A linearized conservative Galerkin–Legendre spectral method for the strongly coupled nonlinear fractional Schrödinger equations

In this paper, based on Galerkin–Legendre spectral method for space discretization and a linearized Crank–Nicolson difference scheme in time, a fully discrete spectral scheme is developed for solving the strongly coupled nonlinear fractional Schrödinger equations. We first prove that the proposed scheme satisfies the conservation laws of mass and energy in the discrete sense. Then a prior bound of the numerical solutions in L∞-norm is obtained, and the spectral scheme is shown to be unconditionally convergent in L2-norm, with second-order accuracy in time and spectral accuracy in space. Finally, some numerical results are provided to validate our theoretical analysis.


Introduction
The space fractional Schrödinger equation (FSE) is a natural extension of the classic Schrödinger equation, and it has been successfully used to describe the fractional quantum phenomena. Laskin [1,2] originally derived the Riesz space FSE via replacing the Brownian trajectories with Levy flights in the Feynman path integrals. Some physical applications of the FSE were presented in [3,4]. For the well-posedness, global attractor, soliton dynamics and ground states related to the FSE, we refer to Refs. [5][6][7] and the references therein.
The motivations of the current work are as follows. Firstly, since the conservative method performs better than the general goal method in long-time simulation, the discrete scheme which can preserve the invariant quantities of the original system is desirable. Moreover, to avoid time-consuming iterative process at each time step, an interesting topic is to construct a linearly implicit scheme for the SCFSEs. Furthermore, we intend to consider the unconditionally convergent spectral method, which takes advantage of spectral accuracy in space. Based on these considerations, the main objective of this paper is to develop a linearized Galerkin-Legendre spectral scheme for solving the SCFSEs. The derived scheme can preserve both the mass-and the energy-conservation laws in the discrete sense. Based on the discrete energy-conservation law, we show that the numerical solutions are bounded in L ∞ -norm. Moreover, the discrete scheme is proved to be unconditionally convergent with second-order accuracy in time and spectral accuracy in space by the energy method.
The outline of this paper is given as follows. In Sect. 2, some useful definitions and lemmas are recalled. In Sect. 3, a linearized Legendre spectral scheme is constructed for the SCFSEs. In Sect. 4, the conservation, boundedness and convergence properties of the proposed scheme are analyzed theoretically. Some numerical results are presented in Sect. 5, and some conclusions are drawn in the last section.

Preliminaries
In this section, before deriving the fully discrete Legendre spectral scheme for the SCFSEs, we first introduce some notations, definitions and lemmas which play an important role in subsequent theoretical analysis.

Notation
Define the inner product in the space L 2 ( ) as (v, u) := vū dx and the associated L 2norm is denoted by · . Besides, define the L p -norm (1 ≤ p < ∞) and L ∞ -norm as follows:
and the norm v H α ( ) := v 2 + |v| 2 and H α 0 ( ) denotes the closure of C ∞ ( ) with respect to · H α ( ) , where ξ andv represent the Fourier transform parameter and the Fourier transform of v, respectively.

Fully discrete Legendre spectral scheme
In this section, we will construct a Legendre spectral method for numerically solving the SCFSEs (1)-(4).

The semi-discrete variational scheme
The Legendre polynomials L k (s) are determined by the following recurrence relation: Denote Then the approximate function space V 0 N is given as The semi-discrete variational scheme for the SCFSEs (1)-(4) is to find where I N represents the Legendre-Gauss-Lobatto (LGL) interpolation operator [43]. The bilinear form B(·, ·) in (19) and (20) is defined as where Lemma 3 has been used in deriving (22). For convenience of theoretical analysis, one can define the following semi-norm and norm: By virtue of Lemma 1, |v| α 2 and v α 2 are equivalent with the semi-norms and norms of J

The fully discrete Galerkin-Legendre spectral scheme
Based on Legendre spectral method for space discretization and a linearized Crank-Nicolson difference scheme in time, we develop a linearized spectral scheme for the Schrödinger system (1)-(4), which is to find u n+1 To obtain the first step approximate solutions u 1 N and v 1 N , we employ the following Crank-Nicolson scheme: with the initial conditions

Theoretical analysis
This section is devoted to discussing the theoretical analysis of the spectral scheme (24)- (28), including the discrete mass-and energy-conservation laws, boundedness and the unconditional convergence.

Conservative properties of the spectral scheme
Theorem 1 The fully discrete spectral scheme (24)- (28) is conservative in the sense that where M n and E n are defined, respectively, as Proof Taking w =ũ n N in (24) gives As a result and then considering the imaginary part of (33) yields It further means that Taking w =ṽ n N in (25), we arrive at Similarly, we take the imaginary part of (38) to get Combining (37) and (39), we can conclude that the discrete mass conservation law (29) holds.
On the other hand, substituting w = δˆtu n N in (24), we arrive at It is easy to get and Re Taking the real part of (40), and combining with (41)-(43), we have Denoting w = δˆtv n N in (25), we obtain Analogously, taking the real part of the above equation yields It is easy to get from (44) and (46) γ u n+1 Noticing the definition of E n , it follows from (47) that E n = E n-1 for 1 ≤ n ≤ M -1, which further implies that (30) holds. Therefore, we complete the proof.

A prior bound
Based on the discrete mass-and energy-conservation laws, we can establish a prior bound for the numerical solutions of the scheme (24)- (28) in both L 2 -and L ∞ -norms.

Theorem 2
The solutions of the fully discrete spectral scheme (24)- (28) are bounded in the sense that Proof It is easy to deduce that Combining with the discrete mass conservation law (29), we have When τ ≤ 1 2| | , it follows from (53) that (50) holds. Noticing the energy-conservation law (30), we have where the Cauchy-Schwartz inequality, (50) and Lemma 5 have been used in deriving the above inequalities. Since the semi-norm | · | α 2 is equivalent to the semi-norm | · | H α 2 , and noticing Lemma 2, it follows that there exists a positive constant C 1 such that In view of (30), (54) and (55), we obtain Noticing that 1 < α ≤ 2, when taking 1 4 < α 0 < α 4 , it follows from (56) that → +∞. However, we can conclude that E 1 is bounded by the discrete conservation law (30). It will lead to a contradiction. Therefore, we can deduce that According Lemma 4, we can further deduce from (57) that (51) holds, which completes the proof.

Convergence analysis
Now we turn to discuss the convergence analysis of the discrete spectral scheme (24)- (28). To this end, we first introduce the projection operator α 2 ,0 The error estimate of the projection operator α 2 ,0 N is given in the following lemma.
v - = v(x, 1 2 ), and we also use u n and v n to represent the analytical solutions u(x, t n ) and u(x, t n ), respectively. In view of (24) and (25), the exact solutions u n and v n satisfy the equations i δˆtu n , wγ B ũ n , w + κ u n 2 + ρ v n 2 ũ n , w + β ũ n , w i δˆtv n , wγ B ṽ n N , w + κ v n 2 + ρ u n 2 ṽ n , w + β ṽ n , w where the local truncation errors R n u and R n v are defined as From (26) and (27), we can also deduce that where the local truncation errors R 0 u and R 0 v are given as By virtue of a Taylor expansion, we can deduce that Next, we focus on a rigorous convergence analysis for the spectral scheme (24)- (28).
where C is a positive constant which is independent of τ and N .
Analogously, it follows from (26), (27), (65) and (66) that where Thanks to Lemma 6 and (69), we obtain Now taking w =θ 1 2 in (81) and w =η 1 2 in (82), and then considering the imaginary part of the resulting equations, we have It is obvious that Im(η Noticing the definition of G 1 2 u , and using Lemma 7 as well as Theorem 2, we observe that where C 4 denotes a positive constant. Following a similar analysis, we also conclude that Therefore, we further deduce that Analogously, we find that Obviously, we can also deduce that Substituting (92)-(94) into (89), we have This, combined with Lemma 6 and (86), gives Moreover, one easily gets Therefore, when the time step τ in (96) is chosen sufficiently small such that τ ≤ 1 (12C 4 +1) , it follows from (96)-(98) that This together with Lemma 6 and the triangle inequality implies that (70) holds for n = 1.
By mathematical induction, we assume that (70) is valid for 1 ≤ n ≤ m. Now we turn to a proof that the stated conclusion still holds for n = m + 1. To this end, taking w =θ n in (78) and w =η n in (79), respectively, and considering the imaginary part of the resulting equations, we have 1 4τ θ n+1 2θ n-1 2 + Im G n u ,θ n + Im η n ,θ n = Im R n u ,θ n , 1 4τ η n+1 2η n-1 2 + Im G n v ,η n + Im θ n ,η n = Im R n v ,η n .
For the case of α = 3 2 , the stated result (71) can be obtained by a similar analysis. Hence, we have completed the proof of Theorem 3.

Numerical experiment
In this section, we present some numerical results to confirm our theoretical analysis of the spectral scheme (24)- (28).
Example 1 Consider the following strongly coupled fractional Schrödinger system: subject to the initial conditions and the homogeneous boundary conditions where the computation domain is chosen sufficiently large as = (-25, 25). The first objective is to check the convergence behavior of the spectral scheme (24)- (28). Since the analytical solutions of the system (116)-(119) are difficult to find, we take the numerical solutions computed by fixed τ = 10 -5 and N = 512 as the "exact" solutions. When fixing N = 512, we present the L 2 -errors with different time steps in Fig. 1. It can be Figure 1 The L 2 -error versus time step with N = 512. It shows that the derived spectral scheme has second-order temporal accuracy Figure 2 The L 2 -error versus N with τ = 10 -5 . It shows that the derived spectral scheme has spectral accuracy in space Figure 3 The values of the mass M n for different α with time evolution. It shows that the spectral scheme preserves the total discrete mass very well and the values of the mass M n are independent of α observed that the derived spectral scheme has second-order temporal accuracy. Moreover, we fix τ = 10 -5 and plot the L 2 -errors with the change of N in Fig. 2. It shows that the errors are exponentially decaying with N increases, and this indicates the spectral accuracy in space.
Now we turn to a validation of the discrete conservation laws of Theorem 1. To the end, we take τ = 0.001 and N = 256 and depict the mass M n and the energy E n as well as corresponding error functions for different α in Figs. 3-6. It can be found that the spec-

Conclusion
In the current work, we have constructed a linearized Galerkin-Legendre spectral method for solving the strongly coupled nonlinear fractional Schrödinger equations. The main novelty of this paper is that the proposed scheme can preserve both the mass-and the energy-conservation laws in the discrete sense, and the optimal error estimate is established rigorously without imposing any restriction on the grid ratio. The discrete scheme is efficient in the sense that only a linear system needs to be solved at each time step. Theoretical results show that our scheme is second-order convergent in time and at the same time has the advantage of spectral accuracy in space. Numerical results show that the de- Figure 8 The Graphs of |u| and |v| for α = 1.95 with time evolution rived scheme is quite efficient and exhibits remarkable mass-and energy-preserving properties. The spectral method and corresponding theoretical analysis for high-dimensional SCFSEs is worth of further investigation.