Spectral Method For A Class Of Evolution Equations

Spectral method is more and more widely used in many fields of science and engineering,for example, fluid mechanics, water conservancy and hydropower, ocean engineering,quantum mechanics, atmospheric sciences, electromagnetic technology and so on. Afterdecades of development, spectral method has been made much progress not only in theory butalso in numerical simulation. Recently, rapidly developing of spectral method perhaps mostlydue to its “spectral accuracy”. Its convergence and the approximation properties only dependon the smoothness of associated problem, that is, the smoother of solution is, the higher rateof convergence is. If the solution is of infinite smooth, then its convergence rate is ofexponential order. By using the spectral method to solve the certain problems, error analysisof the scheme is particularly important, especially for nonlinear problems. This paper isdedicated to spectral method of three nonlinear equations and their convergence and stability.Firstly, we discuss the numerical solution of fractional BBM-Burgers equation withDirichlet boundary condition.The classical BBM-Burgers equation is firstly proposed in theresearch of nonlinear dispersive wave propagation at the1972by T.B.Benjamin, J.L. Bonaand J.J.Mahony. Replacing classical derivative by fractional order derivative with respect totime variable, the fractional order differential equation is obtained. The FractionalBBM-Burgers equation is treated by Legendre-Chebyshev spectral method, the numericalscheme to take the form of Legendre-Galerkin in the whole, the nonlinear terms bycollocation method. Then we prove the existence, stability and convergence of thesemi-discrete scheme in the sense of theH1estimates. From the results of theory andnumerical experiments we can illustrate the superiority of the spectral method.Secondly, we discuss generalized regularized long-wave equation with zero boundarycondition. This equation describes the numerical plasma wave weak spatial transformationunder the action of nonlinear propagation. Making use of the same numerical method in thelast chapter, semi-discrete scheme for the equation is established. We verified the stability andconvergence of the scheme and obtained the optimal convergence order estimation.Fully-discrete scheme is obtained after time discretization. Also stability and convergence areset up for the fully-discrete scheme. Finally, the results of numerical simulation with thisscheme are in good agree with the actual situation.Last, in chapter5, we discuss spectral method to solve the RLW problem with non-zeroboundary. After a certain transform, we change the non zero boundary condition into the zeroboundary condition. Meanwhile, the original equation is transformed into anothor nonlineardifferential equation. We make use of the Legendre-Galerkin method to establishsemi-discrete scheme for transformed equation. Employing Laplace modification for time discretization, two schemes are obtained: backward Euler and C-N scheme. For the nonlinearterm, we use interpolation at Chebyshev-Gauss-Lobatto nodes and Chebyshev-Legendretranformation. With the Legendre polynomials as basis functions, the coefficient matrix ofdiscrete equations is decomposed into two subsystems with tri-diagonal matrix. We also setup convergence rate of approximate solution in the sense ofH1.In this paper we use traditional energy estimation method for the error analysis. Stabilityis in sense of generalized stability which is put forward by Guo Benyu.