Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions

Abstract

In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary condition and initial condition, and the optimal convergent rate, without any restriction on the grid ratio, at the order of $O(h^4+{\tau }^2)$ in the discrete $L^2$-norm with time step $\tau$ and mesh size h is obtained. Besides the standard techniques of the energy method, a new technique and some important lemmas are proposed to prove the high order convergence. In order to avoid the outer iteration in implementation, a linearized compact and energy conservative difference scheme is derived. Numerical examples are given to support the theoretical analysis.

Introduction

The nonlinear Schrödinger (NLS) equation is a famous equation used widely in many fields of physics, such as plasma physics, quantum physics and nonlinear optics. In this paper, we consider the following cubic NLS equation in two dimensions

$$i\frac{\partial u}{\partial t}+\mathrm{\Delta }u+\beta |u{|}^{2}u=0\text{,}(x\text{,}y)\in \mathcal{R}\times \mathcal{R}\text{,}0<t⩽T\text{,}$$

subject to $(l_1\text{,}{l}_2)$-periodic boundary condition

$$u(x\text{,}y\text{,}t)=u(x+l_1\text{,}y\text{,}t)\text{,}u(x\text{,}y\text{,}t)=u(x\text{,}y+l_2\text{,}t)\text{,}(x\text{,}y)\in \mathcal{R}\times \mathcal{R}\text{,}0<t⩽T$$

and initial condition

$$u(x\text{,}y\text{,}0)=\phi (x\text{,}y)\text{,}(x\text{,}y)\in \mathcal{R}\times \mathcal{R}\text{,}$$

where $\mathrm{\Delta }=\frac{\partial }{\partial x^2}+\frac{\partial }{\partial y^2}$ is the Laplacian operator, $\beta \ne 0$ is a given real constant, $\phi (x\text{,}y)$ is a given $(l_1\text{,}{l}_2)$-periodic complex-valued function. The NLS equation (1.1) is focusing for $\beta >0$, and defocusing for $\beta <0$, which 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 [55]. It is possible for solutions of the two-dimensional NLS equation to develop singularities at some finite time [29]. Periodic boundary condition is used for certain problems in a very large or even infinite domain consisting of (infinitely) many small cells where particles pass through the boundary of each cell in their individual periodic pattern. So the domain’s boundary condition will be neglected, and one can solve the problem numerically only on a central cell and its surrounding cells.

Extensive mathematical and numerical studies have been carried out for the NLS equation in the literature. Along the mathematical front, for the derivation, well-posedness and dynamical properties of the NLS equation, we refer to [5], [14], [40], [48] and the references therein. In fact, it is easy to show that the periodic-initial value problem (1.1), (1.2), (1.3) conserves the total mass

$$Q(u(·\text{,}·\text{,}t))≔{\int }_{\mathrm{\Omega }}|u(x\text{,}y\text{,}t){|}^2\mathit{dxdy}\equiv Q(u(·\text{,}·\text{,}0))=Q(\phi )\text{,}t\in {\mathcal{R}}^{+}$$

and the energy

$$E(u(·\text{,}·\text{,}t))≔{\int }_{\mathrm{\Omega }}\left[|\nabla u(x\text{,}y\text{,}t){|}^2-\frac{\beta }{2}|u(x\text{,}y\text{,}t){|}^4\right]\mathit{dxdy}\equiv E(u(·\text{,}·\text{,}0))=E(\phi )\text{,}t\in {\mathcal{R}}^{+}\text{,}$$

where $\mathrm{\Omega }=[0\text{,}{l}_1]\times [0\text{,}{l}_2]$. These conservative laws imply that, $||u(·\text{,}·\text{,}t)|{|}_{H_p^1}<\infty$ for the defocusing case or the focusing case with $||\phi |{|}^2⩽\frac{2-\stackrel{\sim }{\epsilon }}{4(1+\epsilon )\beta }$ where $\epsilon$ and $\stackrel{\sim }{\epsilon }$ are two positive numbers which can be arbitrary small. The theory on existence of global solution of the NLS equation [14], [48] gives that, if $||u(·\text{,}·\text{,}t)|{|}_{H_p^1}<\infty \text{,}u(x\text{,}y\text{,}t)\in L^{\infty }({\mathcal{R}}^{+}\text{,}{H}_p^1(\mathrm{\Omega }))$.

Along the numerical front, different efficient and accurate numerical methods including the time-splitting pseudospectral method [6], [8], [9], [50], finite difference method [16], [21], [26], [47], [52], [53], [56], finite element method [1], [2], [27], [34], discontinuous Galerkin method [33], [54], meshless methods [23], [24] and Runge–Kutta or Crank–Nicolson pseudospectral method [13], [46], [25] have been developed for the NLS equation. Of course, each method has its advantages and disadvantages. For numerical comparisons between different numerical methods for the NLS equation, we refer to [7], [16], [41], [51] and the references therein.

Error estimates for different numerical methods of NLS equation in one dimension have been established in the literature. For the analysis of splitting error of the time-splitting or split-step method for the NLS equation, we refer to [11], [20], [39], [44], [50] and the references therein. For the error estimates of the implicit Runge–Kutta finite element method for NLS, we refer to [2], [45]. Error bounds, without any restriction on the grid ratio, of conservative finite difference (CNFD) method for NLS equation in one dimension was established in [15], [16], [28], [52]. In fact, their proofs for CNFD scheme rely strongly on not only the conservative property of the method but also the discrete version of the Sobolev inequality in one dimension

$$||f|{|}_{L^{\infty }}⩽C||f|{|}_{H^1}\text{,}\forall f\in H_0^1(\mathrm{\Omega })\mathrm{or}f\in H_p^1(\mathrm{\Omega })\mathrm{with}\mathrm{\Omega }\subset \mathcal{R}\text{.}$$

which immediately imply a priori uniform bound for $||f|{|}_{L^{\infty }}$. However, the extension of the discrete version of the above Sobolev inequality is no longer valid in two dimensions. Thus the techniques used in [15], [16], [28] for obtaining $l^{\infty }$ error bounds of the CNFD scheme for the NLS equation only work for conservative schemes in one dimension and they cannot be extended to high dimensions. Due to the difficulty in obtaining the a priori uniform estimate of the numerical solution, few error estimates are available in the literature of finite difference methods for the NLS equation in two dimensions. Gao and Xie [26] proposed a fourth-order alternating direction implicit compact finite difference scheme for two-dimensional Schrödinger equations, and also used induction argument to prove that their scheme was conditionally convergent in the discrete $L^2$-norm. In their analysis, a serious restriction on the grid ratio was necessary. Bao and Cai [3] used different techniques to establish optimal but conditional error bounds of CNFD scheme and semi-implicit finite difference (SIFD) method for the homogeneous initial-boundary value problem of the Gross-Pitaevskii equation with angular momentum rotation in two and three dimensions.

Recently, there has been growing interest in high-order compact methods for solving partial differential equations [10], [17], [18], [19], [21], [22], [26], [31], [32], [37], [38], [42], [43], [49], [53]. It was shown that the high-order difference methods play an important role in the simulation of high frequency wave phenomena. However, because the discretization of nonlinear term in compact scheme is more complicated than that in second-order one, a priori estimate in the discrete $L^{\infty }$-norm is hard to be obtained, so the unconditional convergence of any compact difference scheme for nonlinear equation is difficult to be proved. In fact, for any compact difference scheme of nonlinear equations, especially those in two or three dimensions, there are very few results on unconditional convergence. In [52], the error estimates of two compact difference schemes for one-dimensional nonlinear Schrödinger equation were established without any restrictions on the mesh ratios, but the analysis method used there cannot be extended directly to high dimensions. In this paper, we will give a compact and conservative difference scheme for solving the periodic-initial value problem of the NLS equation in two dimensions, and prove that the proposed scheme is convergent at the order of $O(h^4+{\tau }^2)$ in the discrete $L^2$-norm without any restriction on the mesh ratio.

The remainder of this paper is arranged as follows. In Section 2, some notations are given and a difference scheme is proposed. In Section 3, some auxiliary lemmas are introduced or proved. In Section 4, solvability, discrete conservative laws of the proposed scheme are discussed and a priori estimate is obtained, then the convergence is proved based on the estimation. Lastly, numerical experiments are presented in Section 5 and some remarks are given in the concluding section.