The lattice Boltzmann method (LBM) was originally developed as a numerical scheme in the field of computational fluid dynamics. In this dissertation the LBM is considered and the topic of this work lies on the construction of artificial boundary conditions (ABCs) - well suited boundary conditions for artificial boundaries. Artificial boundaries are introduced when a spatial domain is confined to a smaller computational domain. A boundary condition (BC) not tailored for artificial boundaries, like a fixed pressure or velocity condition, behaves in an unphysical manner and generates spurious reflections. Ideally, a BC does not interact with the fluid. Thus, any fluctuation from the interior should leave the computational domain without generating a reflection when impinging on the artificial boundary.
At the beginning of this dissertation a theoretical embedding of the LBM is given. As an example, the D3Q19 model is analytically derived with a numerical integration approach. Moreover the connection of the LBM to macroscopic fluid models (Navier-Stokes equations) is shown, and it is demonstrated that the application of the LBM is not limited to simulate fluid flows.
Based on the theoretical foundation, the features of an ideal boundary condition for artificial boundaries are elucidated. Approaches from literature for the treatment of artificial boundaries are presented. Here a focus lies on perfectly matched layer (PML) approaches and LODI-based characteristic boundary conditions (CBCs). Then the LODI-based CBCs are extended and thus novel ABCs for one- to three-dimensional problems are developed. CBCs are fully described on a macroscopic level, thus differently to the fluid description the LBM is based on. It is explained that from the perspective of the LBM any CBC eventually describes a Dirichlet BC for macroscopic quantities, whose implementation introduces errors.
Afterwards, in the main part of this dissertation the construction of ABCs, which are completely described on the discrete level of the LBM, is pursued. Firstly, a novel procedure based digraphs is introduced for understanding the (temporal) evolution of the discrete quantities within the LBM. With this new understanding a theoretical basis for the formulation of novel general discrete ABCs is constructed by analytically deriving an exact BC for artificial boundaries in 1D. A consistency condition satisfied by this exact discrete ABC is deduced. The one-dimensional exact discrete ABC is approximated, whereby a new parameter is introduced which controls the accuracy of the approximation. The approximated discrete ABC is interpreted as a separate lattice Boltzmann simulation, and this interpretation is generalized. Hence, a discrete ABC for general lattice Boltzmann simulations (not restricted to models recovering Navier-Stokes equations) in higher dimensions is formulated. Error sources of the discrete ABCs, details for their efficient implementation as well as their computational costs are discussed.
All novel ABCs are implemented and applied for a variety of one- to three-dimensional test scenarios. Their performance is compared to selected ABCs from literature. One numerical simulation explains the working principle of the discrete ABCs visually.
Finally, the results of this dissertation are summarized and additionally possible future research tasks are pointed out.