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

  • Weidong Shi (a1) (a2), Jianjun Xu (a2) and Shi Shu (a1)

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.

Corresponding author
*Corresponding author. Email: (W. D. Shi), (J. J. Xu), (S. Shu)
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[1] Berger, M. J. and Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), pp. 484.
[2] Berger, M. J. and Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82 (1989), pp. 6484.
[3] Berger, M. and Rigoutsos, I., An algorithm for point clustering and grid generation, IEEE Trans. Syst. Man. Cybern, 21 (1991), pp. 12781286.
[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
[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. 243255.
[6] Dziuk, G. and Elliott, C., Finite element methods for surface PDEs, Acta Numer., 22 (2013), pp. 289396.
[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. 647668.
[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.
[9] Grande, J., Eulerian finite element methods for parabolic equations on moving surfaces, SIAM J. Sci. Comput., 36 (2014), pp. B248B271.
[10] Gross, S. and Reusken, A., Numerical Methods for Two-Phase Incompressible Flows, Springer, 2011.
[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. 431461.
[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. 119.
[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. 330354.
[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. 300321.
[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.
[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. 13541377.
[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. 20922120.
[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.
[19] Shu, C., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, Springer, 1998.
[20] Strain, J., Semi-Lagrangian methods for level set equations, J. Comput. Phys., 151 (1999), pp. 498533.
[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. 146159.
[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. 81124.
[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. 10091037.
[24] Teigen, K. E. and Munkejord, S. T., Influence of surfactant on drop deformation in an electric field, Phys. Fluids, 22 (2010), 112104.
[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.
[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. 573594.
[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. 445454.
[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. 590616.
[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. 453473.
[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. 58975909.
[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. 451469.
[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. 7190.
[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).
[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. 179195.
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