Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-11T22:33:06.539Z Has data issue: false hasContentIssue false
  • English
  • Français

A Third-Order Accurate Direct Eulerian GRP Scheme for One-Dimensional Relativistic Hydrodynamics

Published online by Cambridge University Press:  28 May 2015

Kailiang Wu ,
Zhicheng Yang  and
Huazhong Tang
Kailiang Wu*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR., China
Zhicheng Yang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR., China
Huazhong Tang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR., China
*
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
Get access

Abstract

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.

Keywords

65M0676M1276Y05Godunov-type schemeWENOgeneralised Riemann problemRiemann invariantRankine-Hugoniot jump conditionrelativistic hydrodynamics
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014 , pp. 95 - 131
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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