Some Numerical Aspects of the Level Set Method

Abstract

Many practical applications imply the solution of free boundary value problems. If the free boundary is complex and can change its topology, it will be hard to solve such problems numerically. In recent years a new method has been developed, which can handle boundaries with complex geometries. This new method is called the level set method. However, the level set method also has some drawbacks, which are mainly concerning conservation of mass or numerical instabilities of the boundaries. Our aim is to analyze some aspects of the level set method on the basis of two-phase ow in a Hele-Shaw cell. We investigate instabilities of two-phase ow between two parallel plates. A solution of the linearized problem is obtained analytically in order to check whether the numerical schemes compute reasonable results. The developed numerical scheme is based on nite di erence approximations and the level set method. The equations of two-phase Hele-Shaw ow are written in a modi ed formulation using the one-dimensional Dirac delta-function. Since the level set function is not smooth enough after re-initialization, special attention during the computation of curvature is needed. We propose a method that can solve the problems for two-phase Hele-Shaw ow with changing topology. The numerical solution shows good agreement with the analytical solution of the linearized problem. We describe the method below and analyze the results.