Studies Of Numerical Simulations Based On Lattice Boltzmann Model For The Kuramoto-Sivashinsky Equation

Since the Lattice Boltzmann method (LBM) was proposed, it has been concerned by many scientists. Recently, LBM has been developed gradually as an effective tool for computational fluid dynamics (CFD). The LBM has been widely used in many research fields such as multiphase and multi-component fluid, particulate suspensions fluid, flows through porous media, magneto hydrodynamics, reaction-diffusion system, non-Newtonian fluid, turbulent flow, and incompressible flows. Additionally, lattice Boltzmann models have been developed to simulate linear and nonlinear partial differential equations. In the thesis we use the LBM to simulate the Kuramoto- Sivashinsky equation. Details are as follows:In chapter 1, the lattice Boltzmann method is reviewed.In chapter 2, the basic theory of LBM is given. The fundamental ideas of the LBM is to construct simplified kinetic models that incorporate the essential physics of microscopic models and mesoscopic processes so that the macroscopic averaged properties obey the desired macroscopic equations. We discrete the space into regular lattices and the time into the same steps. Boltzmann differential equation is expressed as lattice Boltzmann equation. By expanding the distribution function in the lattice Boltzmann equation using Chapman-Enskog expansion and multiple-dimensioned expansion to time, series of partial differential lattice Boltzmann equation in different time scales is obtained, and thus we obtain the moments of the equilibrium distribution function. By assuming the particles to meet the mass and momentum conservation, we can get the equilibrium distribution function, and thus can carry out iteration.In chapter 3, we construct a LBM for Kuramoto-Sivashinsky equation. We select a 5-bit model. We obtain the Kuramoto-Sivashinsky equation with the four-order accuracy of truncation error by selecting the appropriate higher order moment. The expression of truncation errors can be gained. The Hirt's heuristic stability conditions are given. We also carry out numerical tests on the model through some numerical examples. The results show LBM can simulate Kuramoto- Sivashinsky equation. The effect is satisfying.Finally, give the conclusions. LBM is an effective numerical method to simulate the Kuramoto-Sivashinsky equation. In the future work, we may study the two and three dimension lattice Boltzmann models for the Kuramoto-Sivashinsky equation.