Hostname: page-component-6766d58669-kn6lq Total loading time: 0 Render date: 2026-05-14T22:10:31.236Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient Adaptive Rescaling Scheme for Computing Moving Interface Problems

Part of: Incompressible viscous fluids Fredholm integral equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  07 February 2017

Meng Zhao ,
Wenjun Ying ,
John Lowengrub  and
Shuwang Li
Meng Zhao*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
Wenjun Ying*
Affiliation:
Department of Mathematics, Shanghai JiaoTong University, Shanghai 200240, China
John Lowengrub*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA
Shuwang Li*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
*
*Corresponding author. Email addresses:mzhao8@hawk.iit.edu (M. Zhao), wying@sjtu.edu.cn (W. Ying), lowengrb@math.uci.edu (J. Lowengrub), sli@math.iit.edu (S. Li)
*Corresponding author. Email addresses:mzhao8@hawk.iit.edu (M. Zhao), wying@sjtu.edu.cn (W. Ying), lowengrb@math.uci.edu (J. Lowengrub), sli@math.iit.edu (S. Li)
*Corresponding author. Email addresses:mzhao8@hawk.iit.edu (M. Zhao), wying@sjtu.edu.cn (W. Ying), lowengrb@math.uci.edu (J. Lowengrub), sli@math.iit.edu (S. Li)
*Corresponding author. Email addresses:mzhao8@hawk.iit.edu (M. Zhao), wying@sjtu.edu.cn (W. Ying), lowengrb@math.uci.edu (J. Lowengrub), sli@math.iit.edu (S. Li)
Get access

Abstract

In this paper, we present an efficient rescaling scheme for computing the long-time dynamics of expanding interfaces. The idea is to design an adaptive time-space mapping such that in the new time scale, the interfaces evolves logarithmically fast at early growth stage and exponentially fast at later times. The new spatial scale guarantees the conservation of the area/volume enclosed by the interface. Compared with the original rescaling method in [J. Comput. Phys. 225(1) (2007) 554–567], this adaptive scheme dramatically improves the slow evolution at early times when the size of the interface is small. Our results show that the original three-week computation in [J. Comput. Phys. 225(1) (2007) 554–567] can be reproduced in about one day using the adaptive scheme. We then present the largest and most complicated Hele-Shaw simulation up to date.

Keywords

Boundary integralrescaling schememoving boundary problemHele-Shaw

MSC classification

Secondary: 45B05: Fredholm integral equations 76D27: Other free-boundary flows; Hele-Shaw flows 35R37: Moving boundary problems

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 679 - 691
Copyright
Copyright © Global-Science Press 2017 

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.)

Article purchase

Temporarily unavailable

Footnotes

Communicated by Ming-Chih Lai

References

[1] Ambrose, D. M., Siegel, M., and Tlupova, S.. A small-scale decomposition for 3d boundary integral computations with surface tension. J. Comp. Phys., 247:168–191, 2013.Google Scholar
[2] Barua, A., Li, S., Feng, H., Li, X., and Lowengrub, J.. An efficient rescaling algorithm for simulating the evolution of multiple elastically stressed precipitates. Commun. Comput. Phys., 14(04):940–959, 2013.Google Scholar
[3] Ben-Jacob, E. and Garik, P.. The formation of patterns in non-equilibrium growth. Nature, 343:523–530, 1990.Google Scholar
[4] Carvalho, G. D., Gadêlha, H., and Miranda, J. A.. Stationary patterns in centrifugally driven interfacial elastic fingering. Phys. Rev. E, 90(6):063009, 2014.CrossRefGoogle ScholarPubMed
[5] Chang, Y.-C., Hou, T. Y., Merriman, B., and Osher, S.. A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comp. Phys, 124(2):449–464, 1996.Google Scholar
[6] Dai, W. and Shelley, M.. A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble. Phys. Fluids A, 5:2131–2146, 1993.Google Scholar
[7] dos Reis, L. and Miranda, J. A.. Controlling fingering instabilities in nonflat Hele-Shaw geometries. Phys. Rev. E, 84(6):066313, 2011.Google Scholar
[8] Fast, P. and Shelley, M. J.. Moore's law and the Saffman-Taylor instability. J. Comput. Phys., 212(1):1–5, 2006.CrossRefGoogle Scholar
[9] Feng, H., Barua, A., Li, S., and Li, X.. A parallel adaptive treecode algorithm for evolution of elastically stressed solids. Commun. Comput. Phys., 15:365–387, 2014.CrossRefGoogle Scholar
[10] Greenbaum, A., Greengard, L., and McFadden, G. B.. Laplace's equation and the Dirichlet-Nuemann map in multiply connected domains. J. Comput. Phys., 105(2):267–278, 1993.Google Scholar
[11] Greengard, L. and Rokhlin, V. I.. A fast algorithm for particles summations. J. Comp. Phys., 73:325–348, 1987.Google Scholar
[12] He, A., Lowengrub, J. S., and Belmonte, A.. Modeling an elastic fingering instability in a reactive Hele-Shaw flow. SIAM J. Appl. Math, 72(3):842–856, 2012.Google Scholar
[13] Hele-Shaw, H. S.. Flow of water. Nature, 58(1509):520, 1898.Google Scholar
[14] Hou, T. Y., Li, Z., Osher, S., and Zhao, H.. A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comp. Phys, 134(2):236–252, 1997.CrossRefGoogle Scholar
[15] Hou, T. Y., Lowengrub, J. S., and Shelley, M. J.. Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys., 114(2):312–338, 1994.Google Scholar
[16] Hou, T. Y., Lowengrub, J. S., and Shelley, M. J.. Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys., 169(2):302–362, 2001.CrossRefGoogle Scholar
[17] Ju, L., Zhang, J., Zhu, L., and Du, Q.. Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput., 62(2):431–455, 2015.Google Scholar
[18] Langer, J. S.. Instabilities and pattern formation in crystal growth. Reviews of Modern Physics, 52:1–28, 1980.Google Scholar
[19] Langer, J. S.. Dendrites, viscous fingers, and the theory of pattern formation. Science, 243(4895):1150–1156, 1989.Google Scholar
[20] LeVeque, R. J., and Li, Z.. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput., 18(3):709–735, 1997.Google Scholar
[21] Li, S., Lowengrub, J., and Leo, P.. A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell. J. Comput. Phys., 225(1):554–567, 2007.Google Scholar
[22] Li, S., Lowengrub, J. S., Fontana, J., and Palffy-Muhoray, P.. Control of viscous fingering patterns in a radial Hele–Shaw cell. Phys. Rev. Lett., 102:174501, 2009.Google Scholar
[23] Liu, X. and Nie, Q.. Compact integration factor methods for complex domains and adaptive mesh refinement. J. Comp. Phys., 229(16):5692–5706, 2010.CrossRefGoogle ScholarPubMed
[24] Paterson, L.. Radial fingering in a Hele-Shaw cell. Journal of Fluid Mechanics, 113:513–529, 1981.CrossRefGoogle Scholar
[25] Praud, O. and Swinney, H.. Fractal dimension and unscreened angles measured for radial viscous fingering. Phys. Rev. E, 72(1):011406, 2005.Google Scholar
[26] Saad, Y. and Schultz, M. H.. Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856–869, 1986.Google Scholar
[27] Saffman, P. G. and Taylor, G.. The penetration of a fluid into a porous medium or a Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. London Ser. A, 245(1242):312–329, 1958.Google Scholar
[28] Shelley, M. J., Tian, F.-R., and Wlodarski, K.. Hele-Shaw flow and pattern formation in a time-dependent gap. Nonlinearity, 10:1471–1495, 1997.CrossRefGoogle Scholar
[29] Tian, F. R. and Nie, Q.. Singularities in Hele–Shaw flows. SIAM J. Appl. Math., 58(1):34–54, 1998.Google Scholar
[30] Ying, W., Li, S., and Li, X.. A kernel free boundary integral method for moving interface problems. in preparation.Google Scholar
[31] Zhao, M., Belmonte, A., Li, S., Li, X., and Lowengrub, J.. Nonlinear simulations of elastic fingering in a Hele-Shaw cell. J. Comput. Appl. Math., page in press., 2015.Google Scholar