Calculation of interface curvatures with the level-set method for two-phase flow simulations and a second-order diffuse-domain method for elliptic problems in complex geometries

This thesis considers in the first part the mathematical modelling of incompressible two-phase flow, in particular the calculation of interface curvatures and normal vectors with the level-set method. The main contribution is the development of two new numerical methods that enable a more robust calculation of the curvature and normal vectors in areas where the gradient of the level-set method is discontinuous. Incompressible two-phase flow is in this thesis modelled by the Navier- Stokes equations with a singular source term at the interface between the phases. The singular source term leads to a set of interface jump conditions. These jump conditions are used in the ghost-fluid method to solve two-phase flow in a sharp manner. The interface position is captured and evolved in time with the level-set method. The Navier- Stokes equations for two-phase flow are solved with projection methods and discretized by finite differences in space and Runge-Kutta methods in time. The advective terms in the governing equations are discretized by a weighted essentially non-oscillatory scheme. In the second part, the thesis considers the more general problem of solving partial-differential equations (PDEs) in complex geometries. An extension of a diffuse-domain method is presented, where the accuracy is improved by adding a correction term. The extension is derived for elliptic problems with Neumann and Robin boundary conditions. One of the advantages of the diffuse-domain methods is that they allow the use of standard tools and methods because they are based on solving PDEs reformulated in larger and regular domains