Study On Lattice Boltzmann Model For Some Nonlinear Partial Differential Equations

Nonlinear partial differential equations play an important role in the fields of mathematics,physics and engineering.With the development of computer technology,more and more methods for solving nonlinear partial differential equations have been proposed.As an emerging numerical method,the lattice Boltzmann method has made many contributions to solving nonlinear partial differential equations.In this dissertation,lattice Boltzmann method is applied to numerical simulations of one-dimensional nonlinear differential partial equations and two-dimensional generalized nonlinear damped wave,respectively.At the last,the effectiveness and applicability of the models are test and verified by the numerical simulations of several examples.The main work is as follows:(1)For the one-dimensional fourth-order nonlinear partial differential equation with source,a lattice Boltzmann model D1Q5 with correction term is established,using the Taylor expansion and Chapman-Enskog multi-scale expansion technique,and selecting the appropriate local equilibrium distribution function and correction function,the model is restored to the macro equation.By simulating four numerical examples,comparison charts and absolute value error maps of numerical solutions and exact solutions at different time are shown,respectively.The simulation results show that the numerical solution is basically consistent with the exact solution at different times,but the calculation error is relatively large at the position of the waveform change.(2)For the two-dimensional generalized nonlinear damped wave with source,a lattice Boltzmann model D2Q5 with correction term is established,using the Taylor expansion and Chapman-Enskog multi-scale expansion technique,and selecting the appropriate local equilibrium distribution function and correction function,the model is restored to the macro equation.By simulating three numerical examples,the numerical solution and exact solution at different times,the convergence rate figures and the comparison of the diagonal numerical solution and the exact solution are given.and the convergence rate figures from the simulation results of the new model,the accuracy of the model in space verifies are obtained and the effectiveness and applicability of the new model are verified.