Efficient Difference Methods For The Fractional Schr(o|¨)dinger Equation And Fractional Ginzburg-Landau Equation

Over the past two decades, with the wide applications of fractional partial differential e-quations, it has been shown that the models involving fractional derivatives always give better descriptions of some phenomena in science and engineering. The fractional Schrodinger equa-tion and fractional Ginzburg-Landau equation are two important classes of fractional partial differential equations, which have attracted great attention in physical applications and mathe-matical analysis. Since their analytical solutions can not be derived in general, one must resort to numerical methods and it is the main theme of this dissertation. More precisely, for the frac-tional Schrodinger equation, we will propose some efficient difference schemes which satisfy some conservation properties. We pay main attention to the theoretical analysis and numerical verification of the conservation and convergence. While for the fractional Ginzburg-Landau e-quation, we will construct and analyze two class of difference schemes. We focus on analyzing the unconditional convergence of the schemes. The whole dissertation contains the following seven parts:In Chapter 1, we introduce the two classes of equations and review briefly the research background and the present state of the research about the two equations.From Chapter 2 to Chapter 5, the fractional Schrodinger equation will be considered. In Chapter 2, we present a difference scheme which satisfies both the discrete mass and energy conservation. Then by introducing some new techniques, we rigorously prove the conservation and convergence.Chapter 3 is devoted to overcoming the nonlinear difficulty of the scheme introduced in the last chapter. In view of the Adams-Bashforth extrapolation, we propose a linearized difference scheme. The convergence is also analyzed. In addition, using the proposed scheme, we study numerically the collision of two solitons under the fractional impact.In Chapter 4, the unconditional convergence of some conservative schemes is considered. By introducing the fractional Sobolev norm and establishing some useful inequalities and an equivalence relation, we show the unconditional convergence of the schemes in the lh2 norm, semi-Ha?/2 norm and lh? norm.In Chapter 5, we study the two dimensional problem. For overcoming the nonlinear and multi-dimensional difficulties, we split the original problem into a linear problem and a simple nonlinear problem via the split-step technique so as to solve them sequentially and in particular, we construct some ADI schemes for the linear problem. A theoretical analysis and numerical testification show that, the scheme satisfies the mass and energy conservation in the linear case but conserves the mass in the nonlinear case. Moreover, numerical comparisons with other existing schemes show the effectiveness of the scheme.In Chapter 6 and Chapter 7, the fractional Ginzburg-Landau equation will be considered. We in Chapter 6 construct an implicit midpoint difference scheme, which is second order accurate in both time and space, and analyze the unconditional convergence in the lh2 norm.In Chapter 7, we propose another difference scheme. The scheme is based on a combination of the BDF2 rule and the explicit second order Gear's extrapolation method in time, and the fourth order quasi-compact method for the fractional Laplacian. By introducing the G-stability theory, we obtain the unconditional convergence of the scheme in the lh2 norm, semi-Hh?/2 norm and lh? norm. Numerical tests show the superiority of the scheme in accuracy and efficiency.