Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

: A ﬁnite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L 2-1 σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efﬁciently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to conﬁrm the theoretical claims.


Introduction
The 2003 Nobel prize winning Ginzburg-Landau model in the field of physics is widely used in superconductors, superfluids, and condensation processes of Bose-Einstein type. As a low-temperature superconducting model [1], the Ginzburg-Landau model was first introduced by physicists Ginzburg and Landau in the 1950s. A wide variety of phenomena can be described by the Ginzburg-Landau equation starting from second-order phase transitions to nonlinear waves, and from Bose-Einstein condensation, superconductivity and superfluidity to strings in field theory and liquid crystals [2]. The concept of the fractional Ginzburg-Landau equation, which can be used to describe the dynamical processes in a medium with fractal dispersion was first derived by Tarasov in [3]. The variational Euler-Lagrange equation for fractal media was the generator of the fractional generalization of Ginzburg-Landau equation. A coupled Ginzberg-Landau system was used to describe a class of nonlinear optical fiber materials with active and passive coupled cores [4,5].
The well-posedness was discussed globally, and the long-time dynamics for the nonlinear complex Ginzburg-Landau equation involving fractional Laplacian was tackled in [6]. The dynamics and well-posedness of the coupled fractional Ginzburg-Landau equation, which describes a class of nonlinear optical fiber materials with active and passive coupled cores, was discussed in [7].
Motivated by their vast applications, numerical methods dealing with Ginzberg-Landau problems have gained attention due to the difficulty of obtaining exact solutions for the fractional order form of Ginzberg-Landau models. Armed by that, Galerkin spectral methods of Legendre type would be one of the most appropriate numerical methods for handling this kind of problem well. The Galerkin spectral method is a valuable tool for solving partial differential equations [8] and has been successfully applied to solving various types of fractional-order models [9][10][11][12][13][14].
The authors have already made some contributions to numerically solving various kinds of time-space fractional order problems based on the ideas of Legendre Galerkin spectral and finite difference schemes. In [15], semi-implicit spectral approximations were proposed to solve nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions. A combination of the Legendre Galerkin spectral approximation and the L1 difference approximation formulae over graded and uniform meshes was proposed. The work in [16] was concerned with a numerical treatment of nonlinear fractional Schrödinger equations with Riesz space-and Caputo time-fractional derivatives. The L1 finite difference approximation was used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method was used for the spatial approximation. The contribution in [17] was devoted to coupled nonlinear time-space fractional Schrödinger equations with non-smooth solutions in the time direction. The method combined the L1 scheme with temporal nonuniform mesh and the Galerkin-Legendre spectral approximation. The convergence and the stability estimates were performed using energy estimates and discrete forms of Grönwall inequalities [18][19][20]. More recently, a numerical algorithm was proposed for the time-space fractional Ginzburg-Landau equation by a highorder difference/Galerkin spectral scheme. For the temporal approximation, the smooth Alikhanov difference formula was used to discretize the time fractional derivative of Caputo type, while for the spatial discretization, we hinged on the Legendre-Galerkin spectral method [21]. In [22], a graded mesh finite difference/Galerkin spectral method was used to numerically solve a coupled system of time and space fractional diffusion equations.
In this paper, we propose a high order Alikhanov Legendre-Galerkin spectral method for solving the following nonlinear coupled fractional Ginzburg-Landau equations: with the initial conditions and the homogeneous boundary conditions such that Ω = (a, b) ⊂ R and I = (0, T] ⊂ R. The parameters ν i , η i , k i , ζ i , ε i , µ i and γ i , i = 1, 2 are given real constants, and φ(x) is a given smooth function. The temporal fractional derivative is defined in Caputo sense [23]: The spatial fractional operator of Riesz type of order α with respect to a ≤ x ≤ b, namely, ∂ α Ψ ∂|x| α , is defined as [23] There exists a wide variety of numerical methods which deal with space and/or fractional differential equations [24][25][26][27][28][29][30][31]. The coupled space fractional Ginzburg-Landau system was numerically investigated in [32]. A linearized semi-implicit difference scheme is proposed with unconditional stability and fourth order of convergence. In [33], a discrete difference scheme based on the implicit midpoint in time and a weighted and shifted Grünwald difference scheme with respect to space. The scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent. For the strongly coupled fractional Ginzburg-Landau system, a linearized three time level semi-implicit finite difference scheme in [34] was proposed to solve it. The difference scheme is unconditionally stable, fourth-order accurate in space, and second-order accurate in time.
The main concern of this work is to first design a combined numerical scheme for a coupled system (1) of Ginzburg-Landau with the time Caputo fractional derivative and the Riesz space fractional Laplacian operator. That scheme combines the Alikhanov L2-1 σ differentiation formula [35] with Legendre Galerkin spectral approximation. Error estimates and unconditional convergence of the proposed scheme based on discrete energy estimates are detailed here. Accordingly, the manuscript is organized as follows. The next section is devoted to some preliminaries. The numerical scheme and its implementation are illustrated in the third section. The fourth section focuses on the convergence analysis of the proposed scheme in both semi and full discretized styles. Some numerical experiments are done in the penultimate section while the manuscript ends with a section for conclusion and remarks.

Preliminaries
Some spaces of fractional derivatives are recalled below, see [36]. The notation (·, ·) 0,Ω denotes the inner product on the space L 2 (Ω) with the L 2 -norm · 0,Ω and the maximum norm · ∞ . C ∞ 0 (Ω) denotes the space of non-singular functions with compact support in Ω. H r (Ω) and H r 0 (Ω) are Sobolev spaces with the norm · H r and semi-norm |·| H r . We denote P N (Ω) the space of polynomials. The approximation space V 0 N is defined as Additionally, I N is the interpolation operator of Legendre-Gauss-Lobatto type, Definition 1 (Left fractional derivative space). We define the semi-norm and the norm for η > 0, respectively as Definition 2 (Right fractional derivative space). We define the semi-norm and the norm for η > 0, respectively as Definition 3 (Symmetric fractional derivative space). We define the semi-norm and the norm for η = n − 1 2 , n ∈ N, respectively as , and denote J η s as C ∞ (Ω) closure with respect to · J η s . Definition 4 (Fractional Sobolev space). The Sobolev space H η (Ω) for η > 0, is given as endowed with the semi-norm and norm respectively as . Also, F (Ψ) is the Fourier transformation of the functionΨ and the zero extension of Ψ outside Ω denoted byΨ.

Numerical Scheme
Here, the discretization of problem (1) is done by using the L2-1 σ approximation difference formula for the Caputo time fractional operator side by side to the Legendre-Galerkin spectral method for the Riesz spatial-fractional operator. A detailed implementation of the proposed scheme is proposed here.

Discretization
We partition the temporal domain I by t j = jτ, j = 0, 1, . . . , and Lemma 2 (see [35]). The high order Alikhanov L 2 -1 σ difference formula under the assumption It can be rewritten as where d Definition 6. Let j ∈ Z [0,M−1] , Alikhanov L2-1 σ difference formula at the node t j+σ is defined as The following identity holds directly by Taylor's theorem.

Lemma 3.
The following identity holds: Starting from the L2-1 σ Formula (11) for the discretization of the time Caputo fractional derivative of (1a), this leads to By aid of Lemmas 2 and 3, this semi-scheme is of second order accuracy. Let us introduce the parameters The semi-scheme (13) has following equivalent form: And so, the full discrete Alikhanov L2-1 σ Galerkin spectral scheme for (14) is to get where P N is a suitable projection operator. Its related features are illustrated in Section 4.

Algorithmic Implementation
Via the hypergeometric function, Jacobi polynomials can be presented for α, β > −1 and x ∈ (−1, 1) [8]: such that the notation (·) i represents the symbol of Pochhammer. Armed by (16), we get the equivalent three-term recurrence relation The Legendre polynomial L i (x) is a special case of the Jacobi polynomial, this means The weight function which makes the orthogonality of Jacobi polynomials occur is given as where δ ij is the Dirac Delta symbol, and Lemma 4 (see for example [37]). For α > 0, one has We introduce the following rescale functions: and we write ∧(x) asx. The basis functions selected for the spatial discretization are given by [9,38]: The function space V 0 N can be specified as follows: The approximate solutions ψ j+1 N and φ j+1 N are shown as whereψ j+1 i andφ j+1 i are the unknown expansion coefficients to be determined. Choosing v = ϕ i , 0 ≤ i ≤ N − 2, the matrix representation of the Alikhanov L2-1 σ Legendre-Galerkin spectral scheme has the following representation: where Lemma 5 (see [8,9]). The elements of the stiffness matrix S are given by are Jacobi-Gauss points and their weights with weight function The mass matrix M is symmetric and its nonzero elements are given as Monitoring H j+1,r pq ), p, q = 1, 2, r ≥ 0, the linear system (26) can be solved by the iteration Algorithm 1: Algorithm 1: Iterative algorithm for the problem (1). Set Ψ n = Ψ n,r+1 and Φ n = Φ n,r+1 .

Convergence Analysis
We will present the convergence analysis of the Alikhanov L2-1 σ Galerkin spectral scheme for the generalized fractional coupled Ginzburg-Landau system in both semi and full discretized forms. Any C represents a generic positive constant which can differ from one inequality to the another and is independent of τ, N and n.
The interpolation operator I N achieves the following property: Lemma 7 (see [8]). Suppose that Ψ ∈ H s (Ω) (s ≥ 1), then and C > 0 is a constant has no dependence on N.

Semi-Discrete form Convergence Analysis
Assume that {ψ, φ} and {ψ N , φ N } are the solutions of (1) and (13), respectively, satisfying {ψ, φ} ∈ H 1 (I; H α 2 0 (Λ) ∩ H s (Λ)). Then, we get Proof. The variational formulation comes by taking the inner product of (1a) with v 1 , Let e = ψ − ψ N , ζ e = ψ − P N ψ and η e = P N ψ − ψ N , we get e = ζ e + η e . Also, let We get the following estimate in the case of α = 3 2 by using Lemma 6, Subtract (39) from (15), then we obtain The orthogonality of P N , yields Taking the inner product of (42) with η e . Choosing the real part of the resulting equation, we get Invoking (33), (43) and (45). Using Lemma 8, we obtain Define G(ψ) = |ψ| 2 ψ, and by the use of Cauchy-Schwarz inequality, we deduce By Lemma 7, we have We get as in [40] depending on Lemma 11 that Also, Hence, we get the following estimate Simultaneously, by taking the inner product of each part of (1b) with 2 and following the same steps as before, we also get Then adding (51) and (52) leads to By Lemma 9, we obtain Lemma 10 implies now that t β E β,1+β (Ct β ) ≤ C. Finally we can see that The other inequality of the conclusion can be achieved in a similar fashion if α = 3 2 and 0 < µ < 1 2 .
Proof. The next variational formula is derived by taking the inner product of (1a) with v 1 , Let e = ψ − ψ N , ζ e = ψ − P N ψ and η e = P N ψ − ψ N , we get e j+σ = ζ j+σ e + η j+σ e . Also, Subtract (39) from (15 ), then we obtain The orthogonality of the operator P N , causes After taking the inner product of (42) with η Proceeding as in the proof of Theorem 1, we define G(ψ j+σ ) = |ψ j+σ | 2 ψ j+σ , and using the Cauchy-Schwarz inequality, we get By Lemma 7, we obtain We get as in [40] depending on Lemma 11 that Also, The next estimate flows after some manipulations, Simultaneously, Adding (71) to (72) and applying the L2-1 σ Discrete fractional form of Grönwall inequality in Lemma 13, then the final result (55) is achieved directly. Similarly, we can get the result (56) when α = 3 2 . Then the proof is fulfilled.

Numerical Experiments
In this section, we provide two numerical examples to validate the analysis and the performance of the present scheme for the time-space fractional Ginzburg-Landau equations. All computations and visualizations have been carried out using Mathematica 12.1 on a personal computer with 12 GB memory and 2.3 GHz speed. Moreover, the spatial and the temporal convergence orders are computed using the following formulae: , in time, Example 1 (Convergence test). Consider the following nonlinear coupled system of fractional Ginzburg-Landau equations to test the accuracy of the proposed scheme: with the homogeneous boundary conditions The initial conditions and the source terms f 1 (x, t) and f 2 (x, t) are determined by the exact solutions Tables 1 and 2 list the L 2 -errors and corresponding convergence orders with α = β + 1 = 1.2, 1.5, 1.8 and N = 100 for φ and ψ, respectively. We can see that these results confirm the second-order convergence in time. The convergence orders in space a are depicted for different values of α and β at M = 1600 in Figures 1 and 2. All the convergence results are in agreement with the theoretical results.   Henceforth, we take D 0 = 2, V 0 = 3, x ∈ [−10, 10].
In this test, we select ν 1 = ν 2 =η 1 = η 2 = ζ 1 =ζ 2 = µ 1 = µ 2 = 1, k 1 = k 2 = 1/2 and ε 1 = ε 2 = 0. Moreover, we will set the computational parameter N = 100 and M = 500. Figures 3 and 4 display the numerical solutions for different α and β according to Example 2. We observe that, the fractional parameters α and β will dramatically affect the shape of the soliton, which is completely different from the classical case and shows the nonlocal character of the Caputo fractional derivative and fractional Laplacian. We find from these two figures also that the parameters γ 1 and γ 2 dramatically influence on the wave-shape. It is also observed that the numerical solutions decay fast with time evolution especially when the parameters γ 1 and γ 2 become more smaller and also when the fractional orders become more close to the integer orders, which means α comes close to 2 and β be closer to 1. These results seems to be in a good agreement with those appeared in ( [33], Example 3).

Conclusions
In this work, we provided an efficient high order numerical scheme for the generalized fractional coupled Ginzburg-Landau system. This scheme has a second order of convergence with respect to time and spectral accuracy with respect to space. The convergence analysis of the scheme shows the unconditional convergence of its approximate solution. An algorithmic easy implementation of the scheme is also provided to facilitate its numerical application. The paper ends with a numerical example, it shows the agreement between the theoretical results and numerical ones. Finally, we need to clarify the following issues and present some future work:

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Our proposed high order hybrid numerical scheme is a linearized scheme of second order of convergence with respect to time inspite of the nonlinearity of the problem under consideration. The spectral accuracy is achieved due to the use of Galerkin Legendre approximation. Up to our knowledge, it is the first time that scheme is used to solve that kind of problems, especially noting the appearance of time and space fractional derivatives in the model under study. Unconditional convergence and stability of that scheme is secured, which means the error estimates of the numerical model has no dependence on time and spatial steps. This work reflects the possibility of that kind of schemes to be extended to deal with success with the singularity near the initial values of time fractional Caputo operators appearing in the generalized Ginzburg-Landau system. The latter can be secured by using nonuniform Alikhanov schemes combined with Legendre Galerkin spectral and it would be a near future plan for us.

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Due to the intrinsically nonlocal property and historical dependence of the fractional derivative, numerical applications of the numerical methods are always timeconsuming. Therefore, fast schemes based on local approximations [41,42] can be implemented to avoid the high computational costs coming from the prehistory feature of spatial fractional order operators. Fast L1 and Fast Alikhanov formulas of the Caputo derivative which are based on the sum of exponentials can be used to to reduce the huge storage and computational cost [43,44]. Invoking these approaches to reduce the computational cost of finite difference/Galerkin spectral methods would be a target of our new works in the near future. Funding: The first author wishes to acknowledge the financial support of the National Research Centre of Egypt (NRC). The second author wishes to acknowledge the support of RFBR Grant 19-01-00019.

Data Availability Statement:
Data sharing is not applicable to this article.