Abstract
In this paper we study an Eulerian formulation for solving partial differential equations (PDE) on a moving interface. A level set function is used to represent and capture the moving interface. A dual function orthogonal to the level set function defined in a neighborhood of the interface is used to represent some associated quantity on the interface and evolves according to a PDE on the moving interface. In particular we use a convection diffusion equation for surfactant concentration on an interface passively convected in an incompressible flow as a model problem. We develop a stable and efficient semi-implicit scheme to remove the stiffness caused by surface diffusion.
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Adalsteinsson, D., and Sethian, J. A. (1995). A fast level set method for propagating interfaces. J. Comput. Phys. 118, 269.
Adalsteinsson, D., and Sethian, J. A. (1999). The fast construction of extension velocities in the level set methods. J. Comput. Phys. 148, 2.
Adalsteinsson, D., and Sethian, J. A. Transport and Diffusion of Material Quantities on Propagating Interfaces via Level Set Methods, preprint.
Aris, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid, Prentice–Hall, Englewood Cliffs, NJ.
Batchlor, G. K. (1967). An Introduction to Fluid Dynamics, Cambridge University Press.
Burchard, P., Cheng, L.-T., Merriman, B., and Osher, S. (2001). Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170(2), 720-741.
Ceniceros, H., and Hou, T. Y. (2001). An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 172(2), 1-31.
Bertalmio, M., Cheng, L.-T., Osher, S., and Sapiro, G. (2001). Variational problems and partial differential equations on implicit surfaces: The framework and examples in image processing and pattern formation. J. Comput. Phys. 174, 759-780.
Chen, S., Merriman, B., Osher, S., and Smereka, P. (1997). A simple level set method for solving Stefan problems. J. Comput. Phys. 135, 8.
Herring, C. (1951). Surface diffusion as a motivation for sintering. In The Physics of Powder Metallurgy, McGraw–Hill, New York, NY.
Hunter, J. K., Li, Z., and Zhao, H. K. Reactive autophobic spreading of drops, to appear in J. Comput. Phys.
Hou, T. Y., Li, Z., Osher, S., and Zhao, H. (1997). A hybrid method for moving interface problems with applications to the Hele-Shaw flows. J. Comput. Phys. 134, 236.
James, A., Lowengrub, J., and Seigel, M. Distribution formulation of interfaces with surfactant, preprint.
Jiang, G.-S., and Shu, C.-W. (1996). Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228.
Jiang, G.-S., and Peng, D. (2000). Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126.
Li, X., and Pozrikidis, C. (1997). Effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 441-454.
Li, Z.-L., Zhao, H.-K., and Gao, H.-J. (1999). A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid. J. Comput. Phys. 152, 281.
Mullins, W. W. (1995). Mass transport at interfaces in single component system. Metallurgical and Materials Trans. A 26, 1917-1925.
Nichols, F. A., and Mullins, W. W. (1965). Surface (interface) and volume-diffusion contributions to morphological changes driven by capillarity. Trans. Metall. Soc. AIME 233, 1840-1847.
Osher, S., and Sethian, J. A. (1988). Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12.
Osher, S., and Shu, C. W. (1991). High order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907.
OsherS., and Fedkiw, R. P. (2001). Level set methods: An overview and some recent results. J. Comput. Phys. 169, 463.
Peng, D., Merriman, B., Osher, S., Zhao, H.-K., and Kang, M. (1999). A PDE-based fast local level set method. J. Comput. Phys. 155, 410.
Saad, Y. (1996). Iterative methods for sparse linear systems, PWS.
Scriven, L. E. (1960). Chem. Eng. Sci. 12, 98.
Sethian, J. A. (2001). Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169, 503.
Shu, C.-W. (1988). Total-variation-diminishing time discretization. SIAM J. Sci. Stat. Comput. 9, 1073.
Stone, H. A. (1989). A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2(1), 111.
Strikwerda, J. C. (1989). Finite Difference Schemes and Partial Differential Equations, Wadsworth and Brooks/Cole Advanced Books and Software, Pacific grove, California.
Sussman, M., Smereka, P., and Osher, S. (1994). A levelset approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146-159.
Waxman, A. M. (1984). Stud. Appl. Math. 70, 63.
Wong, H., Rumschitzki, D., and Maldarelli, C. (1996). On the surfactant mass balance at a deforming fluid interface. Phys. Fluids 8(11), 3203.
Zhao, H., Chan, T., Merriman, B., and Osher, S. (1996). A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179.
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Xu, JJ., Zhao, HK. An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface. Journal of Scientific Computing 19, 573–594 (2003). https://doi.org/10.1023/A:1025336916176
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DOI: https://doi.org/10.1023/A:1025336916176