Fast Evaluation of Nonreflecting Boundary Condit ions for the SchrSdinger Equation in One Dimension

-We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the SchrSdinger equation in one dimension. The exact nonrefleeting boundary condition contains a nonloeal term which is a convolution integral in time, with a kernel proportional to 1/v~. The key observation is that this integral can be split into two parts: a local part and a history part, each of which allows for separate treatment. The local part is computed by a quadrature suited for square-root singularities. For the history part, we approximate the convolution kernel uniformly by a sum of exponentials. The integral can then be evaluated reeursively. As a result, the computation of the nonrefleeting boundary conditions is both accurate and efficient. (~) 2004 Elsevier Ltd. All rights reserved. Keywords--Nonreflecting boundary condition, SchrSdinger equation, Fast algorithm. 1. I N T R O D U C T I O N The initial value problem for the time dependent SchrSdinger equation iu t (x , t )=-ux~(x , t )+V(x)u(x , t ) , fort>0, x C • , 0) = (1) in u n b o u n d e d domains arises in q u a n t u m sca t te r ing , unde rw a te r acoust ics , and wave guide sireu la t ion (where it is of ten referred to as the Fresnel equa t ion) . We will assume in th is p a p e r t h a t V(x) and Uo(X) are c o m p a c t l y s u p p o r t e d in the in terval [ 1 , 1]; the so lu t ion u(x, t) at l a te r t imes , of course, is not . March ing schemes based on some d i sc re t i za t ion of (1) i nev i t ab ly require the impos i t i on of ar t i f ic ia l b o u n d a r y condi t ions on a finite c o m p u t a t i o n a l domain . In o the r words, we mus t reduce the p rob l em defined on the real line to one on a f inite interval . tAuthor to whom all correspondence should be addressed. This work was supported by the U.S. Department of Energy under Contract DE-FGO2-00-ER25053. :~Current address: MadMax Optics, Inc., 3035 Whitney Avenue, Hamden, CT 06518. 0898-1221/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by A.A4$-TEX doi:10.1016/j.camwa. 2003.04. 004 956 S. JIANG AND L. GREENGARD There are several different approaches which have been applied to this reduction. One can modify the potential function V(x) outside the interval [-I, I] in some way so that outgoing waves are absorbed and spurious reflections are small. These boundary conditions are generally referred to as absorbing layers [I-3]. Since these conditions are semiempirical, the obtained accuracy, especially for dispersive equations like (I), tends to be low. Another approach is based on exact nonreflecting boundary conditions (ENBC), for which the resulting initial-boundary value problem is equivalent to the original problem [4,5]. The exact conditions, however, are nonlocal. This introduces two difficulties. First, the direct implementation of ENBC is highly inefficient. If we let NT denote the number of time steps in the simulation, then naive implementation requires O(N~) work and O(NT) storage. Second, discretization is a very delicate issue. It is shown, for example, in [6], that the Crank-Nicholson scheme coupled with certain discretizations of the ENBC is only conditionally stable. When coupled to Dirichlet or Neumann conditions, on the other hand, the Crank-Nicholson scheme is unconditionally stable. Various techniques have been proposed to overcome these two difficulties. First, there have been several efforts to obtain local boundary conditions based on local approximations of the ENBC [7,8]. These conditions are straightforward to implement and are computationally efficient. Unfortunately, their accuracy is difficult to analyze and, in practice, tends to depend very sensitively on the initial data. Second, a number of schemes have been developed over the last decade which are collectively referred to as discrete transparent boundary conditions (DTBC). These are designed for specific space-time discretizations of the SchrSdinger equation. The fully discrete case (both space and time) was treated in [9-11], the temporally semidiscrete case was treated in [12-15], and the spatially semidiscrete case was treated in the recent papers [16,17]. While this approach is very promising, the DTBC have certain drawbacks. First, almost all the available DTBC are of second-order accuracy. Second, the underlying grid must be uniform either in space or in time or in both. Third, the DTBC are still nonlocal (although suitable fast algorithms have been proposed in [16-18]). In this paper, we describe a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the one-dimensional SchrSdinger equation. As is well known, the nonlocal part of the exact nonreflecting boundary conditions in time is given by a convolution integral whose kernel has a square root singularity. Following the approach of [19,20], we split the convolution t--5 into two parts: a local part ftt_~ and a history part f0 . The local part is easily computed by a suitable quadrature. For the history part, we approximate the kernel by a sum of exponentials. Since convolution with an exponential function can be evaluated recursively, this approximation allows us to reduce the computation and storage cost dramatically. Rather than the O(N~) work and O(NT) storage required by the direct method, the algorithm in this paper requires only O(NT log NT) work and O(log NT) storage where the constant implicit in the O( ) notation depends only on the precision required. We then couple a simple second-order discretization of the "fast" ENBC with various interior discretization schemes. It is interesting to note that in our numerical experiments, the errors obtained are noticeably worse than those obtained from the DTBC (although both are second-order accurate). Nevertheless, our approach is more flexible; it can be coupled to arbitrary interior discretization schemes with variable time steps or spatial grids. Moreover, the algorithm can be generalized to twoor three-dimensional problems [21,22]. The paper is organized as follows. In Section 2, we derive nonreflecting boundary conditions which are exact, nonlocal in time, and easily expressed in terms of a convolution operator. In Section 3, we develop a sum-of-exponentials approximation for the convolution kernel. Section 4 contains a simple discretization scheme. The performance of the scheme is illustrated in Section 5.