Stochastic Computation of Rarefied Gas Flows Using the Fokker-Planck-DSMC Method: Theory, Algorithms, and Parallel Implementation

Abstract

The topic of this thesis is the analysis and parallel implementation of the Fokker-Planck-DSMC algorithm for the numerical simulation of rarefied gas flows. The most established method for this task is the Direct Simulation Monte Carlo (DSMC) technique. For gas flows in the near-continuum regime, however, its computational cost becomes intractable due to the high number of collisions of (computational) particles that need to be computed. The Fokker-Planck (FP) algorithm, on the other hand, provides accurate numerical predictions for near-continuum gas flows at computational cost independent of the number of collisions. In this method, the trajectories of the computational particles evolve independently along continuous stochastic paths. Since both DSMC and the FP algorithm are stochastic particle methods sharing the same underlying structure, they may be coupled seamlessly: the resulting FP-DSMC algorithm is capable of simulating rarefied gas flows from the near-continuum to the fully rarefied regime. One result of this thesis is a flexible, yet computationally efficient simulation software, which uses both distributed- and shared-memory parallelization to exploit state-of-the-art high-performance computer cluster technologies. It provides the means to conduct computer simulations of flows of diatomic, rarefied gases in complex domains, using many computational particles. The new implementation is used to analyze the accuracy and performance of the FP-DSMC algorithm by means of a variety of simulations. It is shown that given a limited amount of computational resources, using FP-DSMC can provide more accurate results at lower computational cost compared to pure DSMC. Further, the implementation is capable of performing automatic local mesh refinement, as well as parallel load balancing. This is achieved by choosing space-filling curves (SFCs) as a fundamental concept for the ordering of the computational mesh and particle data. SFCs not only allow for an elegant implementation of these features, but have the additional benefit of ensuring cache-friendly computations. The impact on computational performance of using different space-filling curves is analyzed numerically, and the implementation is demonstrated to deliver accurate simulation results for a relevant test case. In order to maximize the efficiency gains due to the FP-DSMC algorithm, the computational mesh should be locally adapted to the flow gradients. With this goal in mind, a general theoretical framework for the estimation of mixed partial derivatives of statistics of scattered data is developed based on the concept of kernel density estimation. The new approach allows for the computation of flow gradients locally in each cell of the mesh as a simple weighted sum of the particle states, and may prove useful beyond the scope of rarefied gas flow simulations.