| This dissertation addresses three different aspects of the level set method originated in the seminal paper [37] of Osher and Sethian.;After a short introductory chapter, we present in Chapter 2 the WENO scheme for the Hamilton-Jacobi equation, which includes the typical evolution equations of level set functions as special cases. This scheme uses as many grid points as its ENO counterpart, yet is 2 orders of accuracy higher than the latter, and is more robust. Numerical examples are presented that demonstrate the advantages of this scheme.;In Chapter 3, we propose a PDE based fast level set method that localizes the standard level set method, and address issues that are intrinsic to this method. namely, extension and reinitialization. This localized method saves one order of magnitude of complexity, is easy to implement. and is as versatile as the global method. Numerical tests based on this method are reported at the end of this chapter which verify our claim.;Chapter 4 is the application of the level set method to a classical problem, the shape of equilibrium Wulff crystals. We demonstrate that the edges and facets that dominate the world of crystals are closely connected with the jumps and rarefactions of gas dynamics, thus relating the field of crystals with that of conservation laws. Numerous 2D and 3D examples that are based on the local level set method and WENO scheme are presented at the end of this part that verify our theoretical results. |
