In this work, new numerical methods are proposed to efficiently resolve interfaces occurring in multiphase incompressible flows. Multiphase flow problems consist of a large class of physical phenomenon from bubbles to bow waves in ships. Over the recent decades, numerical methods are becoming an important tool in addition to pure analytical and experimental methods. However, there is still large room for improvement in existing numerical methods.

Contributions are made in the field of interface advection and the jump conditions for pressure. In the case of advection, a method is developed specifically for implicit interfaces that evolve with the Eulerian advection of a scalar field. The new method is validated by comparison with the interfaces that evolve with Lagrangian advection using a connected set of marker particles.

To accurately capture the jump conditions, a second order accurate method is proposed for solving the variable coefficient Poisson's equation in the discretized Navier-Stokes formulation. This new method assumes both phases exist in the interface cell and that their collective effect can be expressed by a volume fraction weighted average value.

The new capabilities have been integrated to build a dynamic Navier-Stokes equation solver. The new advection scheme scheme is also associated to track the interface. The new solver is tested by applications in several two phase flow problems.