JCM
Journal of Computational Mathematics ›› 2020, Vol. 38 ›› Issue (5): 768-796.DOI: 10.4208/jcm.1905-m2018-0020

Previous Articles     Next Articles

TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS

Yuhuan Yuan, Huazhong Tang   

  1. Center for Applied Physics and Technology, HEDPS, and LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
  • Received:2018-02-08 Revised:2019-03-03 Online:2020-09-15 Published:2021-03-11
  • Supported by:
    The authors were partially supported by the Special Project on Highperformance Computing under the National Key R&D Program (No. 2016YFB0200603), Science Challenge Project (No. JCKY2016212A502), and the National Natural Science Foundation of China (Nos. 91630310 & 11421101).
PDF ( 4 ) Cited

Yuhuan Yuan, Huazhong Tang. TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS[J]. Journal of Computational Mathematics, 2020, 38(5): 768-796.

This paper studies the two-stage fourth-order accurate time discretization[J.Q. Li and Z.F. Du, SIAM J. Sci. Comput., 38 (2016)] and its application to the special relativistic hydrodynamical equations. Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed. With the aid of the direct Eulerian GRP (generalized Riemann problem) methods and the analytical resolution of the local "quasi 1D" GRP, the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations. Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.
Key words: Time discretization, Shock-capturing scheme, GRP method, Relativistic hydrodynamics, Hyperbolic conservation laws

CLC Number: 

[1] M. Anderson, E.W. Hirschmann, S.L. Liebling, and D. Neilsen, Relativistic MHD with adaptive mesh refinement, Class. Quantum Grav., 23(2006), 6503-6524.
[2] D.S. Balsara, Riemann solver for relativistic hydrodynamics, J. Comput. Phys., 114(1994), 284-297.
[3] M. Ben-Artzi and J. Q. Li, Hyperbolic conservation laws:Riemann invariants and the generalized Riemann problem, Numer. Math., 106(2007), 369-425.
[4] Y.P. Chen, Y.Y. Kuang, and H.Z. Tang, Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations, J. Comput. Phys., 349(2017), 300-327.
[5] L. Del Zanna and N. Bucciantini, An efficient shock-capturing central-type scheme for multidimensional relativistic flows I:Hydrodynamics, Astron. Astrophys., 390(2002), 1177-1186.
[6] L. Del Zanna, N. Bucciantini, and P. Londrillo, An efficient shock-capturing central-type scheme for multidimensional relativistic flows I. Hydrodynamics, Astron. Astrophys., 390(2002), 11771186.
[7] R. Donat, J.A. Font, J.M. Ibáñez, and A. Marquina, A flux-split algorithm applied to relativistic flows, J. Comput. Phys., 146(1998), 58-81.
[8] EE Han, J.Q. Li, and H.Z. Tang, An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229(2010), 1448-1466.
[9] EE 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(2011), 577-606.
[10] P. He and H.Z. Tang, An adaptive moving mesh method for two-dimensional relativistic hydrodynamics, Commun. Comput. Phys., 11(2012), 114-146.
[11] P. He and H.Z. Tang, An adaptive moving mesh method for two-dimensional relativistic magnetohydrodynamics, Computers & Fluids, 60(2012), 1-20.
[12] V. Honkkila and P. Janhunen, HLLC solver for ideal relativistic MHD, J. Comput. Phys., 223(2007), 643-656.
[13] B. van der Holst, R. Keppens, and Z. Meliani, A multidimensional grid-adaptive relativistic magnetofluid code, Comput. Phys. Comm., 179(2008), 617-627.
[14] G.S. Jiang and C.-W. Shu, Efficient implementation of Weighted ENO schemes, J. Comput. Phys., 126(1996), 202-228.
[15] A.V. Koldoba, O.A. Kuznetsov, and G.V. Ustyugova, An approximate Riemann solver for relativistic magnetohydrodynamics, Mon. Not. R. Astron. Soc., 333(2002), 932-942.
[16] S.S. Komissarov, A Godunov-type scheme for relativistic magnetohydrodynamics, Mon. Not. R. Astron. Soc., 303(1999), 343-366.
[17] B.J. Lee, E.F. Toro, C.E. Castro, and N. Nikiforakis, Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246(2013), 165-183.
[18] J.Q. Li and Z.F. Du, A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. Hyperbolic conservation laws, SIAM J. Sci. Comput., 38(2016), A3046-A3069.
[19] M.M. May and R.H.White, Hydrodynamic calculations of general-relativistic collapse, Phys. Rev., 141(1966), 1232-1241.
[20] M.M. May and R.H. White, Stellar dynamics and gravitational collapse, in Methods in Computational Physics, Vol. 7, Astrophysics (B. Alder, S. Fernbach, and M. Rotenberg edited), Academic Press, 1967, 219-258.
[21] A. Mignone and G. Bodo, An HLLC Riemann solver for relativistic flows I. Hydrodynamics, Mon. Not. R. Astron. Soc., 364(2005), 126-136.
[22] L. Pan, K. Xu, Q.B. Li, and J.Q. Li, An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations, J. Comput. Phys., 326(2016), 197-221.
[23] S. Qamar and G. Warnecke, A high-order kinetic flux-splitting method for the relativistic magnetohydrodynamics, J. Comput. Phys., 205(2005), 182-204.
[24] T. Qin, C.-W. Shu and Y. Yang, Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics, J. Comput. Phys., 315(2016), 323-347.
[25] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition, Springer, Berlin, 2009.
[26] J.R. Wilson, Numerical study of fluid flow in a Kerr space, Astrophys. J., 173(1972), 431-438.
[27] J.R. Wilson, A numerical method for relativistic hydrodynamics, in Sources of Gravitational Radiation, L.L. Smarr edited, Cambridge University Press, 1979, 423-446.
[28] K. Wu, Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics, Phys. Rev. D, 95(2017), 103001.
[29] K.L. Wu and H.Z. Tang, Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics, J. Comput. Phys., 256(2014), 277-307.
[30] K.L. Wu and H.Z. Tang, A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics, SIAM J. Sci. Comput., 38(2016), B458-B489.
[31] K.L. Wu and H.Z. Tang, High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, J. Comput. Phys., 298(2015), 539-564.
[32] K.L. Wu and H.Z. Tang, Admissible states and physical constraints preserving numerical schemes for special relativistic magnetohydrodynamics, Math. Models and Meth. in Appl. Sci., 27(2017), 1871-1928.
[33] K.L. Wu and H.Z. Tang, Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state, Astrophys. J. Suppl. series, 228(2017), 3.
[34] K.L. Wu and H.Z. Tang, On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state, Z. Angew. Math. Phys., 69(2018), 84.
[35] K.L. Wu, Z.C. Yang, and H.Z. Tang, A third-order accurate direct Eulerian GRP scheme for one-dimensional relativistic hydrodynamics, East Asian J. Appl. Math., 4(2014), 95-131.
[36] 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(2014), 177-208.
[37] Z.C. Yang, P. He, and H.Z. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics:One-dimensional case, J. Comput. Phys., 230(2011), 7964-7987.
[38] Z.C. Yang and H.Z. Tang, A direct Eulerian GRP scheme for relativistic hydrodynamics:Twodimensional case, J. Comput. Phys., 231(2012), 2116-2139.
[39] J. Zhao and H.Z. Tang, Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics, J. Comput. Phys., 343(2017), 33-72.
[40] J. Zhao and H.Z. Tang, Runge-Kutta central discontinuous Galerkin methods for the special relativistic hydrodynamics, Commun. Comput. Phys., 22(2017), 643-682.
[1] Changna Lu, Jianxian Qiu, Ruyun Wang. A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS [J]. Journal of Computational Mathematics, 2010, 28(6): 807-825.
[2] Erich Carelli and Andreas Prohl. A Note on Pressure Approximation of First and Higher Order Projection Schemes for the Nonstationary Incompressible Navier-Stokes Equations [J]. Journal of Computational Mathematics, 2009, 27(2-3): 338-347.
[3] Jianxian Qiu. HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS [J]. Journal of Computational Mathematics, 2007, 25(2): 131-144.
[4] Hua-zhong Tang,Gerald Warnecke. HIGH RESOLUTION SCHEMES FOR CONSERVATION LAWS AND CONVECTION-DIFFUSION EQUATIONS WITH VARYING TIME AND SPACE GRIDS [J]. Journal of Computational Mathematics, 2006, 24(2): 121-140.
[5] Hua Zhong TANG. ON THE CENTRAL RELAXING SCHEME Ⅱ: SYSTEMS OF HYPERBOLIC CONSERVATION LAWS [J]. Journal of Computational Mathematics, 2001, 19(6): 571-582.
[6] Hua Zhong TANG. ON THE CENTRAL RELAXING SCHEMES I:SINGLE CONSERVATION LAWS [J]. Journal of Computational Mathematics, 2000, 18(3): 313-324.
[7] Hua Zhong TANG, Hua Mo WU. ON A CELL ENTROPY INEQUALITY OF THE RELAXING SCHEMES FOR SCALAR CONSERVATION LAWS [J]. Journal of Computational Mathematics, 2000, 18(1): 69-74.
[8] Hua Zhong TANG,Ning ZHAO. A estimate of the rate of entropy dissipation of high resolution MUSCL type Godunov schemes [J]. Journal of Computational Mathematics, 1999, 17(4): 369-378.
Viewed
Full text


Abstract

Journal of Computational Mathematics
Contact Us
Editor-in-Chief: Zhiming Chen
Electronic: 1991-7139
ISSN Print: 0254-9409
Issues: 6 / year
Institute of Computational Mathematics and Scientific/ Engineering Computing
Chinese Academy of Sciences
Zhongguancun East Rd. No.55
Beijing 100190, China
Phone: 86-10-82541418
Email: jcm@lsec.cc.ac.cn
×

扫码分享

The journal is sponsored by ICMSEC
Copyrights © ICMSEC