Arndt, Mirco2025-10-292025-10-292024http://hdl.handle.net/2003/4405110.17877/DE290R-25819The aim of this thesis is to develop, implement and validate a fast time-simultaneous solver for computational fluid dynamic (CFD) problems, which is optimized on high-performance computing (HPC) architectures to efficiently exploit numerous cores through a parallelization in space and time.The sequential-in-time and parallel-in-space projection method approach, which decouples velocity and pressure, serves as the basis for the development of the new solver approach. The Crank-Nicolson scheme is used as time stepping scheme. Since the structure and settings of the new solver approach are not clear, especially with respect to a convective term such as in the Navier-Stokes equations, fundamental investigations must first be carried out. In order to explore the necessary requirements and characteristics of the new solver approach, the first step is to look at the theory for general convergence criteria. From the theory it was determined that a divergence-free velocity is a necessary condition for convergence. Two different divergence-free approaches have been implemented and tested for the new time-simultaneous solver: The augmented Lagrangian method and a newly developed update similar to the projection method update, which only approximately guarantees a divergence-free velocity. Both approaches lead to a reduced number of iterations until convergence, but in terms of computational time, the approximate divergence-free update is significantly more efficient and leads to a considerable reduction in computational time. With the newly developed approximate update approach, thousands of time steps of the Navier-Stokes equations can be computed parallel-in-space and -time. In the time-simultaneous case, the choice of the right-hand side for the pressure Poisson problem is not clear and has a significant importance. Numerical results verify that a divergence-free velocity component from the previous time step with a non-divergence-free component from the current time step ensures the most optimal convergence behavior. By ensuring a approximate divergence-free velocity, even in the case of the incompressible Navier-Stokes equations, it is possible to compute large time intervals time-simultaneously. A multigrid in time approach leads to a faster achievement of a divergence-free velocity for large time-intervals, since the approximate divergence-free velocity is not lost as quickly with larger time steps. It has been investigated that the multigrid in time approach only achieves a speed up in combination with a multigrid in space solver. To compute a large number of time steps, especially in 3D, the memory usage must be taken into account. An embedding of the solver in a memory-optimized local recoupling approach using FGMRES has been investigated, which can further enforce convergence. A local recoupling with FGMRES is not recommended because the performance in terms of the computational time is insufficient. To enforce convergence, a global recoupling seems to be the better approach. Although it ensures fewer iterations and can speed up the computation, it uses too much memory. Embedding the solver in a recoupling approach is not necessary and can be neglected, as it also requires additional programming effort, such as new data types. Another question was whether a fast existing solver for the pressure Poisson problem using prehandling could be integrated in the solver. The fast solver needs the Q2/Q1 finite element with the true Laplacian matrix. The theory already provides that the Q2/Q1 finite element is more prone to problems than the Q2/P1 finite element. Therefore, it is not recommended to integrate the solver, as in this case a recoupling approach using GMRES is necessary to achieve convergence. In conclusion, the newly developed solver is decoupled from velocity and pressure, as well as from time and space. However, to achieve convergence, a divergence-free velocity must be ensured at least approximately at each time point. A time-simultaneous solver is much faster than a sequential-in-time solver, which can only compute parallel-in-space. To have direct access to the different cores on a computer, an implementation in C++ (FEAT3) is recommended.enComputational fluid dynamic (CFD)High performance computing (HPC)Finite element method (FEM)Time simultaneousTime parallelMultigrid in spaceMultigrid in timeAugmented LagrangianApproximate divergence-free updateGMRESFGMRESProjection methodPPTSPPStokes equationsIncompressible Navier-Stokes equationsCrank-Nicolson (CN)510PhDThesisNumerische StrömungssimulationNavier-Stokes-GleichungInkompressible StrömungHochleistungsrechnen