Magnetohydrodynamic turbulence: The development of lattice Boltzmann methods for dissipative systems

Computer simulations of complex phenomena have become an invaluable tool for scientists in all disciplines. These simulations serve as a tool both for theorists attempting to test the validity of new theories and for experimentalists wishing to obtain a framework for the design of new experiments. Lattice Boltzmann Methods (LBM) provide a kinetic simulation technique for solving systems governed by non-linear conservation equations. Direct LBMs use the linearized single time relaxation form of the Boltzmann equation to temporally evolve particle distribution functions on a discrete spatial lattice. We will begin with a development of LBMs from basic kinetic theory and will then show how one can construct LBMs to model incompressible resistive magnetohydrodynamic (MHD) conservation laws. We will then present our work in extending existing models to the octagonal lattice, showing that the increased isotropy of the octagonal lattice produces better numerical stability and higher Reynolds numbers in MHD simulations. Finally, we will develop LBMs that use non-uniform grids and apply them to one dimensional MHD systems.