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Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics

Part of: Quantum hydrodynamics and relativistic hydrodynamics Basic methods in fluid mechanics Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  06 July 2017

Jian Zhao  and
Huazhong Tang
Jian Zhao*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Huazhong Tang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China; School of Mathematics and Computational Science, Xiangtan University, Hunan Province, Xiangtan 411105, P.R. China
*
*Corresponding author. Email addresses: everease@163.com (J. Zhao); hztang@math.pku.edu.cn (H. Z. Tang)
*Corresponding author. Email addresses: everease@163.com (J. Zhao); hztang@math.pku.edu.cn (H. Z. Tang)
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Abstract

This paper develops Runge-Kutta PK -based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux will be avoided. The WENO limiter is adaptively implemented via two steps: the “troubled” cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2K+1) replacing the CDG solutions inside the “troubled” cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells. Because the WENO limiter is only employed for finite “troubled” cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

Keywords

Central discontinuous Galerkin methodWENO limiterRunge-Kutta time discretizationrelativistic hydrodynamics

MSC classification

Secondary: 76M10: Finite element methods 76M25: Other numerical methods 76Y05: Quantum hydrodynamics and relativistic hydrodynamics 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 643 - 682
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Balsara, D.S., Riemann solver for relativistic hydrodynamics, J. Comput. Phys., 114:284–297, 1994.CrossRefGoogle Scholar
[2] Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131:267–279, 1997.CrossRefGoogle Scholar
[3] Biswas, R., Devine, K.D., and Flaherty, J.E., Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math., 14:255–283, 1994.CrossRefGoogle Scholar
[4] Cockburn, B., Hu, S.C., and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp., 54:545–581, 1990.Google Scholar
[5] Cockburn, B., Li, F.Y., and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194:588–610, 2004.CrossRefGoogle Scholar
[6] Cockburn, B., Lin, S.Y., and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84:90–113, 1989.CrossRefGoogle Scholar
[7] Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws II: general framework, Math. Comp., 52:411–435, 1989.Google Scholar
[8] Cockburn, B. and Shu, C.-W., The Runge-Kutta local projection P 1-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 25:337–361, 1991.CrossRefGoogle Scholar
[9] Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35:2440–2463, 1998.CrossRefGoogle Scholar
[10] Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141:199–224, 1998.CrossRefGoogle Scholar
[11] Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16:173–261, 2001.CrossRefGoogle Scholar
[12] Dai, W.L. and Woodward, P.R., An iterative Riemann solver for relativistic hydrodynamics, SIAM J. Sci. Comput., 18:982–995, 1997.CrossRefGoogle Scholar
[13] Dolezal, A. and Wong, S.S.M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120:266–277, 1995.CrossRefGoogle Scholar
[14] Donat, R., Font, J.A., Ibáñez, J.M., and Marquina, A., A flux-split algorithm applied to relativistic flows, J. Comput. Phys., 146:58–81, 1998.CrossRefGoogle Scholar
[15] Duncan, G.C. and Hughes, P.A., Simulations of relativistic extragalactic jets, Astrophys. J., 436:L119–L122, 1994.CrossRefGoogle Scholar
[16] Eulderink, F. and Mellema, G., General relativistic hydrodynamics with a Roe solver, Astrophys. J. Suppl. S., 110:587–623, 1995.Google Scholar
[17] Falle, S.A.E.G. and Komissarov, S.S., An upwind numerical scheme for relativistic hydrodynamics with a general equation of state, Mon. Not. R. Astron. Soc., 278:586–602, 1996.CrossRefGoogle Scholar
[18] He, P. and Tang, H.Z., An adaptive moving mesh method for two-dimensional relativistic hydrodynamics, Commun. Comput. Phys., 11:114–146, 2012.CrossRefGoogle Scholar
[19] Hu, C.Q. and Shu, C.-W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21:666–690, 1999.CrossRefGoogle Scholar
[20] Krivodonova, L., Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226:879–896, 2007.CrossRefGoogle Scholar
[21] Kunik, M., Qamar, S., and Warnecke, G., Kinetic schemes for the relativistic gas dynamics, Numer. Math., 97:159–191, 2004.CrossRefGoogle Scholar
[22] Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, Pergaman Press, 2nd edition, 1987.Google Scholar
[23] Lepsky, O., Hu, C.Q., and Shu, C.-W., Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Numer. Math., 33:423–434, 2000.CrossRefGoogle Scholar
[24] Li, F.Y. and Xu, L.W., Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Phys., 231:2655–2675, 2012.CrossRefGoogle Scholar
[25] Li, F.Y., Xu, L.W., and Yakovlev, S., Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, J. Comput. Phys., 230:4828–4847, 2011.CrossRefGoogle Scholar
[26] Li, F.Y. and Yakovlev, S., A central discontinuous Galerkin method for Hamilton-Jacobi equations, J. Sci. Comput., 45:404–428, 2010.CrossRefGoogle Scholar
[27] Liu, Y.J., Central schemes on overlapping cells, J. Comput. Phys., 209:82–104, 2005.CrossRefGoogle Scholar
[28] Liu, Y.J., Shu, C.-W., Tadmor, E., and Zhang, M.P., Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction, SIAM J. Numer. Anal., 45:2442–2467, 2007.CrossRefGoogle Scholar
[29] Liu, Y.J., Shu, C.-W., Tadmor, E., and Zhang, M.P., L 2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 42:593–607, 2008.CrossRefGoogle Scholar
[30] Liu, Y., Shu, C.-W., Tadmor, E. and Zhang, M., Central local discontinuous Galerkin methods on overlapping cells for diffusion equations, ESAIM Math. Model. Numer. Anal., 45:009–1032, 2011.CrossRefGoogle Scholar
[31] Martí, J.M. and Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Relativity, 6:1–100, 2003.CrossRefGoogle ScholarPubMed
[32] May, M.M. and White, R.H., Hydrodynamic calculations of general-relativistic collapse, Phys. Rev., 141:1232–1241, 1966.CrossRefGoogle Scholar
[33] May, M.M. and White, R.H., Stellar dynamics and gravitational collapse, in Methods in Computational Physics, Vol. 7, Astrophysics (Alder, B., Fernbach, S., and Rotenberg, M. eds.), Academic Press, 219–258, 1967.Google Scholar
[34] Mignone, A. and Bodo, G., An HLLC Riemann solver for relativistic flows I. hydrodynamics, Mon. Not. R. Astron. Soc., 364:126–136, 2005.CrossRefGoogle Scholar
[35] Mignone, A., Plewa, T., and Bodo, G., The piecewise parabolic method for multidimensional relativistic fluid dynamics, Astrophys. J. Suppl. S., 160:199–219, 2005.CrossRefGoogle Scholar
[36] Qin, T., Shu, C.-W. and Yang, Y., Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics, J. Comput. Phys., 315:323–347, 2016.CrossRefGoogle Scholar
[37] Qiu, J.X. and Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26:907–929, 2005.CrossRefGoogle Scholar
[38] Reed, W.H. and Hill, T.R., Triangular mesh methods for neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[39] Remacle, J.-F., Flaherty, J.E., and Shephard, M.S., An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev., 45:53–72, 2003.CrossRefGoogle Scholar
[40] Reyna, M.A. and Li, F., Operator bounds and time step conditions for DG and central DG methods, J. Sci. Comput., 62:532–554, 2015.CrossRefGoogle Scholar
[41] Schneider, V., Katscher, U., Rischke, D.H., Waldhauser, B., Maruhn, J.A., and Munz, C.D., New algorithms for ultra-relativistic numerical hydrodynamics, J. Comput. Phys., 105:92–107, 1993.CrossRefGoogle Scholar
[42] Shao, S.H. and Tang, H.Z., Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model, Discrete Contin. Dyn. Syst. Ser. B, 6:623–640, 2006.Google Scholar
[43] Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51(2009), 82–126.CrossRefGoogle Scholar
[44] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77:439–471, 1988.CrossRefGoogle Scholar
[45] Tang, H.Z. and Warnecke, G., A Runge-Kutta discontinuous Galerkin method for the Euler equations, Computers & Fluids, 34:375–398, 2005.CrossRefGoogle Scholar
[46] van Odyck, D. E. A., Review of numerical special relativistic hydrodynamics, Int. J. Numer. Meth. Fluids, 44:861–884, 2004.CrossRefGoogle Scholar
[47] Wilson, J.R., Numerical study of fluid flow in a Kerr space, Astrophys. J., 173:431–438, 1972.CrossRefGoogle Scholar
[48] 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.CrossRefGoogle Scholar
[49] Wu, K.L. and Tang, H.Z., A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM J. Sci. Comput., 38:B458–B489, 2016.CrossRefGoogle Scholar
[50] Wu, K.L. and Tang, H.Z., High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, J. Comput. Phys., 298:539–564, 2015.CrossRefGoogle Scholar
[51] Wu, K.L. and Tang, H.Z., Admissible states and physical constraints preserving numerical schemes for special relativistic magnetohydrodynamics, arXiv:1603.06660, 2016.Google Scholar
[52] Wu, K.L. and Tang, H.Z., Physical-constraint-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state, Astrophys. J. Suppl. Ser., 228(1), 2017, 3.CrossRefGoogle Scholar
[53] Wu, K.L., Yang, Z.C., and Tang, H.Z., A third-order accurate direct Eulerian GRP scheme for one-dimensional relativistic hydrodynamics, East Asian J. Appl. Math., 4:95–131, 2014.CrossRefGoogle Scholar
[54] 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
[55] Yang, J.Y., Chen, M.H., Tsai, I.N., and Chang, J.W., A kinetic beam scheme for relativistic gas dynamics, J. Comput. Phys., 136:19–40, 1997.CrossRefGoogle Scholar
[56] 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
[57] 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.CrossRefGoogle Scholar
[58] Del Zanna, L. and Bucciantini, N., An efficient shock-capturing central-type scheme for multidimensional relativistic flows I: Hydrodynamics, Astron. Astrophys., 390:1177–1186, 2002.CrossRefGoogle Scholar
[59] Zhang, M.P. and Shu, C.-W., An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Math. Models Meth. Appl. Sci., 13:395–413, 2003.CrossRefGoogle Scholar
[60] Zhang, M.P. and Shu, C.-W., An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods, Computers & Fluids, 34:581–592, 2005.CrossRefGoogle Scholar
[61] Zhao, J., He, P., and Tang, H.Z., Steger-Warming flux vector splitting method for special relativistic hydrodynamics, Math. Meth. Appl. Sci., 37:1003–1018, 2014.CrossRefGoogle Scholar
[62] Zhao, J. and Tang, H.Z., Runge-Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 24:138–168, 2013.CrossRefGoogle Scholar
[63] Zhao, J. and Tang, H.Z., Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics, arXiv: 1610.03404, 2016.Google Scholar
[64] Zhu, J. and Qiu, J.X., Runge-Kutta discontinuous Galerkin method using WENO-type limiters: three-dimensional unstructured meshes, Commun. Comput. Phys., 11:985–1005, 2012.CrossRefGoogle Scholar
[65] Zhu, J., Qiu, J.X., Shu, C.-W., and Dumbser, M., Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes, J. Comput. Phys., 227:4330–4353, 2008.CrossRefGoogle Scholar