Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T17:42:55.641Z Has data issue: false hasContentIssue false
  • English
  • Français

An Adaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics

Published online by Cambridge University Press:  20 August 2015

Peng He  and
Huazhong Tang
Peng He*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
Huazhong Tang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
*
Email:hepengpku@gmail.com
Corresponding author.Email:hztang@math.pku.edu.cn
Get access

Abstract

This paper extends the adaptive moving mesh method developed by Tang and Tang [36] to two-dimensional (2D) relativistic hydrodynamic (RHD) equations. The algorithm consists of two “independent” parts: the time evolution of the RHD equations and the (static) mesh iteration redistribution. In the first part, the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations. The second part is an iterative procedure. In each iteration, the mesh points are first redistributed, and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.

Keywords

35L6565M5074S1076L0576N1576Y05Adaptive moving mesh methodfinite volume methodconservative interpolationrelativistic hydrodynamics
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 1 , January 2012 , pp. 114 - 146
Copyright
Copyright © Global Science Press Limited 2012

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]Balsara, D. S., Riemann solver for relativistic hydrodynamics, J. Comput. Phys., 114 (1994), 284–297.Google Scholar
[2]Brackbill, J. U., An adaptive grid with directional control, J. Comput. Phys., 108 (1993), 38–50.Google Scholar
[3]Brackbill, J. U. and Saltzman, J. S., Adaptive zoning for singular problems in two dimensions, J. Comput. Phys., 46 (1982), 342–368.Google Scholar
[4]Budd, C. J., Huang, W. and Russell, R. D., Adaptivity with moving grids, Acta Numer., 18 (2009), 111–241.Google Scholar
[5]Cao, W. M., Huang, W. Z. and Russell, R. D., An r-adaptive finite element method based upon moving mesh PDEs, J. Comput. Phys., 149 (1999), 221–244.Google Scholar
[6]Ceniceros, H. D. and Hou, T. Y., An efficient dynamically adaptive mesh for potentially singular solutions, J. Comput. Phys., 172 (2001), 609–639.Google Scholar
[7]Chen, G. X., Tang, H. Z. and Zhang, P. W., Second-order accurate Godunov scheme for multi-component flows on moving triangular meshes, J. Sci. Comput., 34(1) (2008), 64–86.Google Scholar
[8]Dai, W. and Woodward, P. R., An iterative Riemann solver for relativistic hydrodynamics, SIAM J. Sci. Stat. Comput., 18 (1997), 982–995.Google Scholar
[9]Davis, S. F. and Flaherty, J. E., An adaptive finite element method for initial-boundary value problems for partial differential equations, SIAM J. Sci. Stat. Comput., 3 (1982), 6–27.Google Scholar
[10]Di, Y. N., Li, R., Tang, T. and Zhang, P. W., Moving mesh finite element methods for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 26 (2005), 1036–1056.Google Scholar
[11]Dolezal, A. and Wong, S. S. M., Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 120 (1995), 266–277.Google Scholar
[12]Donat, R., Font, J. A., Ibáñez, J. M. and Marquina, A., A flux-split algorithm applied to relativistic flows, J. Comput. Phys., 146 (1998), 58–81.Google Scholar
[13]Duncan, G. C. and Hughes, P. A., Simulations of relativistic extragalactic jets, Astrophys. J., 436 (1994), L119–L122.Google Scholar
[14]Dvinsky, A. S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys., 95 (1991), 450–476.Google Scholar
[15]Eulderink, F. and Mellema, G., General relativistic hydrodynamics with a Roe solver, Astron. Astrophys., Suppl., 110 (1995), 587–623.Google Scholar
[16]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 (1996), 586–602.Google Scholar
[17]Font, J. A., Ibáñez, J. M., Marquina, A. and Martí, J. M., Multidimensional relativistic hydrodynamics: characteristic fields and modern high-resolution shock-capturing schemes, Astron. Astrophys., 282(1) (1994), 304–314.Google Scholar
[18]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 (2011), 577–606.Google Scholar
[19]Han, J. Q. and Tang, H. Z., An adaptive moving mesh method for multidimensional ideal magnetohydrodynamics, J. Comput. Phys., 220 (2007), 791–812.Google Scholar
[20]Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Butterworth-Heinemann, second edition, 1987.Google Scholar
[21]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.Google Scholar
[22]Li, R., Tang, T. and Zhang, P. W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170 (2001), 562–588.Google Scholar
[23]Li, R., Tang, T. and Zhang, P. W., A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys., 177 (2002), 365–393.CrossRefGoogle Scholar
[24]Li, S. and Petzold, L., Moving mesh methods with upwinding schemes for time-dependent PDEs, J. Comput. Phys., 131 (1997), 368–377.CrossRefGoogle Scholar
[25]Lucas-Serrano, A., Font, J. A., Ibáñez, J. M. and Martí, J. M., Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations, Astron. Astro-phys., 428(2) (2004), 703–715.Google Scholar
[26]Martí, J. M. and Müller, E., The analytical solution of the Riemann problem in relativistic hydrodynamics, J. Fluid Mech., 258(1) (1994), 317–333.CrossRefGoogle Scholar
[27]Martí, J. M. and Müller, E., Numerical hydrodynamics in special relativity, Living Rev. Relativ., 6(7) (2003), 1–100.Google Scholar
[28]Mignone, A. and Bodo, G., An HLLC Riemann solver for relativistic flows I. hydrodynamics, Mon. Not. R. Astron. Soc., 364(1) (2005), 126–136.Google Scholar
[29]Mignone, A., Plewa, T. and Bodo, G., The piecewise parabolic method for multidimensional relativistic fluid dynamics, Astrophys. J., Suppl., 160(1) (2005), 199–219.Google Scholar
[30]Miller, K. and Miller, R. N., Moving finite element I, SIAM J. Numer. Anal., 18 (1981), 1019– 1032.Google Scholar
[31]Ren, W. Q. and Wang, X. P., An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys., 159 (2000), 246–273.Google Scholar
[32]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 (1993), 92– 107.Google Scholar
[33]Stockie, J. M., Mackenzie, J.A. and Russell, R.D., A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 22 (2001), 1791–1813.CrossRefGoogle Scholar
[34]Tan, Z. J., Zhang, Z. R., Huang, Y.Q. and Tang, T., Moving mesh methods with locally varying time steps, J. Comput. Phys., 200 (2004), 347–367.Google Scholar
[35]Tang, H. Z., A moving mesh method for the Euler flow calculations using a directional monitor function, Commun. Comput. Phys., 1 (2006), 656–676.Google Scholar
[36]Tang, H. Z. and Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservations laws, SIAM J. Numer. Anal., 41 (2003), 487–515.Google Scholar
[37]Tang, H. Z., Tang, T. and Zhang, P. W., An adaptive mesh redistribution method for nonliear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phys., 188 (2003), 543–572.CrossRefGoogle Scholar
[38]Tang, T., Moving mesh methods for computational fluid dynamics, Contemp. Math., 383 (2005), 141–174.Google Scholar
[39]Dam, A. van and Zegeling, P.A., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), 138–170.Google Scholar
[40]Leer, B. van, Towards the ultimate conservative difference scheme III. upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263–299.Google Scholar
[41]Wang, D. S. and Wang, X. P., Three dimensional adaptive method based on iterative grid redistribution, J. Comput. Phys., 199 (2004), 423–436.Google Scholar
[42]Wilson, J. R., Numerical study of fluid flow in a Kerr space, Astrophys. J., 173 (1972), 431–438.Google Scholar
[43]Wilson, J. R., A numerical method for relativistic hydrodynamics, in Smarr, L. L., editor, Sources of Gravitational Radiation, 423-446, Cambridge University Press, 1979.Google Scholar
[44]Winslow, A. M., Numerical solution of the quasi-linear Poisson equation in a nonuniform triangle mesh, J. Comput. Phys., 1 (1966), 149–172.Google Scholar
[45]Zanna, L. D. and Bucciantini, N., An efficient shock-capturing central-type scheme for multidimensional relativistic flows, I. hydrodynamics, Astron. Astrophys., 390(3) (2002), 1177– 1186.Google Scholar
[46]Zhang, W. Q. and MacFadyen, A. I., RAM: a relativistic adaptive mesh refinement hydrodynamics code, Astrophys. J. Suppl., 164(1) (2006), 255–279.Google Scholar