Lattice Boltzmann Method For Wave Equation And Its Numerical Investigation

In the last two decades, the lattice Boltzmann method has attracted the greatest attention as a new numerical method in computational fluid dynamics. It solved many problems that could not be solved by traditional computational methods. We focus on the lattice Boltzmann model for the wave equation in the thesis.In Chapter 1, we reviewed the history of the lattice Boltzmann method. The lattice Boltzmann method originated from a Boolean fluid model known as the lattice gas automata. As the inheritance and the development of the lattice gas automata, it has become an effective tool for simulating many complex problems. In present, lattice Bhatnagar-Gross-Krook(BGK) model is the most important one of all the models in the lattice Boltzmann method field. Recently, the most successful application is based on this simple model. In this Chapter, not only the basic models are introduced, but some fields of the applications of the lattice Boltzmann method are reviewed.In Chapter 2, we used higher- order moment method, multi- scale technique and Chapman- Enskog expansion to obtain the lattice Boltzmann model of wave equation. In order to preserve the shape and frequency in the wave propagation, lattice Boltzmann models for the energy conservation and for the entropy conservation are introduced separately. In addition, to solve the local equilibrium distribution function, the mass of the moving particles and the rest particles is included as parameters in themodel.In the wave propagation problem with energy conservation, we select two kinds of energy definition forms separately: one is defined as the potential energy, possessing the form of the square of the distribution function, and the 5-bit square lattice is used. The other is defined as the multi-energy-level form. The 9-bit double layer square lattice is employed. Three energy levels are defined in the model.In wave propagation problem with entropy conservation, we select two kinds of entropy definition forms too. One of them is defined as the multi-entropy-level form. The 9-bit double layer square lattice is used. We defined three entropy levels in the model.In Chapter 3, we conducted the numerical verification for the models in Chapter 2, some classical examples are simulated, including the interference of two point sources, the interference of the double- slit, Young's experiment, light wave propagation through triangular prism, simulation of rainbow etc.. We compared numerical results of the lattice Boltzmann models to solutions getting classical method. The results show these models can be used to simulate wave propagation.In Chapter 4, as an extension of the contents of the thesis, we obtain a lattice Boltzmann model for the heat wave problem. Three numerical examples are employed to test the validity of the model. The numerical results show the model is effective for solving the heat wave problem too.