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

  • Jian Zhao (a1) and Huazhong Tang (a2)
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.

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Corresponding author
*Corresponding author. Email addresses: everease@163.com (J. Zhao); hztang@math.pku.edu.cn (H. Z. Tang)
References
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[1] Balsara, D.S., Riemann solver for relativistic hydrodynamics, J. Comput. Phys., 114:284297, 1994.
[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., 131:267279, 1997.
[3] Biswas, R., Devine, K.D., and Flaherty, J.E., Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math., 14:255283, 1994.
[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., 54:545581, 1990.
[5] Cockburn, B., Li, F.Y., and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194:588610, 2004.
[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., 84:90113, 1989.
[7] Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws II: general framework, Math. Comp., 52:411435, 1989.
[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., 25:337361, 1991.
[9] Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35:24402463, 1998.
[10] Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141:199224, 1998.
[11] Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16:173261, 2001.
[12] Dai, W.L. and Woodward, P.R., An iterative Riemann solver for relativistic hydrodynamics, SIAM J. Sci. Comput., 18:982995, 1997.
[13] Dolezal, A. and Wong, S.S.M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120:266277, 1995.
[14] Donat, R., Font, J.A., Ibáñez, J.M., and Marquina, A., A flux-split algorithm applied to relativistic flows, J. Comput. Phys., 146:5881, 1998.
[15] Duncan, G.C. and Hughes, P.A., Simulations of relativistic extragalactic jets, Astrophys. J., 436:L119L122, 1994.
[16] Eulderink, F. and Mellema, G., General relativistic hydrodynamics with a Roe solver, Astrophys. J. Suppl. S., 110:587623, 1995.
[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., 278:586602, 1996.
[18] He, P. and Tang, H.Z., An adaptive moving mesh method for two-dimensional relativistic hydrodynamics, Commun. Comput. Phys., 11:114146, 2012.
[19] Hu, C.Q. and Shu, C.-W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21:666690, 1999.
[20] Krivodonova, L., Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226:879896, 2007.
[21] Kunik, M., Qamar, S., and Warnecke, G., Kinetic schemes for the relativistic gas dynamics, Numer. Math., 97:159191, 2004.
[22] Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, Pergaman Press, 2nd edition, 1987.
[23] Lepsky, O., Hu, C.Q., and Shu, C.-W., Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Numer. Math., 33:423434, 2000.
[24] Li, F.Y. and Xu, L.W., Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Phys., 231:26552675, 2012.
[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., 230:48284847, 2011.
[26] Li, F.Y. and Yakovlev, S., A central discontinuous Galerkin method for Hamilton-Jacobi equations, J. Sci. Comput., 45:404428, 2010.
[27] Liu, Y.J., Central schemes on overlapping cells, J. Comput. Phys., 209:82104, 2005.
[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., 45:24422467, 2007.
[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., 42:593607, 2008.
[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., 45:0091032, 2011.
[31] Martí, J.M. and Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Relativity, 6:1100, 2003.
[32] May, M.M. and White, R.H., Hydrodynamic calculations of general-relativistic collapse, Phys. Rev., 141:12321241, 1966.
[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, 219258, 1967.
[34] Mignone, A. and Bodo, G., An HLLC Riemann solver for relativistic flows I. hydrodynamics, Mon. Not. R. Astron. Soc., 364:126136, 2005.
[35] Mignone, A., Plewa, T., and Bodo, G., The piecewise parabolic method for multidimensional relativistic fluid dynamics, Astrophys. J. Suppl. S., 160:199219, 2005.
[36] Qin, T., Shu, C.-W. and Yang, Y., Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics, J. Comput. Phys., 315:323347, 2016.
[37] Qiu, J.X. and Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26:907929, 2005.
[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.
[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., 45:5372, 2003.
[40] Reyna, M.A. and Li, F., Operator bounds and time step conditions for DG and central DG methods, J. Sci. Comput., 62:532554, 2015.
[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., 105:92107, 1993.
[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, 6:623640, 2006.
[43] Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51(2009), 82126.
[44] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77:439471, 1988.
[45] Tang, H.Z. and Warnecke, G., A Runge-Kutta discontinuous Galerkin method for the Euler equations, Computers & Fluids, 34:375398, 2005.
[46] van Odyck, D. E. A., Review of numerical special relativistic hydrodynamics, Int. J. Numer. Meth. Fluids, 44:861884, 2004.
[47] Wilson, J.R., Numerical study of fluid flow in a Kerr space, Astrophys. J., 173:431438, 1972.
[48] Wu, K.L. and Tang, H.Z., Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics, J. Comput. Phys., 256:277307, 2014.
[49] Wu, K.L. and Tang, H.Z., A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM J. Sci. Comput., 38:B458B489, 2016.
[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., 298:539564, 2015.
[51] Wu, K.L. and Tang, H.Z., Admissible states and physical constraints preserving numerical schemes for special relativistic magnetohydrodynamics, arXiv:1603.06660, 2016.
[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.
[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., 4:95131, 2014.
[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., 264:177208, 2014.
[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., 136:1940, 1997.
[56] Yang, Z.C., He, P., and Tang, H.Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: one-dimensional case, J. Comput. Phys., 230:79647987, 2011.
[57] Yang, Z.C. and Tang, H.Z., A direct Eulerian GRP scheme for relativistic hydrodynamics: two-dimensional case, J. Comput. Phys., 231:21162139, 2012.
[58] Del Zanna, L. and Bucciantini, N., An efficient shock-capturing central-type scheme for multidimensional relativistic flows I: Hydrodynamics, Astron. Astrophys., 390:11771186, 2002.
[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., 13:395413, 2003.
[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, 34:581592, 2005.
[61] Zhao, J., He, P., and Tang, H.Z., Steger-Warming flux vector splitting method for special relativistic hydrodynamics, Math. Meth. Appl. Sci., 37:10031018, 2014.
[62] Zhao, J. and Tang, H.Z., Runge-Kutta discontinuous Galerkin methods with WENO limiter for the special relativistic hydrodynamics, J. Comput. Phys., 24:138168, 2013.
[63] Zhao, J. and Tang, H.Z., Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics, arXiv: 1610.03404, 2016.
[64] Zhu, J. and Qiu, J.X., Runge-Kutta discontinuous Galerkin method using WENO-type limiters: three-dimensional unstructured meshes, Commun. Comput. Phys., 11:9851005, 2012.
[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., 227:43304353, 2008.
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