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An Efficient Adaptive Rescaling Scheme for Computing Moving Interface Problems

Part of: Miscellaneous topics - Partial differential equations Incompressible viscous fluids Fredholm integral 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)
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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
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 679 - 691
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Ming-Chih Lai

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