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Spatially adaptive techniques for level set methods and incompressible flow. (English) Zbl 1177.76295
Summary: Since the seminal work of M. Sussman, P. Smereka and S. Osher [J. Comput. Phys. 114, No. 1, 146–159 (1994; Zbl 0808.76077)] on coupling the level set method of S. Osher and J. A. Sethian [J. Comput. Phys. 79, No. 1, 12–49 (1988; Zbl 0659.65132)] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, M. Sussman, P. Smereka and S. Osher [J. Comput. Phys. 114, No. 1, 146–159 (1994; Zbl 0808.76077)] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both of its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton-Jacobi WENO [G. S. Jiang and D. Peng, SIAM J. Sci. Comput. 21, No. 6, 2126–2143 (2000; Zbl 0957.35014)], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [D. Enright et al., J. Comput. Phys. 183, No. 1, 83–116 (2002; Zbl 1021.76044)] and the coupled level set volume of fluid method [M. Sussman, E. G. Puckett, J. Comput. Phys. 162, No. 2, 301–337 (2000; Zbl 0977.76071)], and adaptive gridding techniques such as the octree approach to free surface flows proposed by F. Losasso et al. [Simulating water and smoke with an octree data structure, ACM Trans. Graph. (SIGGRAPH Proc) 23, 457–462 (2004)].

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M28 Particle methods and lattice-gas methods
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
Full Text: DOI
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