Efficient Difference Schemes For Schr(o|¨)dinger Equations

In this thesis, we mainly study initial-boundary valued problems of Schrodinger equations. As we all know, Schrodinger equation is an important kind of models in physics. The numerical methods to research Schrodinger equations are various. Based on the idea to improve the computational efficiency, we propose two more efficient schemes for Schrodinger equations. The first one is originated from eliminating or improving the splitting error due to alternation direction. We propose a modified high order compact alternative direction implicit (ADII) for multidimensional Schrodinger equation. It does not only improve the accuracy of the scheme but also with small amount of extra computational cost. The other scheme is based on the idea of extrapolation which can improve the accuracy of low order multi-symplectic integrators by using extrapolation technique.In Chapter2, we focus on construct an high order compact (HOC) modified alternating direction scheme for Schrodinger equations. Firstly, we discuss high order compact method. Under the same grid stencil, it gets higher order accuracy than general central difference schemes. Then, through the error analysis of the typical alternating direction implicit scheme, we add a modified term to reduce the splitting error and improving the accuracy. At the same time, we extend the scheme to three dimensional equations and nonlinear cases. Numerical experiments confirmed the validity and correctness of the scheme.In Chapter3, We will show that the Schrodinger equations we considering in this paper is a Hamilton system, and the classic Preissmann scheme which conserves the multi-symplectic geometry structure of the original system is applied to the discretion of the Schrodinger Hamilton system. One of the most important advantage of symplectic method is that the numerical simulation can be performed for a long time, however, it is low accuracy. For this reason, we introduce Richardson extrapolation method both in the time and spatial directions. The same discrete method applied to simulate the same problem with different time and spatial mesh sizes. Different numerical solutions are obtained at the same points. Finally, combining linearly the two sets of data can get new results to improve the accuracy. We also use numerical tests to verify.In Chapter4, we summarize our work and state further work and our plan.