The present thesis investigates, and wherever possible improves, the convergence behavior of a multigrid-like solver. This solver is part of a software package developed during a PhD-thesis (Frisch, 2014). The aim of this software is to provide a massive-parallel fluid flow solver, including visualisation capabilities to show the fluid flow and the passed geometry, that can make efficient use of the parallel architecture of modern high-performance computers. The Navier-Stokes equations are solved in two steps. First an intermediate velocity is com- puted via the Chorins projection using pressure values from the previous time step. Second a pressure correction is made by using a pressure Poisson equation which accounts for as much as 90 % of the computing time of a single time step. The actual solution of the Poisson equa- tion consists of a multigrid v-cycle scheme with a Jacobi Method as smoother on all levels except for the coarsest where a Symmetric Gauss-Seidel Method is utilized. It turned out that the progression of residuals flattens out more with each iteration of the v-cycle during each time step. For investigation the test cases Driven Cavity, Taylor Couette, Kármán vortex street and Operating Theatre Klinikum rechts der Isar are used. Every test case is computed for a minimal depth of the multigrid scheme of 2, 3 and 4. Investigating the behavior of the code is done by analysing the global residuals with respect to the v-cycle and by analysing the local residuals with respect to time and space. This revealed an oscillation of the residuals. Trials of the influence of singularities, boundary con- ditions, time step size and different abort criterias on the path of residuals led to no success. Attempts to smooth the residuals via adaption of the relaxation parameters reduced the number of v-cycles by approximately 10%, but the original problem still existed. Finally, the v-cycle is replaced by a full multigrid scheme. Because of the better smoothing of residuals, the relaxation parameters where adapted reducing the computational costs sig- nificantly. The results show, at least for the computed examples, that there is only one cycle necessary for one time step. Thus the original problem does not exist anymore. Furthermore, for all computed examples, the number of iterations of the Gauss-Seidel Method decreases rapidly after around 100 to 400 time steps until there is only one iteration necessary per time step. This leads to a speed up for the Driven Cavity test case computed at a depth of 3 of 3.83 compared with the original code or 9.83 for the Operating Theatre test case at a depth of 3. At the end there is a short tract on how to compute optimal relaxation parameters, although a matrix-free approach is used. It would be crucial to implement it to avoid wrong relaxation parameters and therefore an oscillation of the residuals which, in turn would cause the orig- inal problem again. Furthermore, a depth-optimising algorithm is sketched. As computing time increases with increasing depth of the multigrid scheme, it aims for a minimal depth of the multigrid scheme in order to reduce the computational effort.At last, a kind of new solver is suggested using the amplitude spectrum gained via the fast Fourier transformation to combine precomputed results to get the final result.     «

The present thesis investigates, and wherever possible improves, the convergence behavior of a multigrid-like solver. This solver is part of a software package developed during a PhD-thesis (Frisch, 2014). The aim of this software is to provide a massive-parallel fluid flow solver, including visualisation capabilities to show the fluid flow and the passed geometry, that can make efficient use of the parallel architecture of modern high-performance computers. The Navier-Stokes equations are solve...     »