 1, 267, (1961)
[68] Saltzman, J., An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys., 115, 153, (1994) · Zbl 0813.65111
[69] Schulz-Rinne, C. W.; Collins, J. P.; Glaz, H. M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput., 14, 7, 1394-1414, (1993) · Zbl 0785.76050
[70] Stecca, G.; Siviglia, A.; Toro, E. F., Upwind-biased FORCE schemes with applications to free surface shallow flows, J. Comput. Phys., 229, 6362-6380, (2010) · Zbl 1426.35192
[71] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618, (2002) · Zbl 1024.76028
[72] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[73] Toro, E. F.; Titarev, V. A., Solution of the generalized Riemann problem for advection reaction equations, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 458, 271-281, (2002) · Zbl 1019.35061
[74] Toro, E. F.; Spruce, M.; Speares, W., Restoration of contact surface in the HLL Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053
[75] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten-Lax-Van leer Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053
[76] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL Riemann solver, (June 1992), Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology UK, Technical report CoA 9204
[77] Toro, E.; Hidalgo, A.; Dumbser, M., FORCE schemes on unstructured meshes I: conservative hyperbolic systems, J. Comput. Phys., 228, 9, 3368-3389, (2009) · Zbl 1168.65377
[78] van Leer, B., Toward the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101, (1979) · Zbl 1364.65223
[79] Vides, J.; Nkonga, B.; Audit, E., A simple two-dimensional extension of the HLLE Riemann solver for gas dynamics, (2014), hal-00998235 - version 1 · Zbl 1373.85005
[80] Wendroff, B., A two-dimensional HLLE Riemann solver and associated Godunov-type difference scheme for gas dynamics, Comput. Math. Appl., 38, 175-185, (1999) · Zbl 0984.76064
[81] 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
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.