Study On The Spectral Methods For Nonlinear Sine-Gordon Equation

The Sine-Gordon equation was first introduced in differential geometry,when it was used to describe Gaussian curvature.Because of its volatility characteristics similar to the shallow water wave equation of the fluid,so it is also called the nonlinear wave equation.The nonlinear Sine-Gordon equation is an important nonlinear equation in applied science,plays a very important role in mathematical physics and many practical applications.For a long time,many researchers have tried a variety of methods to solve it,some of them used the finite difference method to give a numerical solution,as well as some used the finite element method,but there is no much good results for solving Sine-Gordon equation by means of spectral methods.The spectral method is a numerical method for solving the partial differential equations developed in the 70's.,which originated in the Ritz-Galerkin method.The biggest advantage of spectral method is called "infinite order convergence" and can be implemented fast Fourier calculations.So it has been widely used in many fields and has become an important numerical method following finite difference method and finite element method.This paper introduce the basic theory and the history viewing of spectral methods,and the development of the early and recently nonlinear Sine-Gordon equation.Afterward,on the basis of analyzing the spectral methods of two kinds of linear basic problems,we construct the semi-discrete and fully discrete schemes for one dimensional and two dimensional nonlinear Sine-Gordon equations.And then we give the error analysis of these two kinds of discrete schemes.Finally,some numerical examples are given.The results show that the method has good accuracy,and it is a practical form.