Highly Accurate Scheme And Compact Alternating Direction Scheme For Schr(?)dinger Equations

Schr?dinger equation is a basic assumption of quantum mechanics. In quantum mechanics, it's just as important as Newton's mechanics in Classical Mechanics. As a result of its importance, the Schr?dinger equation has been studied for a long time. The Coupled nonlinear Schr?dinger equations(CNLSE)is one of these equations, its interesting figure has attracted wide attention in mathematics, physics, chemistry and other fields. In this paper, we proposed two highly accurate finite difference schemes for the CNLSE, the impact of the parameters in the equations was discussed through a large number of numerical experiments.In the first chapter, we introduced the physical meaning of the solution of Schr?dinger equation, the meaning of its parameters, and the theoretical results that have been down by other researchers. The basic lemma and definition of the marks in this paper was given at last.We proposed two different schemes for CNLSE in chapter two. The first scheme satisfies the conservation law and the second has a highly accuracy and stability, both of the natures were proved in this paper. Followed by a large amount of numerical experiment, we compared our results with previous results, observed and analyzed the numerical results. It can be found in our experiment that the scheme of chapter 2.2 is perfect.For two-dimensional linear Schr?dinger equation, we introduced the local one-dimensional compact scheme and the alternating direction compact scheme. The numerical experiment shows that both of the schemes are of good accurate.