Lattice-Boltzmann-Verfahren hoher Ordnung zur Simulation kompressibler Strömungen

In this thesis, a compressible semi-Lagrangian lattice Boltzmann method is newly developed and tested. The lattice Boltzmann method is a rapidly advancing numerical method for computational fluid dynamics. However, in its original form, the lattice Boltzmann method is limited to weakly compressible flows with low Mach number. Previous attempts to extend the lattice Boltzmann method to supersonic flows suffered either from poor stability, from impractically large velocity sets, or from small time step sizes. As an alternative to previous approaches, a semi-Lagrangian streaming step is used in this work. Semi-Lagrangian methods decouple the spatial, time, and velocity space discretization of the original lattice Boltzmann method by interpolation during the streaming step.
Following the introduction, the second and third chapters of this thesis first detail the basics of the lattice Boltzmann method and list previous approaches to simulate compressible flows. Subsequently, the compressible semi-Lagrangian lattice Boltzmann method is developed and described.
In the fourth chapter of the thesis, new cubature-based velocity sets are developed and tested, including a D3Q45 velocity set for the computation of compressible flows, which significantly reduces the computational cost compared to conventional velocity discretizations.
In the fifth chapter of the thesis, simulations of one-dimensional shock tubes, two-dimensional Riemann problems, and shock-vortex interactions are performed for validation. Thereafter, simulations of compressible Taylor-Green vortices as well as wall-bounded problems demonstrate the advantages of the method for compressible flow simulations. The latter include the supersonic flow around a two-dimensional NACA-0012 profile and around a three-dimensional sphere as well as a supersonic channel flow. The simulation section is followed by an extensive discussion of the semi-Lagrangian lattice Boltzmann method in comparison to other methods. The advantages of the method include comparatively large time step sizes, compatibility with body-fitted meshes, and the intrinsic stability of the method even without artificial viscosity.