Parallel implementation and analysis of the discrete-velocity BGK Boltzmann method

Sammanfattning

This project builds upon the work of Gu, Hu, Matharu et al. (2025), who proved that a local weak solution to the d-dimensional incompressible Navier-Stokes equations can be constructed by taking the hydrodynamic limit of a discrete-velocity Boltzmann equation with a simplified collision operator. Additionally, the convergence rate of this was shown numerically to be was shown numerically to be O(ϵ^2) where ϵ is the Knudsen number. This was achieved by implementing the mathematically derived Lattice Boltzmann Method with the lattice structure D2Q9 in two-dimensions and letting the Knudsen number tend to zero while measuring the resulting error compared to a reference solution. In this thesis, a fast parallel code for the Lattice Boltzmann Method in both two and three dimensions was developed. In addition to D2Q9, the lattice structures D2Q7, D2Q13, and D3Q19 were also implemented. For both the additional lattice structures and the higher dimension, the convergence rate with which the hydrodynamic limit is obtained was investigated by running simulations with decreasing Knudsen numbers and comparing to reference solutions. The numerical results in this thesis show that the convergence rate is independent of both the lattice structure and the dimension used. Furthermore, the results are consistent with the results by Gu, Hu, Matharu et al. (2025), i.e. the convergence rate matches closely with a power-law relation a · ϵ^2 where a is a constant. The parallel implementation enables large-scale simulations and speeds up as expected when increasing the number of processors. However, the efficiency of the program decreases with increasing number of processors since more time is spent on communication.