The Numerical Simulation Of The Linear Schr(?)dinger Equation On Unbounded Domains Based On The Compact Finite Difference Method

In this paper, we consider the numerical solution of Schr(?)dinger equation on unbounded domains as follows:,where and m represent the Plank constant divided by 2? and atomic mass, respectively. ?(x, t) is the wave function, and i =?-1 is the imaginary unit. The initial condition ?0(x) and source term f(x, t) are compactly supported functions.The greatest difficulty in solving the numerical solution of the problem is the unbounded nature of the region. At present, the artificial boundary method is one of the powerful methods to solve such difficulty. After introducing the artificial boundary, the unbounded region is divided into two parts: bounded calculation region and unbounded exterior region. Then, we can make artificial boundary conditions on the artificial boundary. Thus, the solution of the initial boundary value problem is a good approximation of the original problem. Therefore, this paper introducestwo methods of constructing artificial boundary conditions: one is that based on Pade approximation and inverse Laplace transform, we review the local absorbing boundary conditions for linear Schrodinger equations in the second section of the second chapter; anther one is that we consider a ”artificial boundary” with a positive thickness. The artificial layer surrounds the calculation area, so that most of the layer's waves can be absorbed. This layer is proposed by Berenger and Maxwell,named the perfectly matched layer(PML). We review the perfect matched layer of linear Schrodinger equation by means of complex coordinate extension method in the second section of the third chapter. Therefore, the Schr(?)dinger equation on unbounded domain is simplified as initial boundary value problem on bounded calculation region.Finite difference method is one of the effective methods for numerical solution of initial boundary value problem. Traditional one order and two order finite difference schemes are difficult to obtain the numerical solution of high accuracy, unless a large number of grid nodes are used. However, this will greatly increase the amount of computation and computing time. In order to solve the disadvantages of the traditional finite difference schemes, a natural method is to construct a compact finite difference scheme. This scheme can not only improve the accuracy of numerical solution under the premise of not increasing the grid nodes, but also can reduce the amount of computation. Therefore, the four order compact finite difference scheme is used to solve the initial boundary value problem in the third section of the second chapter and the third chapter, respectively. With the quotient for the time derivative at the time layer, we obtain two order accuracy in time. In addition, the fourth section of second chapter introduces the four order Runge-Kutta method to obtain the corresponding initial boundary value problem. It is four order accuracy in both time and space. Finally, a numerical example is given to demonstrate theeffectiveness of the compact finite difference scheme.