Space-Time Spectral Method For Solving Fifth-Order Partial Differential Equation

Spectral method has been paid more and more attention by researchers because of its high-order convergence rate.Together with finite difference method and finite element method,spectral method have become important numerical methods for solving partial differential equation.For solving time-dependent partial differential equation,when the exact solution is smooth enough both in time and space,if the difference scheme is used in time direction,it will easily lead to the loss of spectral accuracy.So it's necessary to consider spectral method in time direction.Dividing the time interval into multiple domains and using the spectral method in each domain,this method is called the space-time spectral method of the multiple domains in time.This thesis mainly discusses the space-time spectral method of the fifth-order partial differential equation and the space-time spectral method of the multiple domains in time of the fifth-order partial differential equation.In this thesis,we propose the space-time spectral method for the fifth-order partial differential equation.Legendre-Petrov-Galerkin method is applied in space direction,and Legendre-tau method is applied in time direction.In this thesis,the theoretical analysis of the method on linear problems is given by introducing orthogonal projection operator PN*.At the same time,we choose proper basis functions to make the coefficient matrix sparse.Finally,some numerical examples are given to verify the effectiveness of the algorithm.In addition,the space-time spectral scheme of the multiple domains in time for the fifth-order partial differential equation is given in this thesis,the nonlinear term for the nonlinear equation is computed by interpolation through Chebyshev-Gauss-Lobatto points.At the same time,we also give the theoretical analysis of the method on linear problems.Finally,some numerical examples are given to verify the effectiveness of the algorithm.