Lattice Boltzmann methods are a fully discrete model and numerical method for simulating fluid dynamics, historically they have been developed as a continuation of lattice gas systems. Another route to a lattice Boltzmann system is a discrete approximation to the Boltzmann equation. An analysis of lattice Boltzmann systems is usually performed from one of these directions. In this thesis the lattice Boltzmann method is presented ab initio as a fully discrete system in its own right. Using the Invariant Manifold hypothesis the microscopic and macroscopic fluid dynamics arising from such a model are found. In particular this analysis represents a validation for lattice Boltzmann methods far from equilibrium. Far from equilibrium, at high Reynolds or Mach numbers, lattice Boltzmann methods can exhibit stability problems. In this work a conditional stability theorem for lattice Boltzmann methods is established. Furthermore several practical numerical techniques for stabilizing lattice Boltzmann schemes are tested.

The Lattice Boltzmann Method (LBM) is a discrete velocity method which involves a single particle distribution function with two repeating procedures propagation and collision. When the Bhatnagar-Gross-Krook operator is applied as the collision operator for LBM, this is called lattice Bhatnagar-Gross-Krook method (LBGK). In comparison with the traditional computation methods, LBM appears as an efficient alternative computational approach for simulating complex fluid systems. However, LBM suffers numerical stability deficiencies when applied in low-viscosity fluid flow, such as local blow-ups and spurious oscillations where sharp gradients appear. The development of LBM has taken a further step to resolve the stability problem with applying a discrete entropy H-theorem. However, the stability and accuracy problems are not completely dealt with by the entropic lattice Boltzmann method. One of the remedies for the stability deficiencies is to construct nonequilibrium entropy limiters for LBM. The original concepts with the construction of nonequilibrium entropy limiters are based on flux filters (also called flux-corrected transport) by Boris and Book. The principal idea of the nonequilibrium entropy limiters is to control a scalar quantity, the nonequilibrium entropy. In this thesis, there are 6 limiters are developed and tested in 1D athermal shock tubes in uniformed discretized space lattice sites. Among these limiters, two new nonequilibrium limiters are constructed. All the median entropy limiters are tested with different stencils, which also have an effect on removing spurious oscillations. Apart from the test on a three-velocity set, we use five-velocity sets for the applications of nonequilibrium entropy limiters of LBM. The five-velocity sets are {-3, -1, 0, 1, 3}, {-5, -2, 0, 2, 5} and {-7, -3, 0, 3, 7}. The performance of LBGK without limiters provides a frame of reference for comparison with the performance of LBGK which uses the nonequilibrium entropy limiters. The computations of the LBGK on different velocity sets have shown that the nonequilibrium entropy limiters are able to efficiently remove spurious oscillations for both post-shock and shock regions for high Reynolds number. Among the suggested limiters, we recommend the median nonequilibrium entropy limiter.

No description available.