Spectral Method For A Class Of Parabolic Equations

The contents of this dissertation are divided into three chapters.In chapter one, the Legendre-Galerkin method for the variable coefficient parabolic equation is devised. The Laplace modified method is used in time step and the backward Euler scheme and Crank- Nicolson scheme are developed. By applying the collocation method for the nonlinear term, the computation time is greatly saved. By choosing appropriate basic functions, the coefficient matrix of the discrete system can be not only sparse but also decomposed to two tridiagonal subsystems. At last, the optimal order in H1norm is derived for the error in the approximation solution.In chapter two, the Legendre-Galerkin Chebyshev collocation method for the regularized long wave equation is used and the semi-discrete and Crank-Nicolson full discrete method are derived. For the discretization in space, the scheme is basically formulated in the Legendre spectral method but the nonlinear term is computed by the Chebyshev collocation method at the Chebyshev-Gauss-Lobatto points. By choosing the same base functions as the previous chapter, the similar coefficient matrix is obtained. At last, the optimal order in H1norm is derived for the error in the approximation solution.Chapter three studies the linearized implicit Fourier pseudo-spectral method for the generalized Burgers equations. The Crank-Nicolson implicit scheme is used in time step and the nonlinear term is linearized by the Taylor's formula. Because it need to solve a big algebra system in every time step, the use of iteraton scheme in this chapter can avoid the circumstance. At last, the error estimate in L2-norm is derived for the approximation solution.