Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T19:56:06.085Z Has data issue: false hasContentIssue false
  • English
  • Français

A Simple Implementation of the Semi-Lagrangian Level-Set Method

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Two-phase and multiphase flows

Published online by Cambridge University Press:  11 October 2016

Weidong Shi ,
Jian-Jun Xu  and
Shi Shu
Weidong Shi*
Affiliation:
School of Mathematical and Computational Sciences, Xiangtan University, Xiangtan, Hunan 411105, China Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China
Jian-Jun Xu*
Affiliation:
Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China
Shi Shu*
Affiliation:
School of Mathematical and Computational Sciences, Xiangtan University, Xiangtan, Hunan 411105, China
*
*Corresponding author. Email:weidongshi123@xtu.edu.cn (W. Shi), xujianjun@cigit.ac.cn (J. Xu), shushi@xtu.edu.cn (S. Shu)
*Corresponding author. Email:weidongshi123@xtu.edu.cn (W. Shi), xujianjun@cigit.ac.cn (J. Xu), shushi@xtu.edu.cn (S. Shu)
*Corresponding author. Email:weidongshi123@xtu.edu.cn (W. Shi), xujianjun@cigit.ac.cn (J. Xu), shushi@xtu.edu.cn (S. Shu)
Get access

Abstract

Semi-Lagrangian (S-L) methods have no CFL stability constraint, and are more stable than the Eulerian methods. In the literature, the S-L method for the level-set re-initialization equation was complicated, which may be unnecessary. Since the re-initialization procedure is auxiliary, we propose to use the first-order S-L scheme coupled with a projection technique to improve the accuracy at the grid points just adjacent to the interface. Standard second-order S-L method is used for evolving the level-set convection equation. The implementation is simple, including on the block-structured adaptive mesh. The efficiency of the S-L method is demonstrated by extensive numerical examples including passive convection of interfaces with corners/kinks/large deformation under given velocity fields, a geometrical flow with topological changes, simulations of bubble/ droplet dynamics in incompressible two-phase flows. In terms of accuracy it is comparable to the other existing methods.

Keywords

Semi-Lagrangian methodlevel-set methodinterface motiontwo-phase flowbubble/ droplet dynamicsblock-structured adaptive mesh.

MSC classification

Secondary: 65M06: Finite difference methods 65M20: Method of lines 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 104 - 124
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.)

References

[1] Berger, M. J. and Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), pp. 484–512.Google Scholar
[2] Berger, M. J. and Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE Trans. Syst. Man. Cybern, 21 (1991), pp. 1278–1286.Google Scholar
[3] Courant, R., Issacson, E. and Rees, M., On the solution of nonlinear hyperbolic differential equations by finite difference, Commun. Pure Appl. Math., 5 (1952), pp. 243–255.Google Scholar
[4] Dupont, T. F. and Liu, Y., Back and forth error compensation and corrections methods for removing errors induced by uneven gradients of the level set function, J. Comput. Phys., 190 (2003), pp. 311–324.Google Scholar
[5] Dupont, T. F. and Liu, Y., Back and forth error compensation and correction methods for semi-Lagrangian schemes with applications to level set interface computations, Math. Comput., 76 (2007), pp. 647–668.Google Scholar
[6] Enright, D., Fedkiw, R., Ferziger, J. and Mitchell, I., A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183 (2002), pp. 83–116.Google Scholar
[7] Enright, D., Losasso, F. and Fedkiw, R., A fast and accurate Semi-Lagrangian particle level set method, Comput. Structures, 83 (2005), pp. 479–490.Google Scholar
[8] Jiang, G.-S. and Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126.Google Scholar
[9] Leveque, R., Finite difference methods for ordinary and partial differential equations, SIAM, 2007.Google Scholar
[10] Min, C. and Gibou, F., A second order accurate level set method on non-graded adaptive cartesian grids, J. Comput. Phys., 225 (2007), pp. 300–321.Google Scholar
[11] Osher, S. and Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12.Google Scholar
[12] Osher, S. and Merriman, B., The Wulff shape as the asymptotic limit of a growing crystal interface, Asian J. Math., 1 (1997), 560.Google Scholar
[13] Peng, D., Merriman, B., Osher, S., Zhao, H. and Kang, M., A PDE-based fast local level set method, J. Comput. Phys., 155 (1999), 410.Google Scholar
[14] Selle, A., Fedkiw, R., Kim, B., Liu, Y. and Rossignac, J., An unconditionally stable MacCormack method, J. Sci. Comput., 35 (2008), pp. 350–371.Google Scholar
[15] Shu, C.-W., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, Springer-Verlag, 1998.Google Scholar
[16] Staniforth, A. and Cote, J., Semi-Lagrangian schemes for atmospheric models–a review, Monthly Weather Rev., 119 (1991), pp. 2206–2223.Google Scholar
[17] Strain, J., Semi-Lagrangian methods for level set equations, J. Comput. Phys., 151 (1999), 498.Google Scholar
[18] Strain, J., Tree methods for moving interfaces, J. Comput. Phys., 151 (1999), 616.Google Scholar
[19] Strain, J., A fast modular semi-lagrangian method for moving interfaces, J. Comput. Phys., 161 (2000), pp. 512–536.Google Scholar
[20] Strain, J., A fast semi-lagrangian contouring method for moving interfaces, J. Comput. Phys., 170 (2001), pp. 373–394.Google Scholar
[21] Strain, J., Fast tree-based redistancing for level set computations, J. Comput. Phys., 152 (1999), pp. 664–686.Google Scholar
[22] Sussman, M., Smereka, P. and Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), pp. 146–159.Google Scholar
[23] Wang, Y., Simakhina, S. and Sussman, M., A hybrid level set-volume constraint method for incompressible two-phase flow, J. Comput. Phys., 231 (2012), pp. 6438–6407.Google Scholar
[24] Xiu, D. and Karniadakis, G. E., A semi-Lagrangian high-order method for Navier-Stokes equations, J. Comput. Phys., 172 (2001), 658.CrossRefGoogle Scholar
[25] Xiu, D., Sherwin, S. J., Dong, S. and Karniadakis, G. E., Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows, J. Sci. Comput., 25 (2005), pp. 323–346.Google Scholar
[26] Xu, J.-J., Yang, Y. and Lowengrub, J., A level-set continuum method for two-phase flows with insoluble surfactant, J. Comput. Phys., 231 (2012), pp. 5897–5909.Google Scholar
[27] Xu, J.-J. and Ren, W., A level-set method for two-phase flows with moving contact line and insoluble surfactant, J. Comput. Phys., 263 (2014), pp. 71–90.Google Scholar
[28] Zalesak, S. T., Full multidimensional flux-corrected transport, J. Comput. Phys., 31 (1979), pp. 335–362.Google Scholar
[29] Zerroukat, M., Wood, N. and Staniforth, A., Application of the parabolic spline method to a multi-dimensional conservative semi-Lagrangian transport scheme, J. Comput. Phys., 225 (2007), pp. 935–948.CrossRefGoogle Scholar