High Order Space-Time Spectral Methods For Some Kinds Of Evolution Equations

Evolution equations(parabolic equations, wave equations and hyperbolic equations,etc.) have wide applications in practical engineering problems. These equations appear in material, mechanical, optical, thermal conductivity, vibration, fluid motion, control systems and biological, etc.In this thesis, the discrete schemes spectral collocation method is given for first-order nonlinear ordinary differential equations, thus this method is extended to solve secondorder nonlinear ordinary differential equations by matrix transformation. Then spectral method in both temporal and spatial discretizations is proposed to solve one-dimensional semilinear parabolic equations, one-dimensional Sine-Gordon equation, two-dimensional semilinear parabolic equations and two-dimensional generalized Sine-Gordon equation.A priori error estimates is derived for the semidiscrete formulation. The proposed methods have spectral accuracy in both space and time.The main contents can be summarized as followsFirstly, the discrete schemes spectral collocation method is given for the nonlinear first-order initial value problems. This method is extended to solve the nonlinear secondorder initial value problems by matrix transformation.Then classical methods(boundary value method, etc) are also extended to solve the nonlinear second-order initial value problems by analogous matrix transformation. These new methods preserve the accuracy of the original methods and the main advantages of these new methods are low storage requirements and high efficiency. The stability of our methods are also discussed.Secondly, a Chebyshev–Galerkin spectral method is applied for discretizing spatial derivatives, thus a spectral collocation method or a block spectral collocation method is used for the temporal discretization. Optimal a priori error bound is derived in the weighted L2ωnorm for the semidiscrete formulation. Comparisons of the CPU times and errors in numerical tests with other methods are given. Numerical results of the numerical experiments show that space-time spectral method is an efficient algorithm. Also numerical results confirm the exponential convergence of the space-time spectral method in both space and time.Thirdly, a space-time spectral collocation method is constructed for solving the onedimensional Sine-Gordon equation. Optimal a priori error bounds are derived in the L2 norm for the semidiscrete formulation.Next, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. The method is based on a Legendre-Galerkin spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of first-order ordinary differential equations.Optimal a priori error bound is derived in the L2 norm for the semidiscrete formulation.Finally, a space–time spectral method is constructed to solve generalized two-dimensional Sine-Gordon equation. The proposed method is based on the Legendre-Galerkin spectral method in space and the spectral collocation method(or block spectral collocation method) in time. Optimal a priori error bounds are derived in the L2 and H1norms for semidiscrete formulation.