# Parallele Verfahren höherer Ordnung zur Lösung von Zweiphasen-Strömungen

• Parallel higher order methods for solving twophase flows

Numerical methods for the simulation of twophase fluid flows are discussed within this work. We focus on the efficiency of these methods: higher order approximation of the error bounds is achieved, as well as fast results on modern computer architectures. The underlying model consists of the incompressible twophase Navier-Stokes equations. The phase boundary is described implicitly by a Levelset technique. The surface tension is modeled as a localized force term on the phase boundary.To set up the problem, we first intoduce the governing Navier-Stokes equations. We proceed afterwards with the numerical treatment of stationary twophase flows to give an overview of the used numerical techniques, e.g. spatial discretization of the surface tension force with the Laplace-Beltrami approach. During this overview we present the important technique "Extended Finite Elements" (XFEM), which helps by achieving an accurate spatial discretization of the jump in the pressure variable.Following this introduction, the work is structured in two main parts: first, the development and analysis of a suitable time discretization method, based on well known $\theta$-methods; second, the transfer of the numerical methods on modern parallel computers.Regarding the time integration, we restrict ourselves to a reduced simplified model problem, which allows us to achieve optimal error bounds (in a suitable weak norm). These error bounds are also valid within realistic numerical simulations, the results of a simulated rising butanol droplet in water are presented. Additionally, we show some modification of these methods, which ensures better numerical properties, e.g. better stability.The second part of this work deals with the parallelization and the modification/adaptation of numerical methods for twophase flows on recent hardware architectures. In cooperation with the center of scientific computing of the RWTH Aachen, we have implemented the extension DiST (Distributed Simplex Types) of the solver DROPS, developed at the chair for numerical mathematics (RWTH Aachen). The aim of DiST is the management of distributed simplexes. With this library, it is possible to add MPI-parallel components to DROPS, like efficient parallel hierarchical adaptive grid structures or preconditioners of the iterative solvers. These concepts are successfully applied up to 65.000 threads. Eventually, the shared memory parallelization of DROPS is also discussed.