Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T21:50:27.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Remapping-Free Adaptive GRP Method for Multi-Fluid Flows I: One Dimensional Euler Equations

Published online by Cambridge University Press:  03 June 2015

Jin Qi ,
Yue Wang  and
Jiequan Li
Jin Qi*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Yue Wang*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Jiequan Li*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
*
Email:qi_jin@iapcm.ac.cn
Corresponding author.Email:yue.wang0828@gmail.com
Email:jiequan@bnu.edu.cn
Get access

Abstract

In this paper, a remapping-free adaptive GRP method for one dimensional (1-D) compressible flows is developed. Based on the framework of finite volume method, the 1-D Euler equations are discretized on moving volumes and the resulting numerical fluxes are computed directly by the GRP method. Thus the remapping process in the earlier adaptive GRP algorithm [17,18] is omitted. By adopting a flexible moving mesh strategy, this method could be applied for multi-fluid problems. The interface of two fluids will be kept at the node of computational grids and the GRP solver is extended at the material interfaces of multi-fluid flows accordingly. Some typical numerical tests show competitive performances of the new method, especially for contact discontinuities of one fluid cases and the material interface tracking of multi-fluid cases.

Keywords

76M1276N1576T99The GRP methodmulti-fluid flowsthe Euler equationsthe adaptive mesh method
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1029 - 1044
Copyright
Copyright © Global Science Press Limited 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]Abgrall, R., A reviewof residual distribution schemes for hyperbolic and parabolic problems: the July 2010 state of the art, Commun. Comput. Phys., 11 (2012), 1043–1080.Google Scholar
[2]Abgrall, R. and Karni, S., Computations of compressible multifluids, J. Comput. Phys., 169 (2001), 594–623.Google Scholar
[3]Benson, D., Computations methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng., 99(2-3) (1992), 235–394.Google Scholar
[4]Ben-Artzi, M. and Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55 (1984), 1–32.Google Scholar
[5]Ben-Artzi, M., The generalized Riemann problem for reactive flows, J. Comput. Phys., 81 (1989), 70–101.Google Scholar
[6]Ben-Artzi, M. and Falcovitz, J., An upwind second-order scheme for compressible duct flows, SIAM J. Sci. Stat. Comput., 7 (1986), 744–768.Google Scholar
[7]Ben-Artzi, M. and Falcovitz, J., Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge University Press, 2003.Google Scholar
[8]Ben-Artzi, M. and Li, J., Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math., 106(3) (2007), 369–425.Google Scholar
[9]Ben-Artzi, M., Li, J. and Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 218 (2006), 19–34.Google Scholar
[10]Brackbill, J. and Saltzman, J., Adaptive zoning for singular problems in two-dimensions, J. Comput. Phys., 46 (1982), 342–368.Google Scholar
[11]Brackbill, J., An adaptive grid with directional control, J. Comput. Phys., 108 (1993), 38–50.Google Scholar
[12]Cao, W., Huang, W. and Russell, R., A study of monitor functions for two-dimensional adaptive mesh generation, SIAM J. Sci. Comput., 20 (1999), 1978–1999.Google Scholar
[13]Dam, A. van and Zegeling, P., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), 138–170.Google Scholar
[14]Deng, X., Mao, M., Tu, G., Zhang, H. and Zhang, Y., High-order and high accurate CFD methods and their applications for complex grid problems, Commun. Comput. Phys., 11 (2012), 1081–1102.Google Scholar
[15]Falcovitz, J., Alfandary, G. and Hanoch, G., A 2-D conservation laws scheme for compressible ows with moving boundaries, J. Comput. Phys., 138 (1997), 83–102.Google Scholar
[16]Falcovitz, J. and Birman, A., A singularities tracking conservation laws scheme for compressible duct flows, J. Comput. Phys., 115 (1994), 431–439.Google Scholar
[17]Han, -E., J.Li andH. Tang, An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229 (2010), 1448–1466.Google Scholar
[18]Han, E., Li, J. and Tang, H., 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]He, P. and Tang, H., An adaptive moving mesh method for two-dimensional relativistic hydrodynamics, Commun. Comput. Phys., 11(2012), 114–146.CrossRefGoogle Scholar
[20]Hirt, C., Amsden, A. and Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 135 (1997), 203–216.Google Scholar
[21]Huang, W., Variational mesh adaptation: isotropy and equidistribution, J. Comput. Phys., 174(2) (2001), 903–924.Google Scholar
[22]Di, Y., Li, R., Tang, T. and Zhang, P., Moving mesh finite element methods for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 26 (2005), 1036–1056.Google Scholar
[23]Li, J. and Chen, G., The generalized Riemann problem method for the shallow water equations with bottom topography, Int. J. Numer. Methods Eng., 65 (2006), 834–862.Google Scholar
[24]Li, J., Liu, T. and Sun, Z., Implementation of the GRP scheme for computing radially symmetric compressible fluid flows, J. Comput. Phys., 228 (2009), 5867–5887.Google Scholar
[25]Li, J. and Sun, Z., Remark on the generalized Riemann problem method for compressible fluid flows, J. Comput. Phys., 222(2) (2007), 796–808.CrossRefGoogle Scholar
[26]Maron, M. and Lopez, R., Numerical Analysis, Wadsworth, 1991.Google Scholar
[27]Margolin, L., Introduction to “an arbitrary Lagrangian-Eulerian computing method for all flow speeds”, J. Comput. Phys., 135(2) (1997), 198–202.CrossRefGoogle Scholar
[28]Ni, G., Jiang, S. and Wang, S., A remapping-free, efficient Riemann-solvers based, ALE method for multi-material fluids with general EOS, Comput. Fluids, 71 (2013), 19–27.Google Scholar
[29]Ni, G., Jiang, S. and Xu, K., Remapping-free ALE-type kinetic method for flow computations, J. Comput. Phys., 228 (2009), 3154–3171.Google Scholar
[30]Tang, H. and Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM. J. Numer. Anal., 41(2) (2003), 487–515.Google Scholar
[31]Tang, T., Moving mesh methods for computational fluid dynamics, Contemp. Math., 383 (2005), 185–218.Google Scholar
[32]Teng, Z., Modified equation for adaptive monotone difference schemes and its convergent analysis, Math. Comput., 77 (2008), 1453–1465.Google Scholar
[33]Tian, B., Shen, W., Jiang, S., Wang, S. and Liu, Y., An arbitrary Lagrangian-Eulerian method based on the adaptive Riemann solvers for general equations of state, Int. J. Numer. Mech. Fluids, 59 (2009), 1217–1240.Google Scholar
[34]Tian, B., Shen, W., Jiang, S., Wang, S. and Liu, Y., A Global arbitrary Lagrangian-Eulerian method for stratified Richtmyer-Meshkov instability, Comput. Fluids, 46(1) (2011), 113–121.Google Scholar
[35]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009.Google Scholar
[36]Wang, C., Tang, H. and Liu, T., An adaptive ghost fluid finite volume method for compressible gasCwater simulations, J. Comput. Phys., 227 (2008), 6385–6409.Google Scholar
[37]Winslow, A., Numerical solution of the quasi-linear Poisson equation, J. Comput. Phys., 1 (1967), 149–172.Google Scholar