Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-10T18:40:17.101Z Has data issue: false hasContentIssue false
  • English
  • Français

An Adaptive Semi-Lagrangian Level-Set Method for Convection-Diffusion Equations on Evolving Interfaces

Part of: Numerical analysis: Ordinary differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  28 November 2017

Weidong Shi ,
Jianjun 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
Jianjun 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. D. Shi), xujianjun@cigit.ac.cn (J. J. Xu), shushi@xtu.edu.cn (S. Shu)
*Corresponding author. Email:weidongshi123@xtu.edu.cn (W. D. Shi), xujianjun@cigit.ac.cn (J. J. Xu), shushi@xtu.edu.cn (S. Shu)
*Corresponding author. Email:weidongshi123@xtu.edu.cn (W. D. Shi), xujianjun@cigit.ac.cn (J. J. Xu), shushi@xtu.edu.cn (S. Shu)
Get access

Abstract

A new Semi-Lagrangian scheme is proposed to discretize the surface convection-diffusion equation. The other involved equations including the the level-set convection equation, the re-initialization equation and the extension equation are also solved by S-L schemes. The S-L method removes both the CFL condition and the stiffness caused by the surface Laplacian, allowing larger time step than the Eulerian method. The method is extended to the block-structured adaptive mesh. Numerical examples are given to demonstrate the efficiency of the S-L method.

Keywords

Convection-diffusion equationsemi-Lagrangian methodlevel-set methodblock-structured adaptive meshfinite difference method

MSC classification

Secondary: 65L50: Mesh generation and refinement 65M06: Finite difference methods 65M20: Method of lines
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1364 - 1382
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.CrossRefGoogle Scholar
[2] Berger, M. J. and Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82 (1989), pp. 64–84.CrossRefGoogle Scholar
[3] Berger, M. and Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE Trans. Syst. Man. Cybern, 21 (1991), pp. 1278–1286.CrossRefGoogle Scholar
[4] Colella, P., Graves, D. T., Ligocki, T. J., Martin, D. F., Modiano, D., Serafini, D. B. and Straalen, B. V., CHOMBO: software package for AMR applications: design document, Technical report, Lawrence Berkeley National Laboratory, Applied Numerical Algorithms Group, NERSC Division; CA, USA, 2003 Google Scholar
[5] 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.CrossRefGoogle Scholar
[6] Dziuk, G. and Elliott, C., Finite element methods for surface PDEs, Acta Numer., 22 (2013), pp. 289–396.Google Scholar
[7] 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.CrossRefGoogle Scholar
[8] Elliott, C. M., Stinner, B., Styles, V. and Welford, R., Numerical computation of advection and diffusion on evolving diffuse interfaces, IMA J. Numer. Anal., 31 (2011), pp. 786.Google Scholar
[9] Grande, J., Eulerian finite element methods for parabolic equations on moving surfaces, SIAM J. Sci. Comput., 36 (2014), pp. B248–B271.CrossRefGoogle Scholar
[10] Gross, S. and Reusken, A., Numerical Methods for Two-Phase Incompressible Flows, Springer, 2011.CrossRefGoogle Scholar
[11] Hansboa, P., Larsonb, M. G. and Zahedi, S., Characteristic cut finite element methods for convection-diffusion problems on time dependent surfaces, Comput. Meth. Appl. Mech. Eng., 293 (2015), pp. 431–461.CrossRefGoogle Scholar
[12] Lowengrub, J., Xu, J.-J. and Voigt, A., Surface phase separation and flow in a simple model of multicomponent drops and vesicles, Fluid Dyn. Material Pro., 3 (2007), pp. 1–19.Google Scholar
[13] MacNeice, P., Olson, K. M., Mobarry, C., Defainchtein, R. and Packer, C., PARAMESH: A parallel adaptive mesh refinement community toolkit, Comput. Phys. Commun., v126 (2000), pp. 330–354.Google Scholar
[14] 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.CrossRefGoogle Scholar
[15] Mitran, S., BEARCLAW: a code for multiphysics applications with embedded boundaries: users manual, Technical report, Dept. of Math., Univ. of North Carolina, NC, USA, 2006.Google Scholar
[16] Olshanskii, M. A., Reusken, A. and Xu, X., An Eulerian space-time finite element method for diffusion problems on evolving surfaces, SIAM J. Numer. Anal., 52 (2014), pp. 1354–1377.Google Scholar
[17] Olshanskii, M. A. and Reusken, A., Error analysis of a space-time finite element method for solving PDEs on evolving surfaces, SIAM J. Numer. Anal., 52 (2014), pp. 2092–2120.Google Scholar
[18] Osher, S. and Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12.CrossRefGoogle Scholar
[19] Shu, C., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, Springer, 1998.CrossRefGoogle Scholar
[20] Strain, J., Semi-Lagrangian methods for level set equations, J. Comput. Phys., 151 (1999), pp. 498–533.Google Scholar
[21] 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
[22] Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. and Welcomey, M. L., An adaptive level set approach for incompressible two-Phase flows, J. Comput. Phys., 184 (1999), pp. 81–124.Google Scholar
[23] Teigen, K. E., Li, X., Lowengrub, J., Wang, F. and Voigt, A., A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7 (2009), pp. 1009–1037.Google Scholar
[24] Teigen, K. E. and Munkejord, S. T., Influence of surfactant on drop deformation in an electric field, Phys. Fluids, 22 (2010), 112104.Google Scholar
[25] 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
[26] Xu, J.-J. and Zhao, H., An Eulerian formulation for solving partial differential equations along a moving interface, J. Sci. Comput., 19 (2003), pp. 573–594.Google Scholar
[27] Xu, J.-J., Yuan, H. Z. and Huang, Y. Q., A 3D level-set method for solving convection-diffusion along moving surfaces (in Chinese), Sci. Sinna Math., 42(5) (2012), pp. 445–454.Google Scholar
[28] Xu, J.-J., Li, Z., Lowengrub, J. and Zhao, H., A level set method for solving interfacial flows with surfactant, J. Comput. Phys., 212 (2006), pp. 590–616.Google Scholar
[29] Xu, J.-J., Li, Z., Lowengrub, J. and Zhao, H., Numerical study of surfactant-laden drop-drop interactions, Commun. Comput. Phys., 10 (2011), pp. 453–473.CrossRefGoogle Scholar
[30] 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.CrossRefGoogle Scholar
[31] Xu, J.-J., Huang, Y., Lai, M.-C. and Li, Z., A coupled immersed interface and level set method for three-dimensional interfacial flows with insoluble surfactant, Commun. Comput. Phys., 15 (2014), pp. 451–469.Google Scholar
[32] 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
[33] Shi, W. D., Xu, J.-J. and Shi, S., A simple implementation of the semi-Lagrangian level-set method, Adv. Appl. Math. Mech., 2016 (in press).CrossRefGoogle Scholar
[34] Zhao, H., Chan, T., Merriman, B. and Osher, S., A variational level set approach to multi-phase motion, J. Comput. Phys., 127 (1996), pp. 179–195.Google Scholar