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A third-order accurate direct Eulerian GRP scheme for one-dimensional relativistic hydrodynamics. (English) Zbl 1309.76145
Summary: A third-order accurate direct Eulerian generalised Riemann problem (GRP) scheme is derived for the one-dimensional special relativistic hydrodynamical equations. In our GRP scheme, the higher-order WENO initial reconstruction is employed, and the local GRPs in the Eulerian formulation are directly and analytically resolved to third-order accuracy via the Riemann invariants and Rankine-Hugoniot jump conditions, to get the approximate states in numerical fluxes. Unlike a previous second-order accurate GRP scheme, for the non-sonic case the limiting values of the second-order time derivatives of the fluid variables at the singular point are also needed for the calculation of the approximate states; while for the sonic case, special attention is paid because the calculation of the second-order time derivatives at the sonic point is difficult. Several numerical examples are given to demonstrate the accuracy and effectiveness of our GRP scheme.

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
Full Text: DOI
[1] [1] M. Ben-Artzi and J. Falcovitz, A second-order Godunov-type scheme for compressible fluid dy- namics, J. Comput. Phys.55, 1–32 (1984). · Zbl 0535.76070
[2] [2] M. Ben-Artzi and J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge University Press, 2003. · Zbl 1017.76001
[3] [3] M. Ben-Artzi and J.Q. Li, Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math.,106, 369–425 (2007). · Zbl 1123.65082
[4] [4] M. Ben-Artzi, J.Q. Li, and G. Warnecke, A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys.218, 19–43 (2006).
[5] [5] J.A. Font, Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relativity11, 7 (2008).
[6] [6] E. Han, J.Q. Li, and H.Z. Tang, An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys.229, 1448–1466 (2010). · Zbl 1329.76205
[7] [7] E. Han, J.Q. Li, and H.Z. Tang, Accuracy of the adaptive GRP scheme and the simulation of 2-D Riemann problems for compressible Euler equations, Commun. Comput. Phys.10, 577–606 (2011). · Zbl 1373.76130
[8] [8] J.Q. Li and G.X. Chen, The generalized Riemann problem method for the shallow water equa- tions with bottom topography, Int. J. Numer. Meth. in Eng.65, 834–862 (2006). · Zbl 1178.76249
[9] [9] J.Q. Li, Q.B. Li, and K. Xu, Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations, J. Comput. Phys.230, 5080–5099 (2011). · Zbl 1416.76250
[10] [10] J. Luo and K. Xu, A high-order multidimensional gas-kinetic scheme for hydrodynamic equa- tions, Sci. China Tech. Sci.56, 2370–2384 (2013).
[11] [11] J.M. Martí and E. Müller, The analytical solution of the Riemann problem in relativistic hydro- dynamics, J. Fluid Mech.258, 317–333 (1994). Third-Order Accurate GRP Scheme for 1D RHD131
[12] [12] J.M. Martí and E. Müller, Extension of the piecewise parabolic method to one dimensional relativistic hydrodynamics, J. Comput. Phys.123, 1–14 (1996).
[13] [13] J.M. Martí and E. Müller, Numerical hydrodynamics in special relativity, Living Rev. Relativity 6, 7 (2003).
[14] [14] J.Z. Qian, J.Q. Li, and S.H. Wang, The generalized Riemann problems for compressible fluid flows: Towards high order, J. Comput. Phys.259, 358–389 (2014). · Zbl 1349.76379
[15] [15] H.Z. Tang and T. Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic con- servation laws, SIAM J. Numer. Anal.41, 487–515 (2003).
[16] [16] J.R. Wilson, Numerical study of fluid flow in a Kerr space, Astrophys. J. 173, 431–438 (1972).
[17] [17] J.R. Wilson and G.J. Mathews, Relativistic Numerical Hydrodynamics, Cambridge University Press, 2003.
[18] [18] K.L. Wu and H.Z. Tang, Finite volume local evolution galerkin method for two-dimensional relativistic hydrodynamics, J. Comput. Phys.256, 277–307 (2014). · Zbl 1349.76408
[19] [19] K.L. Wu, Z.C. Yang, and H.Z. Tang, A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics, J. Comput. Phys.264, 177-208 (2014). · Zbl 1349.76551
[20] [20] J.P. Xu, M. Luo, J.C Hu, S.Z. Wang, B. Qi, and Z.G. Qiao, A direct Eulerian GRP Scheme for the prediction of gas-liquid two-phase flow in HTHP transient wells, Abs. Appl. Anal.2013, 171732(2013). · Zbl 1426.76580
[21] [21] Z.C. Yang, P. He, and H.Z. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics: One-dimensional case, J. Comput. Phys.230, 7964–7987 (2011). · Zbl 1408.76597
[22] [22] Z.C. Yang and H.Z. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics: Two- dimensional case, J. Comput. Phys.231, 2116–2139 (2012). · Zbl 1408.76598
[23] [23] L.D. Zanna and N. Bucciantini, An efficient shock-capturing central-type scheme for multidi- mensional relativistic flows, I: hydrodynamics, Astron. Astrophys.390, 1177–1186 (2002). · Zbl 1209.76022
[24] [24] J. Zhao and H.Z. Tang, Runge-Kutta discontinuous Galerkin methods with WENO limiters for the special relativistic hydrodynamics, J. Comput. Phys.242, 138–168 (2013). · Zbl 1314.76035
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