Spatially Adaptive Techniques for Level Set Methods and Incompressible Flow

Since the seminal work of [92] on coupling the level set method of [69] 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, [92] 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 its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton-Jacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56]. ∗Research supported in part by an ONR YIP award and a PECASE award (ONR N00014-01-1-0620), a Packard Foundation Fellowship, a Sloan Research Fellowship, ONR N00014-97-1-0027, ONR N00014-03-1-0071, ONR N00014-02-1-0720, ARO DAAD19-031-0331, NSF DMS-0106694, NSF ITR-0121288, NSF IIS-0326388, NSF ACI-0323866, NSF ITR-0205671 and NIH U54 RR021813. †Computer Science Department, Stanford University, Stanford, CA 94305. ‡Department of Mathematics, University of California Los Angeles, Los Angeles, CA