Numerical analysis of a fourth-order linearized di ﬀ erence method for nonlinear time-space fractional Ginzburg-Landau equation

: An e ﬃ cient di ﬀ erence method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the L 2-1 σ scheme to handle Caputo fractional derivative, while a fourth-order di ﬀ erence method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the di ﬀ erence scheme is unconditionally convergent in pointwise sense with the rate of O ( τ 2 + h 4 ), where τ and h are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis.


Introduction
Progress in the past decades has indicated that fractional partial differential equations can be used to model many physical phenomena with non-locality and long-range interaction [1][2][3][4][5][6][7]. The fractional Ginzburg-Landau equation is regarded as an important model to describe many dynamical processes [8], and various theoretical researches have been carried out, such as global well-posedness [9,10] and long-time asymptotic behavior [11]. Besides, based on the Banach fixed point theorem, Shen et al. [12] analyzed the well-posedness of mild solution of time-space fractional Ginzburg-Landau equation driven by Gaussian noise. Xu et al. [13] studied the well-posedness of time-space fractional Ginzburg-Landau equation with fractional Brownian motion.
However, there is limited work on numerical research for the TSFGLE (1.1)-(1.3). Recently, Zaky et al. [32] constructed a nonlinear method for solving TSFGLE based on L2-1 σ scheme and Legendre spectral discretization, and the error estimate in L 2 -norm is established. Since this method is fully implicit, a nonlinear equation needs to be solved iteratively at each time layer. Subsequently, the authors [33] further studied the Alikhanov Legendre-Galerkin spectral method for the coupled TSFGLEs. To the best of our knowledge, there is not linearized numerical method for model (1.1), and the optimal convergence analysis in the pointwise sense has not been considered. The main novelty of this work is that a high-order linearized difference method is proposed for solving the TSFGLE based on L2-1 σ scheme for time discretization and compact difference method for space discretization, and optimal error estimate in the pointwise sense is established rigorously without imposing any restriction on the time-space grid ratio in one-dimensional case. Theoretical results show that our method has second-order accuracy in time and fourth-order accuracy in space. The discrete scheme is efficient in the sense that only a linear system needs to be solved at each time step.
In addition, the proposed method is extended to solve the two-dimensional problem, and the corresponding convergence theorem is also analyzed.
The remainder of this article is organized as follows. A linearized high-order difference scheme is constructed in Section 2. The unique solvability, boundedness property and error estimate of the proposed scheme are rigorously analyzed in Section 3. The numerical scheme and corresponding convergence analysis for the two-dimensional problem are studied in Section 4. Some numerical tests are presented in Section 5. Finally, we give some conclusions in Section 6.

The fully discrete linearized difference scheme
This section is devoted to constructing a linearized difference scheme for solving TSFGLE (1.1)-

L2-1 σ formula for the Caputo fractional derivative
Many efforts have been devoted to designing difference approximations of Caputo fractional derivative, including the well-known L1 formula [34], L1-2 formula [35] and L2-1 σ method [36]. Moreover, Li' group constructed and analyzed a series of higher-order numerical methods [37,38]. In order to construct a high-order linearized difference scheme for TSFGLE and establish convergence theorem, in this work the L2-1 σ formula will be invoked to discretize the Caputo fractional derivative.

Compact difference method for spatial discretization
In general, high-order difference scheme performs better than low-order method in terms of computational accuracy, and high-order difference approximations for Riesz fractional derivative have been analyzed, see literatures [39][40][41][42][43] for more details. In this paper, we use a fourth-order difference scheme to approximate the Riesz fractional derivative. For a fixed h, denote .
According to Ref. [44], the fractional central difference scheme (2.2) is second-order convergent for discretizing Riesz fractional derivative as h → 0, that is Based on the second-order fractional central difference scheme (2.2), a fourth-order compact difference scheme was constructed in Ref. [39].

The linearized discrete scheme
For simplicity, we denote u n j as the numerical approximation of the exact solution u(x j , t n ), (1 + σ)u n j − σu n−1 j . Considering model (1.1) at the grid point (x j , t n+σ ), applying the L2-1 σ formula for the approximation of Caputo fractional derivative and the fourth-order difference scheme (2.4) for discretizing Riesz fractional derivative, a linearized high-order discrete method for 1 ≤ n ≤ N − 1 is constructed as Here we mention that an extrapolation technique with second-order accuracy has been used to handle the nonlinear term in (1.1). Besides, the first level approximation u 1 j is chosen as u 1 j =û m α j , which is computed by the following iterative method The initial value u 0 j is taken as

Some useful lemmas
For any v, w ∈ V h , the discrete inner product and corresponding l 2 -norm as well as the maximum norm are defined respectively as where w denotes the conjugate of w. For a given constant ∈ [0, 1], we give the definition of the fractional Sobolev semi-norm |v| H and the norm v H as where v(η) denotes the semi-discrete Fourier transform of v. By Parseval's theorem, where C = C( ) is a positive constant independent of h.
Proof. It is equivalent to proving that the resulting homogeneous system of the proposed scheme has only a trivial solution at each time layer. The homogeneous equation of the iterative scheme (2.6) can be expressed as Taking the inner product of both sides withû m , then considering the real part gives where Lemma 2.4 has been used in deriving (3.2). According to Lemma 3.1 we know that c (0,σ) ζσ . Now we turn to study the homogeneous system of the discrete scheme (2.5), i.e., Taking the discrete inner product of (3.3) with u n+1 , then considering the real part yields Similarly, we can conclude from (3.4) that there exists a unique solution for difference scheme (2.5) when τ ≤ α c (n,σ) 0 ζσ . Therefore, we have completed the proof of Theorem 3.1.

Boundedness
Lemma 3.2. [36,49] For a given sequence of complex functions {ω n } N n=0 , we have ) (Discrete fractional Grönwall inequality) Suppose that {w n } N n=0 is a nonnegative sequence and p n > 0, and for 0 ≤ n ≤ N − 1 the following inequality is satisfied Theorem 3.2. The solutions of the fully discrete scheme (2.5)-(2.7) are bounded in discrete L 2 -norm, and it holds for 0 ≤ n ≤ N that Proof. Computing the inner product of (2.5) withū n+σ , then considering the real part gives Taking the discrete inner product of (2.6) for m = m α withû σ m , then taking the real part of the resulting system and noticing that u 1 =û m α , thus we have Due to ν > 0, κ > 0, we can deduce from (3.7) and (3.8) that If ζ ≤ 0, by using Lemma 3.2, we can further deduce from (3.9) that Here we use mathematical induction to prove the conclusion u n ≤ u 0 . For n = 0, it obvious follows from (3.10) that u 1 ≤ u 0 . Assuming that then we need to prove that u n+1 ≤ u 0 . With the help of Lemma 3.1 and (3.11), we can derive from (3.10) that which further indicates that u n+1 ≤ u 0 . Therefore, we can derive that On the other hand, if ζ > 0, it holds from Lemma 3.2 and (3.9) that (3.14) By using Lemma 3.3, we can deduce from (3.14) that In view of (3.13) and (3.15), we have completed the proof of this theorem.

Convergence analysis
To discuss the local truncation error of the proposed method, we first deduce from (2.6) that With the help of Taylor's formula, we obtain Combined with Lemmas 2.1 and 2.2, we can further deduce that where C F is a positive constant independent of τ and h. Now we start to discuss the error estimate of the discrete scheme (2.5)-(2.7). First of all, we denote thatê Let {û s } m α s=1 be solutions of the discrete scheme (2.6) , then there is a positive τ * such that when τ ≤ τ * , we have the error estimates

21)
where C F andC F are positive constants which are independent of τ and h.
Proof. Firstly, subtracting (2.6) from (3.16) for m = 1, then we have the error equation Here mathematical induction is used to prove that To this end, computing the inner product of (3.22) withê 1 , and it follows that Considering the real part of (3.25), and combined with ν > 0, we have (3.26) this together with the Cauchy-Schwarz inequality further implies that From the definition of Ĝ 1 , we obtain Combining (3.27) and (3.28), we have , α/2 c (0,σ) 0 2 , 1 , it follows from (3.18) and (3.29) that Taking the discrete inner product of (3.22) with 1 ν−iξê 1 , and considering the real part of resulting equation yields Furthermore, by using the Cauchy-Schwarz inequality and the Young inequality, we have It holds from (3.18), (3.28) and (3.30)-(3.33) that where whereC F = C β/2 C F 1 + ( π 2 ) 2 C 1 . Therefore, it follows from (3.30) and (3.35) that the conclusion (3.24) is true for m = 1. Assume that the conclusion (3.24) is valid for 1 ≤ k ≤ m − 1, that is We start to prove that the conclusion (3.24) is still valid for k = m. Subtracting (2.6) from (3.16) yields the error equation It follows from the induction assumption (3.36) that when τ and h are taken to be sufficiently small such that τ 1+ kα 2 + h 4 ≤ 1/C F , we have Noticing the definition ofĜ m j , we have Ĝ m ≤ σC G ( ê m + ê m−1 ). Taking the inner product of (3.37) withê m gives Similarly, considering the real part of (3.40) as well as the Cauchy-Schwarz inequality, we obtain , we have Taking the discrete inner product of (3.37) with 1 ν−iξê m , then considering the real part of the resulting equation gives For the first and the third items in the right hand of (3.43), the following estimates hold In view of (3.42) and (3.46) as well as Lemmas 2.3 and 2.4, we can also arrive at From (3.42) and (3.47), we can conclude that (3.24) still holds for k = m. Therefore, the induction is closed. Noticing that u 1 j =û m α j (m α = 2 α ), then we further derive from (3.24) that when τ ≤ min{τ 1 , τ 2 }, it holds Therefore, we have completed the proof of this theorem.
To discuss the local truncation error of proposed method (2.5)-(2.7), we first deduce from (2.5) that Now we use mathematical induction to prove this theorem. We can deduce from Theorem 3.3 that the convergence result (3.52) is valid for n = 1. Assuming that the conclusion (3.52) holds for n ≤ k, then when τ 2 + h 4 ≤ 1/C, we have u n ∞ ≤ U n ∞ + e n ∞ ≤ C u +C(τ 2 + h 4 ) ≤ C u + 1, n ≤ k. (3.55) Now we turn to prove that the estimate (3.52) is still valid for n = k + 1. Taking inner product of (3.53) for n = k withē k+σ gives Thus we can get G k ≤ C G ( ē k+σ + ẽ k+σ ). (3.58) Since ν > 0, considering the real part of (3.56) and Cauchy-Schwarz inequality, it follows that According to Lemma 3.4 , we can deduce from (3.51) and (3.59) that there is a constant τ 3 such that when τ ≤ τ 3 , it follows that e k+1 ≤Ĉ(τ 2 + h 4 ). (3.60) To further establish the estimate of e k+1 in pointwise sense, we take the inner product of (3.53) for n = k with 1 (ν−iξ) ∇ α t e k+σ , then taking the real part yields ν In view of the Cauchy-Schwarz inequality as well as Re(u, v) = Re(v, u), we can further deduce from (3.61) that For three items in the right hand of the inequality (3.62), we have (3.65) From (3.58) and Lemma 2.5, we get It is obvious from (3.62)-(3.66) that (3.67) By using Lemma 3.4, it follows from (3.51) and (3.67) that there is a constant τ 4 such that when τ ≤ τ 4 , we obatin Λ β e k+1 2 ≤ C * (τ 2 + h 4 ) 2 . (3.68) In view of (3.60), (3.68) and Lemma 2.3, we can further deduce that e k+1 ∞ ≤C(τ 2 + h 4 ), (3.69) whereC = C β/2 Ĉ + ( π 2 ) 2 C * is a positive constant independent of τ and h. Combining (3.60) and (3.69), it indicates that the stated result (3.52) still holds for n = k + 1. Therefore, the induction is closed, and we have completed the proof of this theorem.

The stability of the linearized difference method
Denote v n j |1 ≤ j ≤ M − 1, 0 ≤ n ≤ N} as the solution of the following difference scheme for TSFGLE with another initial value function, i.e., with initial value v 0 j = ϕ(x j ), 1 ≤ j ≤ M − 1, then the stability theorem of the discrete scheme (2.5)-(2.7) can be established by a similar analysis as in Theorem 3.4.
Theorem 3.5. Let u n j and v n j as the difference solutions of TSFGLE corresponding to initial value functions ψ(x) and ϕ(x) respectively. When h and τ are taken sufficiently small, it holds that whereC is a positive constant independent of τ and h.
Proof. Since the proof process of this theorem is similar to Theorem 3.4, we omit it here.

Numerical simulation for the two-dimensional problem
Similar to the literature [51], the proposed difference method can be extended to solve multi-dimensional TSFGLE. This section is devoted to constructing a difference scheme for the following two-dimensional problem with parameters 0 < α ≤ 1, 1 < β 1 , β 2 ≤ 2, ν > 0, κ > 0.

Linearized difference method for 2D TSFGLE
For given integers M 1 and M 2 , we define x j = a + jh x (0 ≤ j ≤ M 1 ), y k = c + kh y (0 ≤ k ≤ M 2 ), with h x = b−a M 1 and h y = d−c M 2 . Denote u n jk as the numerical approximation of u(x j , y k , t n ). In view of Lemmas 2.1 and 2.2, a three-level linearized difference method is derived for numerically solving the TSFGLE (4.1)-(4.3) The numerical solution u 1 jk =û m α jk (m α = 2 α ) can be obtained by the iterative processes with initial value u 0 jk = ψ(x j , y k ). (4.6)

Convergence analysis
For any v, w ∈ V h , the discrete inner product and l 2 -norm can defined similarly as in the one-dimensional case (2.8). From Ref. [39] and Lemma 2.4, it is easy to show that there exists a symmetric positive operator Λ 1 2 such that Now we study the local truncation error of difference method (4.4)-(4.6), and it holds from (4.4) that Similarly, we can deduce from (4.5) that =F m jk , m = 1, · · · , m α , (4.10) By using Lemmas 2.1, 2.2 and Taylor's formula, we have Denote e n jk = U n jk − u n jk , then we have the error estimate for discrete scheme (4.4)-(4.6). whereC is a positive constant which is independent of τ, h x and h y .
We mention that the well-posedness, boundness property and stability theorem of the difference scheme (4.4)-(4.6) can be also established by similar analysis as in the one-dimensional case.

Numerical experiment
Since the exact solution (5.2) decays exponentially as |x| → +∞, we take a sufficiently large computational domain Ω = [− 20,20] such that the truncation error can be ignored. The main task is to check the convergence rates of the derived method. To this end, we first fix h = 0.01, and plot the errors at t = 1 in the maximum norm with different time steps in the left side of Figure 1. It follows that the difference method (2.5)-(2.7) has second-order accuracy in time. Similarly, we take τ = 0.0002, and plot errors in the maximum norm with different space steps in the right side of Figure  1. We can observe that the proposed method is fourth-order convergent in space. 3) The parameters are taken as ν = ξ = κ = γ = 1, ζ = −1. The initial value ψ(x), boundary condition and source term f (x, t) are chosen such that u(x, t) = t 3 (1 − x 2 ) 4+ β 2 . Now we continue to validate the convergence rates of proposed method (2.5)-(2.7). First, by fixing h = 1/400, numerical errors at t = 1 and convergence rates with various time step for different α and β are depicted in Table 1. Besides, we fix τ = 1/5000, and present errors in maximum norm as well as convergence rates of the proposed method for various space steps in Table 2. These results show that numerical solutions are convergent with second-order accuracy in time and fourth-order accuracy in space, and also validate the correctness of theoretical analysis in Theorem 3.4. Besides, the graph of the numerical solution is displayed in Figure 2, and we can find that numerical solution approximates the analytical solution very well.    Here we take parameters ν = 1, ξ = −1, κ = 2, γ = 1 2 , and the initial value is taken as ψ(x) = sech(x) exp(−ix).
We take τ = 0.004, h = 0.05, and present the evolution of u n with time t for different ζ in Figure  3. It shows that u n is bounded in finite time, and indicates that the stated conclusion in Theorem 3.2 is valid. In particular, we find that when ζ < 0, the numerical approximation solution in l 2 -norm decays quickly to zero as time t increases, and the decay trend is more rapidly with ζ going smaller. Besides, the graphs of the numerical solutions for ζ = 0.4 is presented in Figure 4, and it indicates that the values of α and β have significant influence on the evolution of numerical solutions.
Example 4. Consider two-dimensional time-space fractional Ginzburg-Landau equation The parameters are taken as ν = ξ = 1 2 , κ = γ = 1.5, ζ = −1. The initial value ψ(x, y), boundary condition and source term f (x, y, t) are chosen such that u(x, y, t) = t 3 (1 − x 2 ) 4+ β 1 2 (1 − y 2 ) 4+ β 2 2 . The l 2 -errors and corresponding convergence rates of scheme (4.4)-(4.6) with the relationship of τ and h = h x = h y are displayed in Tables 3 and 4. It can be observed again that the proposed method  has second-order accuracy in time direction and fourth-order accuracy in space direction, which is accordance with the result as stated in Theorem 4.1.

Conclusions
In this article, based on L2-1 σ formula together with a second-order extrapolation technique for time discretization and a fourth-order difference method for space discretization, a high-order linearized discrete method is proposed for solving the time-space fractional Ginzburg-Landau equation. We show that the discrete scheme is uniquely solvable, and obtain a priori bound of numerical solution in the discrete L 2 -norm. The difference scheme is rigorously proved to be convergent in the pointwise sense with the accuracy of O(τ 2 + h 2 ). Besides, the difference method for the two-dimensional problem is constructed, and the convergence analysis is also analyzed. Numerical results are also presented to confirm our theoretical results.