Local discontinuous Galerkin methods for nonlinear Schrödinger equations

Abstract

In this paper we develop a local discontinuous Galerkin method to solve the generalized nonlinear Schrödinger equation and the coupled nonlinear Schrödinger equation. L² stability of the schemes are obtained for both of these nonlinear equations. Numerical examples are shown to demonstrate the accuracy and capability of these methods.

Introduction

In this paper we develop a local discontinuous Galerkin method to solve the generalized nonlinear Schrödinger (NLS) equation

$$\mathrm{i}{u}_t+u_{\mathrm{xx}}+\mathrm{i}(g(|u{|}^2)u{)}_x+f(|u{|}^2)u=0\text{,}$$

the two-dimensional version

$$\mathrm{i}{u}_t+\mathrm{\Delta }u+f(|u{|}^2)u=0\text{,}$$

and the coupled nonlinear Schrödinger equation

$$\left\{\begin{array}{c}\mathrm{i}{u}_t+\mathrm{i}\alpha u_x+u_{\mathrm{xx}}+\beta u+\kappa v+f(|u{|}^2\text{,}|v{|}^2)u=0\text{,}\hfill \\ \mathrm{i}{v}_t-\mathrm{i}\alpha v_x+v_{\mathrm{xx}}-\beta u+\kappa v+g(|u{|}^2\text{,}|v{|}^2)v=0\text{,}\hfill \end{array}\right.$$

where f(u) and g(u) are arbitrary (smooth) nonlinear real functions and α, β, κ are real constants.

The cubic nonlinear Schrödinger equation

$$\mathrm{i}{u}_t+u_{\mathrm{xx}}+|u{|}^{2}u=0\text{,}$$

which is a special case of Eq. (1.1), describes many phenomena and has important applications in fluid dynamics, nonlinear optics, and plasma physics [3], [4], [15]. Its structure is reminiscent of the Schrödinger equation in quantum physics, where |u|² has the significance of a potential. In Eq. (1.4) the complex function u(x, t) describes the evolution of slowly varying wave trains in a stable dispersive physical system with no dissipation, for example waves in deep water. Various kinds of numerical methods can be found for simulating solutions of the NLS problems [5], [17], [20], [21], [24], [25], [29]. In [5], [29], several important finite difference schemes are tested, analyzed and compared. In [24], a pseudospectral solution of GNLS equation is considered. A numerical solution of the NLS equation is obtained by using the quadratic B-spline finite element method in [17]. The convergence of a class of space–time finite element method for the nonlinear (cubic) Schrödinger equation is analyzed in [20], [21]. The discontinuous Galerkin method considered in [20] refers to a discontinuous Galerkin discretization in time, hence is different from our approach of using a local discontinuous Galerkin discretization for the spatial variables.

The two-dimensional nonlinear Schrödinger equation (1.2) is a generic model for the slowly varying envelop of a wave-train in conservative, dispersive, mildly nonlinear wave phenomena. It is also obtained as the subsonic limit of the Zakharov model for Langmuir waves in plasma physics [34]. It is possible for solutions of the two-dimensional nonlinear Schrödinger equation to develop singularities at some finite time t₀ [18]. The linearized Crank–Nicolson finite difference scheme was used to compute the two-dimensional NLS equation in [27].

The coupled nonlinear Schrödinger equation

$$\left\{\begin{array}{c}\mathrm{i}{u}_t+\mathrm{i}\alpha u_x+\frac{1}{2}{u}_{\mathrm{xx}}+(|u{|}^2+\beta |v{|}^2)u=0\text{,}\hfill \\ \mathrm{i}{v}_t-\mathrm{i}\alpha v_x+\frac{1}{2}{v}_{\mathrm{xx}}+(\beta |u{|}^2+|v{|}^2)v=0\text{,}\hfill \end{array}\right.$$

were first derived 30 years ago by Benney and Newell [2] for two interacting nonlinear packets in a dispersive and conservative system. The classification of the solitary waves is considered in [33]. Ismail and Taha [19] introduced a finite difference method for the numerical simulation of the coupled nonlinear Schrödinger equation. In [28], a multi-symplectic formulation is considered.

The discontinuous Galerkin (DG) method we discuss in this paper is a class of finite element methods using a completely discontinuous piecewise polynomial space for the numerical solution and the test functions in the spatial variables, coupled with explicit and nonlinearly stable high order Runge–Kutta time discretization [26]. It was first developed for hyperbolic conservation laws containing first derivatives by Cockburn et al. [11], [10], [8], [12] in a series of papers. For a detailed description of the method as well as its implementation and applications, we refer the readers to the lecture notes [7], the survey paper [9], other papers in that Springer volume, and the review paper [14].

These discontinuous Galerkin methods were generalized to solve a convection diffusion equation (containing second derivatives) by Cockburn and Shu [13]. Their work was motivated by the successful numerical experiments of Bassi and Rebay [1] for the compressible Navier–Stokes equations. Later, Yan and Shu [31] developed a local discontinuous Galerkin method for a general K dV type equation containing third derivatives, and they generalized the local discontinuous Galerkin method to PDEs with fourth and fifth spatial derivatives in [32]. Levy, Shu and Yan [22] developed local discontinuous Galerkin methods for solving nonlinear dispersive equations that have compactly supported traveling wave solutions, the so-called “compactons”. Recently, Xu and Shu [30] further developed the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general K dV-Burgers type equations, the general fifth-order K dV type equations and the fully nonlinear K(n, n, n) equations.

These discontinuous Galerkin methods have several attractive properties. It can be easily designed for any order of accuracy. In fact, the order of accuracy can be locally determined in each cell, thus allowing for efficient p adaptivity. It can be used on arbitrary triangulations, even those with hanging nodes, thus allowing for efficient h adaptivity. The methods have excellent parallel efficiency. It is extremely local in data communications. The evolution of the solution in each cell needs to communicate only with the immediate neighbors, regardless of the order of accuracy. Finally, it has excellent provable nonlinear stability. One can prove a strong L² stability and a cell entropy inequality for the square entropy, for the general nonlinear cases, for any orders of accuracy on arbitrary triangulations in any space dimension, without the need for nonlinear limiters.

The paper is organized as follows. In Section 2, we present and analyze the local discontinuous Galerkin methods for the NLS equations. In Section 2.1, we present the methods for the generalized NLS equations. We prove a theoretical result of L² stability for the nonlinear case as well as an error estimate for the linear case. In Section 2.2, we present the local discontinuous Galerkin methods for the two-dimensional NLS equations and give a theoretical result of L² stability. In Section 2.3, we present a local discontinuous Galerkin method for the coupled NLS equations and give a theoretical result of L² stability. Section 3 contains numerical results for the nonlinear problems to demonstrate the accuracy and capability of the methods. Concluding remarks are given in Section 4.