Stutdy On The Spectral Collocation Method Of Nonlinear Schrodinger Eqution

With the rapid development of modern computer and the emergence of the fast Fourier transform,spectral method developed rapidly.It has become one of the powerful tools for the research numerical solution of nonlinear partial differential equation.”Exponentially decaying order”is one of the most favoured superiority of spectral method.In other words,spectral method can get the error of order 10-10 with very few number of nodes.Therefore,making full use of the characteristic of spectral method to solve higher order or more complex the partial differential equation,amount of calculation can be greatly reduced.Nonlinear schr(?)dinger equation is one of many important nonlinear evolution equations in mathematical physics.It has a very important role in modern mathematics,quantum physics,quantum chemistry.The research of its solution is of great significance.In this thesis,we can solve schr(?)dinger equations,poisson equations,nonlinear schr(?)dinger-poisson equations of 1d,2d,3d by combining the time splitting method with the spectral collocation methods of Legendre,chebyshev and Fourier.First,the derivation of solving process is given.Then we enumerate the specific numerical example,and we get the numerical solution of partial differential equations by matlab code.By comparing with its numerical solution and exact solution,we draw out of error analysis of form in long out of sync.It can be confirmed that spectral method is exponentially approximation accuracy for smooth function from calculating results.