1 Introduction
We consider the nonlinear Schrödinger equation (NLSE) given as follows
(1.1) 
where is the time variable, is the spatial variable, is the complex unit, is the complexvalued wave function, is the usual Laplace operator, is a given real constant, and is a given initial data. The nonlinear Schrödinger equation (1.1) satisfies the following two invariants, that is mass
(1.2) 
and Hamiltonian energy
(1.3) 
A numerical scheme that preserves one or more invariants of original systems is called an energypreserving method. The earlier attempts to develop energypreserving methods for NLSE can date back to 1981, when Delfour, Fortin, and Payre [22] constructed a twolevel finite difference scheme (also called CrankNicolson finite difference (CNFD) scheme) which can satisfy the discrete analogue of conservation laws (1.2) and (1.3). Some comparisons with nonconservation schemes are investigated by SanzSerna and Verwer [49], which show the nonenergypreserving schemes may easily show nonlinear blowup. Furthermore, it can be proven rigorously in mathematics that the CNFD scheme is secondorder accurate in time and space [6, 7, 54]. In particular, we note that the discrete conservative laws play a crucial role in numerical analyses for numerical schemes of NLSE. However, it is fully implicit and at every time step, a nonlinear equation shall be solved by using a nonlinear iterative method and thus it may be time consuming. Zhang et al. [58] firstly proposed a linearly implicit CrankNicolson finite difference (LICNFD) scheme in which a linear system is to be solved at every time step. Thus it is computationally much cheaper than that of the CNFD scheme. The LICNFD scheme can satisfy a new discrete analogue of conservation laws (1.2) and (1.3) and its stability and convergence were analyzed in [54]. Up to now, the energypreserving CrankNicolson schemes for NLSE are extensively extended and analyzed [3, 2, 17, 23, 26, 41, 56]. Another popular linearly implicit energypreserving method is the relaxation finite difference scheme [9]
. In addition, the energypreserving schemes for NLSE provided by the averaged vector field method
[47] and the discrete variational derivative method [21, 24] can be founded in [16, 38, 45, 25]. More recently, the scalar auxiliary variable (SAV) approach [51, 52] have been a particular powerful tool for the design of linearly implicit energypreserving numerical schemes for NLSE [5]. Nevertheless, to our best knowledge, all of the schemes can achieve at most secondorder in time.It has been proven that no RungeKutta (RK) method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [14]. Thus, over that last decades, how to develop highorder energypreserving methods for general conservation systems attracts a lot of attention. The notable ones include, but are not limited to, highorder averaged vector field (AVF) method [44, 47], Hamiltonian Boundary Value Methods (HBVMs) [11, 12] and energypreserving continuous stage RungeKutta methods (CSRKs) [19, 29, 46, 55]. Actually, the HBVMs and CSRKs have been shown to be an efficient method to develop highorder energypreserving schemes for NLSE [8, 43], however, the proposed schemes are fully implicit. In fact, at every time step, one needs to solve a large fully nonlinear system and thus it might lead to high computational costs.
Due to the high computational cost of the highorder fully implicit schemes, in the literatures, ones are devoted to construct energypreserving explicit schemes for NLSE. Based on the the invariant energy quadratization (IEQ) approach [57] or the SAV approach and the projection method [13, 30, 28], the authors proposed some explicit energypreserving schemes for NLSE [34, 59]. Such scheme is extremely easy to implement, however, it requires very small time step sizes. Especially, this limitation is more serious in 2D and 3D. The linearly implicit one which needs solve a linear equations at every time step, can remove this limitation. Li et al. proposed a class of linearly implicit schemes for NLSE, which can preserve the discrete analogue of conservation law (1.2) [42]. It involves solving linear equations with complicated variable coefficients at every time step and thus it may be very time consuming. Another way to achieve this goal is to combine the SAV approach with the extrapolation technique or the predictioncorrection strategy [10].
In the past few decades, many energypreserving exponential integrators have been done for conservative systems. The exponential integrator often involves exact integration of the linear part of the target equation, and thus it can achieve high accuracy, stability for a very stiff differential equation such as highly oscillatory ODEs and semidiscrete timedependent PDEs. For more details on the exponential integrator, please refer to the excellent review article provided by Hochbruck and Ostermann [32]. Based on the projection approach, Celledoni et al. developed some symmetry and energypreserving implicit exponential schemes for the cubic Schrödinger equation [15]. By combining the exponential integrator with the AVF method, Li and Wu constructed a class of secondorder implicit energypreserving exponential schemes for canonical Hamiltonian systems and successfully applied it to solve NLES [44]. Further analysis and generalization is investigated by Shen and Leok [53]. More recently, Jiang et al. showed that the SAV approach is also an efficient approach to develop secondorder linearly implicit energypreserving exponential scheme for NLSE [33]. Overall, there exist very few works devoted to development of highorder linearly implicit exponential schemes with energypreserving property for NLSE, which motivates this paper.
In this paper, following the idea of the SAV approach, we firstly reformulate the original system (1.1) into a new system by introducing a new auxiliary variable, which satisfies a modified energy. The spatial discretization is then performed with the standard Fourier pseudospectral method [18, 50]. Subsequently, the extrapolation technique is employed to the nonlinear term of the semidiscrete system and a linearized system is obtained. Based on the Lawson transformation, the semidiscrete system is rewritten as an equivalent form and the fully discrete scheme is further obtained by using the symplectic RK method in time. We show that the proposed scheme can preserve the modified energy in the discretized level and at every time step, only a linear equations with constant coefficients is solved for which there is no existing references [10, 40] considering this issue.
The rest of this paper is organized as follows. In Section 2, we use the idea of the SAV approach to reformulate the system (1.1) into a new reformulation, which satisfies a modified energy. In Section 3, the linearly implicit highorder energypreserving exponential scheme is proposed and its energypreservation is discussed. In Section 4, several numerical examples are investigated to illustrate the efficiency of the proposed scheme. We draw some conclusions in Section 5.
2 Model reformulation
In this section, we employ the SAV approach to recast the NLS equation (1.1) into a new reformulation which satisfies a quadratic energy conservation law. The resulting reformulation provides an elegant platform for developing highorder linearly implicit energypreserving schemes.
Based on the idea of the SAV approach [51, 52], we introduce an auxiliary variable
where is a constant large enough to make welldefined for all . Here, is the inner product defined by where represents the conjugate of . Then, the Hamiltonian energy functional is rewritten as the following quadratic form
According to the energy variational principle, we obtain the following SAV reformulated system
(2.1) 
with the consistent initial condition
(2.2) 
where represents the real part of .
Theorem 2.1.
The SAV reformulation (2.1) preserves the following quadratic energy
(2.3) 
Proof.
It is clear to see
This completes the proof. ∎
3 The construction of the highorder linearly implicit exponential integrator
In this section, a class of highorder linearly implicit energypreserving integrators are proposed for the SAV reformulated system (2.1). For simplicity, in this paper, we shall introduce our schemes in two space dimension, i.e., in (2.1). Generalizations to or are straightforward.
3.1 Spatial discretization
We set computational domain and let and . Choose the mesh sizes and with and two even positive integers, and denote the grid points by for and for ; let be the numerical approximation of for , and
be the solution vector; we also define discrete inner product and norm as
In addition, we denote ’ as the element product of vectors and , that is
For brevity, we denote and as and , respectively.
Denote the interpolation space as
where and are trigonometric polynomials of degree and , given, respectively, by
with , and . We then define the interpolation operator as [18]:
where .
Taking the derivative with respect to and , respectively, and then evaluating the resulting expressions at the collocation points (), we have
(3.1)  
(3.2) 
where
Remark 3.1.
We should note that [50]
where
and
is the discrete Fourier transform (DFT) and
represents the conjugate transpose of .Then, we use the standard Fourier pseudospectral method to solve (2.1)
(3.3) 
where is the spectral differentiation matrix and represents the Kronecker product.
Theorem 3.1.
The semidiscrete system (3.3) preserves the following semidiscrete quadratic energy
(3.4) 
Proof.
The proof is similar to Theorem 2.1, thus, for brevity, we omit it.
Remark 3.2.
When the standard Fourier pseudospectral method is employed to the system (1.1) for spatial discretizations, we can obtain semidiscrete Hamiltonian energy as
(3.5) 
We note that however, the quadratic energy (3.4) is only equivalent to the Hamiltonian energy (3.5) in the continuous sense, but not for the discrete sense.
3.2 Temporal exponential integration
Denote and , where is the time step and are distinct real numerbers (usually ). The approximations of the function at points and are denoted by and and the approximations of the function at points and are denoted by and .
We first apply the extrapolation technique to the nonlinear term of (3.3) and a linearized system is obtained, as follows:
(3.6) 
where and are approximations of and , respectively over time interval . Here, is an (explicit) extrapolation approximation to of the order . For the more details on the construction of , please refer to Refs. [27, 40].
By using the Lawson transformation [39], we multiply both sides of the first equation of (3.6) by the operator , and then introduce to transform (3.6) into an equivalent form
(3.7) 
with the consistent initial condition
(3.8) 
We first apply an RK method to the linearized system (3.7), and then the discretization is rewritten in terms of the original variables to give a class of linearly implicit exponential integrations (LIEIs) for solving (1.1)
(3.9) 
where . Then are updated by
(3.10) 
Remark 3.3.
According to the definition of matrixvalued function [31], it holds

and ;

, which can be efficiently implemented by the matlab functions fftn.m and ifftn.m.
Remark 3.4.
We consider a Hamiltonian PDEs system given by
(3.11) 
with a Hamiltonian energy
(3.12) 
where
is a skewadjoint operation and
is a selfadjoint operation and is bounded from below. By introducing a scalar auxiliary variablewhere is a constant large enough to make welldefined for all . On the basis of the energyvariational principle, the system (3.11) can be reformulated into
(3.13) 
which preserves the following quadratic energy
(3.14) 
The proposed linearly implicit schemes can be easily generalized to solve the above system.
Theorem 3.2.
Proof.
Remark 3.5.
Remark 3.6.
The Gauss collocation method is symplectic (see Refs. [30, 48] and references therein), thus, it can preserve the discrete quadratic energy (see (3.16)). In particular, the coefficients of Gauss collocation methods of order 4 and 6 can be given explicitly by (see [30])
, .
Besides its energypreserving property, a most remarkable thing about the above scheme (3.9)(3.10) is that it can be solved efficiently. For simplicity, we take as an example.
We denote
(3.20) 
and rewrite
(3.21) 
With (3.20), we have
(3.22)  
(3.23) 
Then it follows from the first equality of (3.9) and (3.22)(3.23) that
(3.24)  
(3.25) 
where
Multiply both sides of (3.2) and (3.2) with and we then take discrete inner product with and , respectively, to obtain
(3.26)  
(3.27) 
where
(3.28)  
(3.29)  
(3.30)  
(3.31) 
Eqs. (3.26) and (3.27) form a linear system for the unknowns
Solving from the linear system (3.26) and (3.27), and and are then updated from (3.21)(3.2), respectively. Subsequently, and are obtained by (3.10).
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