A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Abstract

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.

Introduction

Wave problems in unbounded domains exist in many fields of application, such as quantum mechanics, optics, acoustics, geophysics, electromagnetics and aerodynamics [24]. For many such instances, the aim of numerical simulation is to compute the wave function around a small bounded part of domain which bears special physical interest. A common practice of accomplishing this is to limit the computational domain and solve a “truncated” problem with a suitable domain-based method, such as finite element or finite difference. The computational domain is usually tailored as small as possible, but at least it should contain the physically interested part. To make complete the truncated problem, special boundary conditions should be designed and applied at the domain boundary. In principle, these boundary conditions should mimic the absorption of waves into the unbounded exterior, and right in this sense, they are usually referred to as absorbing boundary conditions (ABCs) in the literature. Other names of same spirit are also popularly used, such as non-reflecting boundary conditions, transmitting boundary conditions and transparent boundary conditions.

In terms of the taxonomy of Guddati and Lim [26], most of the existing ABCs for wave problems can be classified into two categories: PDE-based and material-based. PDE-based ABCs are applied at prescribed artificial boundaries, and obtained either by factorizing the field equation and allowing only outgoing waves, or by solving the exterior wave problems with an analytical or semi-analytical method. In terms of the forms of formulation, PDE-based ABCs can be further classified into two sub-categories: local and global. Local ABCs are presented with a set of differential relations. They are computationally efficient, but usually contain two contradictory elements: accuracy and stability. Comparatively, global ABCs, which attempt to realize the complete absorption of waves, usually lead to well-approximated and well-posed truncated problems. However, from the implementational point of view, these ABCs are expensive for large-scale wave problems. This is typically because global ABCs are usually formulated in a pseudodifferential form, which contains convolution operations either in space or time, or both. Another great disadvantage of global ABCs is that they are difficult to derive for nonlinear problems. Up to now it is only possible for some specific nonlinear equations, see [30], [53], [23], [54]. On the PDE-based ABCs for wave problems, Givoli [25], Tsynkov [51] and Hagstrom [27], [28] are some good reviews.

Material-based ABCs employ a different philosophy. Instead of using artificial boundaries to limit the computational domain, they resort to a finite-thickness lossy material surrounding the physically interested part to annihilate the outgoing waves. These ABCs became extremely popular and successful after Bérenger proposed perfectly matched layer (PML) in 1994 for computational electromagnetics [14], and Chew et al. [16] offered a wonderful interpretation that PML was in fact equivalent to a continuation of the real coordinate into the complex plane. This interpretation together with many later developments [18], [41] makes it possible to extend PML to more complicated problems. For example, Hu [34] has applied PML to the nonlinear Euler equations recently.

The designing of ABCs for the linear Schrödinger (LS) equation dates back to the early 1980’s in [40], where the exact ABC was presented by Papadakis for the one-dimensional (1D) LS equation in underwater acoustics. Later, it was rederived independently by many authors from different application fields [8], [13], [31]. This exact ABC is global in time and involves a temporal half-order derivative operator. Discretization of this half-order derivative operator turns out to be a delicate issue, since any unsuitable choice, as Mayfield [39] pointed out, may destroy the underlying stability of the interior Crank–Nicolson scheme and lead to an ill-posed problem. Many authors have worked on remedying this problem. For example, instead of discretizing the exact continuous ABC for the 1D LS equation, Schmidt et al. [20], [22], [45], Antoine and Besse [3] derived the exact ABC for the semi-discretized equation in time. The advantage of this treatment lies in the fact that the unconditional stability of the Crank–Nicolson scheme can be maintained automatically. Arnold and Ehrhardt [8] used a different strategy. They first performed discretization in both time and space, and then derived an exact completely-discrete ABC. Again, the stability of the Crank–Nicolson scheme is preserved. Besides, their numerical experiments showed the accuracy of numerical solutions could be greatly improved. Sun and Wu [48] proposed another idea. Instead of changing discretization of the exact continuous ABC in [13], they modified the interior discretization scheme. A special finite difference scheme was designed, and shown to be unconditionally stable in concert with the discretization in [13].

For the two-dimensional (2D) LS equation, Schädle [44] presented an exact ABC in a form of integral equation and designed a numerical scheme. Han and Huang [29] derived an exact ABC in a series form at a circular artificial boundary. Arnold, Ehrhardt and Sofronov [9] studied the completely-discrete exact ABC for the LS equation in waveguides. The above work is concerned with global ABCs. Local ABCs have also been studied. For example, Antoine, Besse and Mouysset [5], Szeftel [50] designed local ABCs for the 2D LS equation at arbitrary curved artificial boundary with the pseudodifferential operator theory. Fevens and Jiang [21] extended Higdon’s idea [32], [33] of designing high-order local ABCs to the LS equation in any dimension. The fast evaluation of global ABCs can be also classified into the category of local ABCs. On this subject, Arnold, Ehrhardt and Sofronov [9], Jiang and Greengard [35], Lubich and Schädle [37], and Alonso-Mallo and Reguera [2] have made great contributions.

During the last several years, increasing attention has been attracted on ABCs for the nonlinear Schrödinger (NLS) equation. The first result on this subject, to the author’s knowledge, is given by Fokas [23]. By the inverse scattering theory, Fokas presented an exact ABC for the 1D cubic NLS equation. Since a double integral gets involved which makes it unsuitable for numerical purposes, Zheng [54] modified Fokas’ ABC and designed a numerical scheme. By the tool of pseudodifferential operators, Antoine, Besse and Descombes [4] constructed a series of integro-differential ABCs for the same equation. Relying on the pseudodifferential calculus and paradifferential calculus, Szeftel [49] derived two families of ABCs for a general 1D NLS equation. Very recently, Soffer and Stucchio [46] presented a new algorithm of solving a general semilinear Schrödinger equation. The idea involved is to solve the semilinear Schrödinger equation on a box with periodic boundary conditions, and decompose the solution into a family of coherent states. The absorption of waves is then realized by deleting those outgoing coherent states.

In comparison to the great deal of work on PDE-based ABCs, the work on material-based ABCs for the LS and NLS equations is rather little [17], [28]. In this paper, we intend to apply PML to the nonlinear Schrödinger wave equations. The idea is stimulated by the good performance of PML for the 1D LS equation with constant potentials, and the time-transverse invariant property of the nonlinear Schrödinger wave equations. Its effectiveness will be demonstrated through numerical tests under different settings of problem.

The rest of this paper is as follows. In Section 2, we explain the PML idea for the LS equation with constant potentials, and in Section 3, we extend it for the LS equation with variable potentials. In Section 4, the PML is applied to the 1D NLS equation, and a time-splitting central difference scheme is proposed. Application of PML for the high-dimensional NLS equation is considered in Section 5. The alternating direction implicit (ADI) scheme is used to overcome the difficulty arising from high dimensions. In Section 6, the PML technique is further extended for the Schrödinger-coupled systems. A time-splitting fast sine transformation (FST) scheme is presented for the model Schrödinger-Poisson system in waveguide. This paper concludes in Section 7.