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The space-time CESE method for solving special relativistic hydrodynamic equations. (English) Zbl 1426.76555
Summary: The special relativistic hydrodynamic equations are more complicated than the classical ones due to the nonlinear and implicit relations that exist between conservative and primitive variables. In this article, a space – time conservation element and solution element (CESE) method is proposed for solving these equations in one and two space dimensions. The CESE method has capability to capture sharp propagating wavefront of the relativistic fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space – time domain without resorting to interpolation or extrapolation. The scheme is efficient, robust, and gives results comparable to those obtained with more sophisticated algorithms, even in highly relativistic two-dimensional test problems.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
Full Text: DOI
[1] Aloy, M.A.; Ibáñez, J.M^{a}.; Martí, J.M^{a}.; Müller, E., GENESIS: a high-resolution code for 3d relativistic hydrodynamics, Astrophys. J., 122, 151-166, (1999)
[2] Chang, S.C., The method of space time conservation element and solution element—a new approach for solving the Navier Stokes and Euler equations, J. comput. phys., 119, 295-324, (1995) · Zbl 0847.76062
[3] S.C. Chang, X.Y. Wang, C.Y. Chow, New developments in the method of space-time conservation element and solution element—applications to two-dimensional time-marching problems, NASA TM 106758, 1994.
[4] Chang, S.C.; Wang, X.Y.; Chow, C.Y., The space-time conservation element and solution element method. A new high resolution and genuinely multidimensional paradigm for solving conservation laws, J. comput. phys., 156, 89-136, (1999) · Zbl 0974.76060
[5] Chang, S.C.; Wang, X.Y.; To, W.M., Application of the space-time conservation element and solution element method to one-dimensional convection-diffusion problems, J. comput. phys., 165, 189-215, (2000) · Zbl 0973.65085
[6] Csernai, L.P., Introduction to relativistic heavy ion collisions, (1994), Springer New York
[7] Del Zanna, L.; Bucciantini, N., An efficient shock-capturing central-type scheme for multidimensional relativistic flows, I. hydrodynamics, A&a, 390, 1177-1186, (2002) · Zbl 1209.76022
[8] Donat, R.; Marquina, A., Capturing shock reflections: an improved flux formula, J. comput. phys., 125, 42-58, (1996) · Zbl 0847.76049
[9] van Dyke, M., An album of fluid motion, (1982), The Parabolic press Stanford, California
[10] Eulderink, F.; Mellema, G., General relativistic hydrodynamics with a roe solver, Astron. astrophys. suppl., 110, 587-623, (1995)
[11] Jaing, G.-S.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. sci. comput., 19, 1892-1917, (1998) · Zbl 0914.65095
[12] Kunik, M.; Qamar, S.; Warnecke, G., Kinetic schemes for the ultra-relativistic Euler equations, J. comput. phys., 187, 572-596, (2003) · Zbl 1061.76068
[13] Kunik, M.; Qamar, S.; Warnecke, G., BGK-type kinetic flux vector splitting schemes for ultra-relativistic Euler equations, SIAM J. sci. comput., 26, 196-223, (2004) · Zbl 1071.82049
[14] Kunik, M.; Qamar, S.; Warnecke, G., Kinetic schemes for the relativistic gas dynamics, Numer. math., 97, 159-191, (2004) · Zbl 1098.76056
[15] C.Y. Loh, L.S. Hultgren, S.C. Chang, P.C.E. Jorgenson, Noise computation of a shock-containing supersonic axisymmetric jet by the CE/SE method, AIAA Paper 2000-0475, presented at the 38th AIAA Aerospace Sciences Meeting, January 10-13, Reno, NV, 2000.
[16] Loh, C.Y.; Hultgren, L.S.; Chang, S.C., Wave computation in incompressible flow using the space-time conservation element and solution element method, Aiaa j., 39, 794-801, (2001)
[17] Liu, M.; Wang, J.B.; Wu, K.-Q., The direct aero-acoustics simulation of flow around a square cylinder using the CE/SE scheme, J. algorithms comput. technol., 1, 525-537, (2007)
[18] Loh, C.Y.; Zaman, K.B.M.Q., Numerical investigation of transonic resonance with a convergent-divergent nozzle, Aiaa j., 40, 2393-2401, (2002) · Zbl 1008.76501
[19] Martí, J.M^{a}.; Müller, E.; Font, J.A.; Ibáñez, J.Ma., Morphology and dynamics of highly supersonic relativistic jets, Astrophys. J., 448, L105-L108, (1995)
[20] Martí, J.M^{a}.; Müller, E., Extension of the piecewise parabolic method to one-dimensional relativistic hydrodynamics, J. comput. phys., 123, 1-14, (1996) · Zbl 0839.76056
[21] Martí, J.M^{a}.; Müller, E., Numerical hydrodynamics in special relativity, Living rev. relativity, 2, 1-101, (1999)
[22] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408-448, (1990) · Zbl 0697.65068
[23] Piran, T.; Shemi, A.; Narayan, R., Hydrodynamics of relativistic fireballs, Mon. not. R. astron. soc., 263, 861-867, (1993)
[24] Qamar, S.; Warnecke, G., A high order kinetic flux-splitting method for the special relativistic hydrodynamics, Ijcm, 2, 49-74, (2005) · Zbl 1189.76371
[25] Qamar, S.; Warnecke, G., Application of space-time CE/SE method to shallow water magnetohydrodynamics equations, J. comput. appl. math., 196, 132-149, (2006) · Zbl 1168.76385
[26] Qamar, S.; Mudasser, S., On the application of a variant CE/SE method for solving two-dimensional MHD equations, J. appl. numer. math., 60, 587-696, (2010) · Zbl 1425.76309
[27] 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
[28] X.Y. Wang, C.L. Chen, Y. Liu, The space-time CE/SE method for solving Maxwell’s equations in time domain, in: 2002 IEEE International Symposium on Antennas and Propagation, National Radio Science Meeting, June 16-21, San Antonio, TX, 2002.
[29] 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
[30] Zhang, Z.C.; Yu, S.T.; Chang, S.C., A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. comput. phys., 175, 168-199, (2002) · Zbl 1168.76339
[31] S.C. Chang, A new approach for constructing highly stable high order CESE schemes, AIAA-2010-543, NASA/TM-2010-216766, 2010.
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