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Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics. (English) Zbl 1349.76408
Summary: The paper proposes a second-order accurate finite volume local evolution Galerkin (FVLEG) method for two-dimensional special relativistic hydrodynamical (RHD) equations. Instead of using the dimensional splitting method or solving one-dimensional local Riemann problem in the direction normal to cell interface, the FVLEG method couples a finite volume formulation with the (genuinely) multi-dimensional approximate local evolution operator, which is derived by evolving the solutions of corresponding locally linearized RHD equations along all bicharacteristic directions. Several numerical examples are given to demonstrate the accuracy and the performance of the proposed FVLEG method.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
Software:
WHAM; GENESIS; RAM
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[1] Aloy, M. A.; Ibáñez, J. M.; Martí, J. M.; Müller, E., GENESIS: A high-resolution code for 3D relativistic hydrodynamics, Astrophys. J., 122, 151-166, (1999)
[2] Balsara, D. S., Riemann solver for relativistic hydrodynamics, J. Comput. Phys., 114, 284-297, (1994) · Zbl 0810.76062
[3] Bollermann, A.; Noelle, S.; Lukáčová-Medvidʼová, M., Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10, 371-404, (2011) · Zbl 1364.76109
[4] Butler, D. S., The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. R. Soc. Lond., 255, 232-252, (1960) · Zbl 0099.41501
[5] Dai, W.; Woodward, P. R., An iterative Riemann solver for relativistic hydrodynamics, SIAM J. Sci. Stat. Comput., 18, 982-995, (1997) · Zbl 0892.35008
[6] Dolezal, A.; Wong, S. S.M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120, 266-277, (1995) · Zbl 0840.76047
[7] Donat, R.; Font, J. A.; Ibáñez, J. M.; Marquina, A., A flux-split algorithm applied to relativistic flows, J. Comput. Phys., 146, 58-81, (1998) · Zbl 0930.76054
[8] Dumbser, M.; Uuriintsetseg, A.; Zanotti, O., On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Commun. Comput. Phys., 14, 301-327, (2013) · Zbl 1373.76126
[9] Duncan, G. C.; Hughes, P. A., Simulations of relativistic extragalactic jets, Astrophys. J., 436, L119-L122, (1994)
[10] Eulderink, F.; Mellema, G., General relativistic hydrodynamics with a roe solver, Astron. Astrophys. Suppl. Ser., 110, 587-623, (1995)
[11] He, P., Numerical simulations of relativistic hydrodynamics and relativistic magneto-hydrodynamics, (2011), School of Mathematical Sciences, Peking University, Ph.D. thesis
[12] He, P.; Tang, H. Z., An adaptive moving mesh method for two-dimensional relativistic hydrodynamics, Commun. Comput. Phys., 11, 114-146, (2012) · Zbl 1373.76354
[13] He, P.; Tang, H. Z., An adaptive moving mesh method for two-dimensional relativistic magnetohydrodynamics, Comput. Fluids, 60, 1-20, (2012) · Zbl 1365.76337
[14] Huang, L. C., Conservative bicharacteristic upwind schemes for hyperbolic conservation laws II, Comput. Math. Appl., 29, 91-107, (1995) · Zbl 0820.65056
[15] Johnston, R. L.; Pal, S. K., The numerical solution of hyperbolic systems using bicharacteristics, Math. Comput., 26, 377-392, (1972) · Zbl 0245.65041
[16] Kunik, M.; Qamar, S.; Warnecke, G., Kinetic schemes for the relativistic gas dynamics, Numer. Math., 97, 159-191, (2004) · Zbl 1098.76056
[17] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 241-282, (2000) · Zbl 0987.65085
[18] Landau, L. D.; Lifshitz, E. M., Fluid mechanics, (1987), Pergamon Press
[19] Lukáčová-Medvidʼová, M.; Morton, K. W., Finite volume evolution Galerkin methods: A survey, Indian J. Pure Appl. Math., 41, 329-361, (2010) · Zbl 1203.65161
[20] Lukáčová-Medvidʼová, M.; Morton, K. W.; Warnecke, G., Evolution Galerkin methods for hyperbolic systems in two space dimensions, Math. Comput., 69, 1355-1384, (2000) · Zbl 0951.35076
[21] Lukáčová-Medvidʼová, M.; Morton, K. W.; Warnecke, G., Finite volume evolution Galerkin methods for Euler equations of gas dynamics, Int. J. Numer. Methods Fluids, 40, 425-434, (2002) · Zbl 1023.76026
[22] Lukáčová-Medvidʼová, M.; Morton, K. W.; Warnecke, G., Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems, SIAM J. Sci. Comput., 26, 1-30, (2004) · Zbl 1078.65562
[23] Lukáčová-Medvidʼová, M.; Noelle, S.; Kraft, M., Well-balanced finite volume evolution Galerkin methods for the shallow water problems, J. Comput. Phys., 221, 122-147, (2007) · Zbl 1123.76041
[24] Lukáčová-Medvidʼová, M.; Saibertová, J.; Warnecke, G., Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comput. Phys., 183, 533-562, (2002) · Zbl 1090.65536
[25] Martí, J. M.; Müller, E., Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics, J. Comput. Phys., 123, 1-14, (1996) · Zbl 0839.76056
[26] Martí, J. M.; Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Relativ., 6, 7, (2003) · Zbl 1068.83502
[27] Mendza, M.; Boghosian, B. M.; Herrmann, H. J.; Succi, S., Derivation of the lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev. D, 82, 105008, (2010)
[28] Mendza, M.; Boghosian, B. M.; Herrmann, H. J.; Succi, S., Fast lattice Boltzmann solver for relativistic hydrodynamics, Phys. Rev. Lett., 105, 014502, (2010)
[29] Mignone, A.; Bodo, G., An HLLC Riemann solver for relativistic flows, I: hydrodynamics, Mon. Not. R. Astron. Soc., 364, 126-136, (2005)
[30] Mignone, A.; Plewa, T.; Bodo, G., The piecewise parabolic method for multidimensional relativistic fluid dynamics, Astrophys. J. Suppl. Ser., 160, 199-219, (2005)
[31] Morton, K. W., On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal., 35, 2195-2222, (1998) · Zbl 0927.65119
[32] Qamar, S.; Warnecke, G., A high-order kinetic flux-splitting method for the relativistic magnetohydrodynamics, J. Comput. Phys., 205, 182-204, (2005) · Zbl 1087.76090
[33] Qamar, S.; Yousaf, M., The space-time CESE method for solving special relativistic hydrodynamic equations, J. Comput. Phys., 231, 3928-3945, (2012) · Zbl 1426.76555
[34] Radice, D.; Rezzolla, L., Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes, Phys. Rev. D, 84, 024010, (2011)
[35] Reddy, A. S.; Tikekar, V. G.; Prasad, P., Numerical solution of hyperbolic equations by the method of bicharacteristics, J. Math. Phys. Sci., 16, 575-603, (1982) · Zbl 0516.65078
[36] Roe, P., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. Comput. Phys., 63, 458-476, (1986) · Zbl 0587.76126
[37] Schneider, V.; Katscher, U.; Rischke, D. H.; Waldhauser, B.; Maruhn, J. A.; Munz, C. D., New algorithms for ultra-relativistic numerical hydrodynamics, J. Comput. Phys., 105, 92-107, (1993) · Zbl 0779.76062
[38] Sun, Y. T.; Ren, Y. X., The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws, J. Comput. Phys., 228, 4945-4960, (2009) · Zbl 1169.65331
[39] Tchekhovskoy, A.; McKinney, J. C.; Narayan, R., WHAM: a WENO-based general relativistic numerical scheme, I: hydrodynamics, Mon. Not. R. Astron. Soc., 379, 469-497, (2007)
[40] van Leer, B., Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection, J. Comput. Phys., 23, 276-299, (1977) · Zbl 0339.76056
[41] Wilson, J. R., Numerical study of fluid flow in a Kerr space, Astrophys. J., 173, 431-438, (1972)
[42] Wilson, J. R.; Mathews, G. J., Relativistic numerical hydrodynamics, (2003), Cambridge University Press · Zbl 1095.83003
[43] Yang, J. Y.; Chen, M. H.; Tsai, I. N.; Chang, J. W., A kinetic beam scheme for relativistic gas dynamics, J. Comput. Phys., 136, 19-40, (1997) · Zbl 0889.76053
[44] Yang, Z. C.; He, P.; Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: one-dimensional case, J. Comput. Phys., 230, 7964-7987, (2011) · Zbl 1408.76597
[45] Yang, Z. C.; Tang, H. Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case, J. Comput. Phys., 231, 2116-2139, (2012) · Zbl 1408.76598
[46] Zanna, L. D.; Bucciantini, N., An efficient shock-capturing central-type scheme for multidimensional relativistic flows, I: hydrodynamics, Astron. Astrophys., 390, 1177-1186, (2002) · Zbl 1209.76022
[47] Zhang, W. Q.; Macfadyen, A. I., RAM: a relativistic adaptive mesh refinement hydrodynamics code, Astrophys. J. Suppl. Ser., 164, 255-279, (2006)
[48] Zhao, J.; He, P.; Tang, H. Z., Steger-warming flux vector splitting method for special relativistic hydrodynamics, Math. Methods Appl. Sci., (2013) · Zbl 1298.35161
[49] Zhao, J.; Tang, H. Z., Runge-Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 242, 138-168, (2013) · Zbl 1314.76035
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