Legendre Space-time Spectral Methods For Two Types Of Time-dependent Partial Differential Equations

Spectral method,as a high precision numerical method,has been widely used for partial differential equations to approximate the exact solution.The biggest characteristic of spectral method is its spectral accuracy.When the solution of the original equation is smooth enough,the numerical solution obtained by proper spectral method can achieve exponential convergence rate.For the time-dependent problems,spectral method is usually used for spatial approximation,and finite difference scheme is applied in time.Such practice will lead to mismatch convergence accuracy in space and time,and the overall accuracy of numerical solution will be reduced.The key idea of space-time spectral method is that spectral approximation is employed in space and time,simultaneously,then exponential convergence can be obtained in both space and time.In addition,compared with single interval spectral method,the multi-interval method can not only reduce the scale of the problem,but also allow the parallel computation.In this thesis,Legendre space-time spectral method is investigated for solving the linear Sobolev equations and third order differential equations with nonperiodic boundary conditions.In Chapter 3 and Chapter 4,single interval Legendre space-time spectral fully discrete scheme and multi-interval scheme are established for multi-dimensional and two-dimensional Sobolev equations,respectively.In theoretical analysis,the rigorous stability and convergence of the numerical schemes are proved.In addition,based on the proper basis functions in time,the Fourier-like basis functions being different from the traditional one are selected in space to present the algorithm implementation.Lots of numerical examples verify that: Firstly,the numerical solutions obtained by our numerical methods achieve the exponential convergence rate in both space and time;Secondly,compared with traditional basis functions,the employment of Fourier-like basis functions succeed in promoting the effect of computing time and memory considerably;Thirdly,based on the proper time interval subdivision,the multi-interval spectral method can get better numerical result than the single interval method.In Chapter 5,Legendre PetrovGalerkin space-time spectral fully discrete scheme is constructed for the third order linear differential equation.Stability conclusion and error estimate of numerical scheme are given and algorithm implementation is presented by proper base functions in space and time.In numerical experiment,the exponential convergence rate is achieved in both space and time,and the numerical results are compared with those calculated in other study to further verify the efficiency of our methods.