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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
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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

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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