| The Lattice Boltzmann Method (LBM) is a simple and highly efficient method for computing nearly incompressible fluid flow. However, it is well known to suffer from numerical instabilities for low values of the transport coefficients. This dissertation examines a number of methods for increasing the stability of the LBM over a wide range of parameters. First, we consider a simple transformation that renders the standard LB equation implicit. It is found that the stability is largely unchanged. Next, we consider a stabilization method based on introducing a Lyapunov function which is essentially a discrete-time H-function. The uniqueness of an H-function that appears in the literature is proven, and the method is extended to stabilize some of the more popular LB models. We also introduce a new method for implementing boundary conditions in the LBM. The hydrodynamic fields are imposed in a transformed moment space, whereas The non-hydrodynamic fields are shifted over from neighboring nodes. By minimizing population gradients, this method exhibits superior numerical stability over other widely employed schemes when tested on the widely-used benchmark of incompressible flow over a backwards-facing step. |
