Conservative Methods For Some Types Of Nonlinear Partial Differential Equations

Non-linear partial differential equations play an important role in the description of non-linear phenomena,which have been widely used in the fields of mechanics,physical chemistry,engineering technology and atmospheric science.Because of the existence of non-linear terms,it is difficult to obtain or express the analytical solution of non-linear problems,it is necessary to construct numerical methods to approximate the solution.In the numerical experiments,it is of great physical significance to design a numerical scheme which can keep conservation of integral invariants as much as possible for the accurate simulation of particle motion of non-linear partial differential equations.In this paper,we mainly study the spectral methods of several kinds of non-linear partial differential equations,including generalized normal long wave(MRLW)equation,coupled Schr?dinger Kd V equation and two-dimensional non-linear Schr?dinger equation The specific research work can be divided into the following parts:1. Consider the spectral method of mrlw equation under periodic boundary conditions.Firstly,the problem is discretized in space by Fourier spectrum collocation point method,and mrlw equation is approximated to matrix form ordinary differential equation on collocation point,and it is proved that this form has two kinds of semi discrete conservation laws.Then,using the time exponential crank Nicholson difference method in time direction,a kind of absolutely stable numeri-cal scheme based on Fourier spectrum method is constructed.The mrlw equation is completely discretized into a set of nonlinear equations about U(x,t)and solved by Newton-Raphson iterative method.The experimental results show that the proposed scheme has second-order convergence in the sense of L_∞.2. The spectral method of coupled Schr?dinger-Kd V equation under peri-odic boundary conditions is studied.Combining Fourier spectrum collocation point method and time exponential three-layer implicit difference scheme,a cou-pled implicit numerical scheme with three kinds of semi discrete conservation laws for coupled Schr?dinger-Kd V equation is constructed.By using the decoupling technique,the coupling system obtained by the full discrete scheme is approximated to a set of nonlinear equations,and the approximate solution is obtained by the fixed point iteration method.3. The spectral method of two-dimensional nonlinear Schr?dinger equation with homogeneous Dirichlet boundary condition is considered.Based on the Laplace operator’s spectral discretization and the exponential Crank-Nicholson difference method,a method with spectral accuracy in space and second-order convergence in time is constructed.Also numerical experiments show that the method can simulate the singular solution effectively.