Spectral Method For The Generalized Sine-Gordon Equation

This thesis is an investigate of the spectral method of the two-dimensional generalized sine-Gordon equation.It mainly studies how to use the knowledge of the spectral method to solve the two-dimensional generalized sine-Gordon equation,and presents the numerical solution method for solving the two-dimensional generalized sine-Gordon equation.And after the numerical solution method for solving the two-dimensional generalized sine-Gordon equation is given,the numerical solution and the analytical solution are compared,and the error analysis of the numerical solution and the analytical solution is calculated.Since the equations studied in this article contain time and space terms,the research in this article is divided into three parts.The first part is the time discretization of the two-dimensional generalized sine-Gordon equation.Because the equation in this article contains a second derivative in the time direction,you can choose to apply the configuration method in the spectral method.By selecting the configuration point,the second-order ordinary differential equation Converted into the form of first-order ordinary differential equations,and then solved;you can also choose to apply the Runge-Kutta method to solve.The second part is the spatial discretization of the two-dimensional generalized sine-Gordon equation.There are many methods for spatial discretization,such as finite element method,configuration method,etc.However,this article chooses to apply the Galerkin method in the spectral method,because the Galerkin method can Make full use of the orthogonality of polynomials,and have more advantages in the design of numerical programs,and the Galerkin method has higher convergence accuracy(it can converge according to exponential accuracy).The third part compares the numerical and analytical solutions of the two-dimensional generalized sine-Gordon equation in this thesis,and obtains an error analysis.The error analysis in this article is the error analysis of the semi-discrete format,that is,only the error analysis after spatial discretization.This article will use relevant knowledge and lemmas to apply the Chebyshev-Galerkin method to the numerical solution obtained after spatial discretization.The error analysis is carried out under the space.