×

zbMATH — the first resource for mathematics

Multidimensional Riemann problem with self-similar internal structure. Part II: Application to hyperbolic conservation laws on unstructured meshes. (English) Zbl 1351.76091
Summary: Multidimensional Riemann solvers that have internal sub-structure in the strongly-interacting state have been formulated recently [the first author, ibid. 231, No. 22, 7476–7503 (2012; Zbl 1284.76261); ibid. 277, 163–200 (2014; Zbl 1349.76303)]. Any multidimensional Riemann solver operates at the grid vertices and takes as its input all the states from its surrounding elements. It yields as its output an approximation of the strongly interacting state, as well as the numerical fluxes. 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. To compute this strongly interacting state and its higher order moments we propose the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables. The use of substructure in the Riemann problem reduces numerical dissipation and, therefore, allows a better preservation of flow structures, like contact and shear waves. In this second part of a series of papers we describe how this technique is extended to unstructured triangular meshes. All necessary details for a practical computer code implementation are discussed. In particular, we explicitly present all the issues related to computational geometry. Because these Riemann solvers are Multidimensional and have Self-similar strongly- Interacting states that are obtained by Consistency with the conservation law, we call them MuSIC Riemann solvers. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.){
}The MuSIC framework is sufficiently general to handle general nonlinear systems of hyperbolic conservation laws in multiple space dimensions. It can also accommodate all self-similar one-dimensional Riemann solvers and subsequently produces a multidimensional version of the same. In this paper we focus on unstructured triangular meshes. As examples of different systems of conservation laws we consider the Euler equations of compressible gas dynamics as well as the equations of ideal magnetohydrodynamics (MHD). Several stringent test problems are solved for both PDE systems.

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:
RIEMANN; MUSIC; HLLE
PDF BibTeX XML 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] Balsara, D. S., Multidimensional HLLE Riemann solver; application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[5] 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
[6] Balsara, D. S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 119, (1998)
[7] 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
[8] 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
[9] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648, (2001) · Zbl 1157.76369
[10] Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149-184, (2004)
[11] Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228, 5040-5056, (2009) · Zbl 1280.76030
[12] 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
[13] Balsara, D. S., Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231, 7504-7517, (2012)
[14] 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
[15] 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
[16] Balsara, D. S., Multidimensional Riemann problem with self-similar internal structure. part I - application to hyperbolic conservation laws on structured meshes, J. Comput. Phys., 277, 163-200, (2014) · Zbl 1349.76303
[17] Barth, T. J.; Frederickson, P. O., Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, (January 1990), AIAA Paper No. 90-0013. 28th Aerospace Sciences Meeting
[18] 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
[19] 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
[20] 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
[21] 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
[22] Chakraborty, A.; Toro, E. F., Development of an approximate Riemann solver for the steady supersonic Euler equations, Aeronaut. J., 98, 325-339, (1994)
[23] Chorin, A. J., Random choice solutions of hyperbolic systems, J. Comput. Phys., 22, 517, (1976) · Zbl 0354.65047
[24] 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
[25] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comput., 6, 104, (1985) · Zbl 0562.76072
[26] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 171, (1990) · Zbl 0694.65041
[27] Colella, P.; Woodward, P. R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54, 174-201, (1984) · Zbl 0531.76082
[28] 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
[29] 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
[30] 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
[31] Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, J. Sci. Comput., 48, 70-88, (2011) · Zbl 1220.65110
[32] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 3, 294-318, (1988) · Zbl 0642.76088
[33] 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
[34] Fey, M., Multidimensional upwinding, 1. the method of transport for solving the Euler equations, J. Comput. Phys., 143, 159, (1998) · Zbl 0932.76050
[35] Fey, M., Multidimensional upwinding, 2. decomposition of the Euler equation into advection equation, J. Comput. Phys., 143, 181, (1998) · Zbl 0932.76051
[36] Gilquin, H.; Laurens, J.; Rosier, C., Multidimensional Riemann problems for linear hyperbolic systems, Notes Numer. Fluid Mech., 43, 284, (1993) · Zbl 0921.35090
[37] 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
[38] Godunov, S. K., Numerical solution of multi-dimensional problems in gas dynamics, (1976), Nauka Press Moscow
[39] Gurski, K. F., An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics, SIAM J. Sci. Comput., 25, 2165, (2004) · Zbl 1133.76358
[40] 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
[41] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065
[42] LeVeque, R. J., Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys., 131, 327, (1997) · Zbl 0872.76075
[43] Li, S.-T., An HLLC Riemann solver for magnetohydrodynamics, J. Comput. Phys., 203, 344, (2005) · Zbl 1299.76302
[44] 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
[45] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315-344, (2005) · Zbl 1114.76378
[46] Orszag, S. A.; Tang, C. M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 129, (1979)
[47] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 158, 339, (1982) · Zbl 0483.65055
[48] Roe, P. L., Approximate Riemann solver, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[49] Roe, P. L., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. Comput. Phys., 63, 458, (1986) · Zbl 0587.76126
[50] Roe, P. L.; Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. Appl. Math., 56, 57, (1996) · Zbl 0845.35092
[51] 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
[52] Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR, 1, 267, (1961)
[53] Saltzman, J., An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys., 115, 153, (1994) · Zbl 0813.65111
[54] 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
[55] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618, (2002) · Zbl 1024.76028
[56] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[57] 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
[58] 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
[59] 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
[60] 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
[61] 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
[62] Vides, J.; Nkonga, B.; Audit, E., A simple two-dimensional extension of the HLLE Riemann solver for gas dynamics, (2014), INRIA Research Report No. 8540 · Zbl 1373.85005
[63] 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
[64] 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.