The finite volume particle method is a meshless discretization technique, which generalizes the classical finite volume method by using smooth, overlapping and moving test functions applied in the weak formulation of the conservation law. The method was originally developed for hyperbolic conservation laws so that the compressible Euler equations particularly apply.In the present work we analyze the discretization error and enforce consistency by a new set of geometrical quantities. Furthermore, we introduce a discrete Laplace operator for the scheme in order to extend the method to second order partial differential equations. Finally, we transfer Chorin's projection technique to the finite volume particle method in order to obtain a meshless scheme for incompressible flow.

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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.

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