Numerical Analysis And Simulation Of Lattice Boltzmann Models For Some Nonlinear Wave Equations

Nonlinear partial differential equations (NPDEs) can be used to describe many important mathematical physical problems. However, it's well known that only very few NPDEs have analytical solutions. Therefore, alternative numerical solutions are of special importance. Up to date, many sophisticated and efficient numerical meth-ods have been developed, including the finite difference method, the finite volume method, the finite element method, the spectral method and so on. Recently, the lattice Boltzmann method (LBM) was proposed as a novel mesoscopic numerical method. Due to its particular natures, such as clear physical background, easy pro-gramming, simple calculation, accessibility to complex boundaries, excellent parallel performance and flexible scalability, this method is becoming a promising technology for numerical simulations. As a result, the LBM has achieved great sucess in various areas, such as solving the NPDEs, multiphase flow, porous media flow, turbulent flow and micro-fluid flow. In this thesis, three lattice Boltzmann models were pro-posed for solving three kinds of NPDEs, the contributions of which are summarized as follows.Firstly, we develop a coupled double-distribution-function lattice Boltzmann model to solve the initial boundary value problem of the nonlinear coupled viscous Burgers equation. Because of the nonlinear term (?)(uv)/(?)x involved in the macro-scopic equation, it is difficult to recover such a nonlinear term within the framework of the standard lattice Boltzmann equation. To the best of our knowledge, there are no open references concerning this topic. The non-standard amended lattice Boltz-mann equation is applied in this thesis. Specifically, local equilibrium distribution functions and amending functions satisfying some conditions of certain moments are chosen, and then the nonlinear term (?)(uv)/(?)x in the nonlinear equation can be recovered correctly. Based on the results of grid independent analysis and nu-merical simulation, it can be concluded that the constructed model is convergent with two-order space accuracy. Some initial boundary value problems with analyt-ical solutions are simulated. The results are compared with those obtained by the improved finite difference method and the Chebyshev spectral collocation method. It is shown that the errors obtained by the Boltzmann model are smaller than the other two schemes. Furthermore, some problems without analytical solutions are nu-merically studied by the present model and the improved finite difference method. The results show that the numerical solutions of the lattice Boltzmann model are in good agreement with those obtained by the improved finite difference method, which can reflect the nonlinear typical characteristics of the wave varying with time, and further validate the effectiveness and stability of the proposed model.Secondly, we focus on the initial boundary value problem of the generalized nonlinear damped wave equation, and construct a non-standard lattice Boltzman-n model with the proper amending functions. Different evolution equation, local equilibrium distribution functions and amending functions are selected, and the nonlinear damped wave equation can be recovered by using the Chapman-Enskog multi-scale analysis. In the construction of this model, only the distribution func-tion needs to be expanded, which greatly simplifies the theoretical derivation of the model and extends the scope of applications. Some initial boundary value problems with analytical solutions, including the second-order telegraph equation, the non-linear Klein-Gordon equation and the damped, driven sine-Gordon equation, are studied. The numerical results indicate that the present method is more efficient than the improved finite difference method and the radial-basis-functions method. Meanwhile, some problems without analytical solutions are studied by the present model and the improved finite difference method, which shows that the numerical solutions of the two schemes are coincided with each other. The numerical solution clearly reflects the propagation characteristic of the nonlinear wave.Finally, A D1Q5non-standard lattice Boltzmann model with the proper a-mending functions is constructed to solve the initial boundary value problem of the generalized nonlinear second-order Benjamin-Ono equation. Due to the higher order derivative (?)4u/(?)x4and the nonlinear term (?)2u2/(?)x2in the macroscopic equation, some reasonable modifications of the local equilibrium distribution functions and the amending functions in the existing model are made. Through the Chapman-Enskog multi-scale analysis, the macroscopic nonlinear equation can be exactly recovered. The initial boundary value problems of "good" Boussinesq and "bad" Boussinesq equations with analytical solutions, are studied. The numerical results show that the present model can be used to simulate the generalized nonlinear second-order Benjamin-Ono equation within a certain range. The proposed model provides ref-erences for further applications of the LBM to other NPDEs.In conclusion, our results not only enrich the application of the LBM in solving NPDEs, but also help to provide valuable references for solving more complicated nonlinear problems in the future. Therefore, this research has important theoretical significance and application value.