Research On Solving Several Nonlinear Partial Differential Equations Based On The Lattice Boltzmann Method

Nonlinear partial differential equations play an important role in mathematics,physics and engineering technology.With the development of computer techniques,there are more and more methods to solve nonlinear partial differential equations.Lattice Boltzmann Method(LBM),as an emerging numerical method,makes many contributions to solving nonlinear partial differential equations.This thesis uses LBM to solve the generalized KdV-Burgers equation with variable coefficient and construct a unified model framework for solving nonlinear partial differential equations based on previous studies.The numerical experiments verify the model is self-consistent.The main works of this research are as follows:1.Recover the continuity equations,the Euler equations,and the Navier-Stokes equations.First,the development process,basic theory,and basic model of the LBM are introduced.Then,a series of macroscopic equations are recovered based on the CE expansion technique,and the conditions for satisfying the model stability are given.2.Study the numerical calculation problems of a class of generalized KdVBurgers equations with variable coefficients.First,the compatibility of the discrete latticeBoltzmann equation and the macroscopic KdV-Burgers equation is proved.Then,the parameter regulating the equilibrium distribution function and the parameter k regulating the relaxation factor are introduced in the thesis,which ensure the stability of the model.Numerical results show that the numerical accuracy in this study is better than that of Ref [46].3.Construct a model framework for solving nonlinear partial differential equations.First,The compatibility of the evolution equation and the macroscopic equation is proved by using the CE method and the direct Taylor expansion method.And the similarities and differences between both methods are concluded.For the unstable Burgers' equation,we compare the numerical results of the LBM with the HOC algorithm in the PHOEBE Solver.As a result,the model uses less CPU time and the computational efficiency was improved by 78%-92% under the condition that the spatial accuracy is of the fourth order.Then the LBM is applied to the KdV and KS equations,which shows the numerical solutions agree well with the analytical solutions.