A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with O(τC4+τF2+h4) in the discrete L2-norm is obtained. Here, τC and τF are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.


Introduction
In this paper, we consider the one-dimensional cubic nonlinear Schrödinger (NLS) equation where i = √ −1 is the complex unit, the subscripts x and t denote the spatial and temporal variable, respectively; η and q are positive real constants; u = u(x, t) is an unknown complex-valued wave function. The initial condition gous to the Gauss-Seidel method for linear systems, of the initial guess in the linearization process, where the available fine grid solution is used as well. Such modification can improve the calculation accuracy of fine grid solutions. In addition, through numerical schemes and experiments, it can find that our method has the advantages of the simple numerical calculation. In addition, compared to the standard nonlinear implicit (SNI) CD scheme, it can reduce the CPU time without loss of accuracy.
The new TT-MCD algorithm includes three steps: firstly, a nonlinear implicit CD scheme, which will be solved by an iterative method on the time coarse mesh, is established; secondly, utilizing the numerical data obtained from the first step, the Lagrange's linear interpolation formula is employed to obtain rough solutions on the time fine mesh; finally, one Newton iteration is applied on the time fine mesh to linearize the nonlinear CD scheme using the mean value of the rough solution and available fine mesh solution as the initial guess. We then solve the linear system to obtain the finial numerical solutions. The main contributions or contents of this article are as follows: • A fast numerical algorithm, which is formulated by combining the fourth-order CD method with the TT-M method, is proposed and developed to solve the NLS equation. • A modification, analogous to the Gauss-Seidel method for linear systems, of the initial guess in the linearization process is taken to improve the calculation accuracy of fine grid solutions. • By using the discrete energy, the detailed proof of the convergence result with O(τ 4 C + τ 2 F + h 4 ) in the discrete L 2 -norm is given. • Numerical experiments on some model problems, including single soliton, interaction of two solitons and birth of standing soliton, are conducted to demonstrate efficiency and accuracy of the TT-MCD algorithm. • It is easy to see from the numerical results that the proposed TT-MCD algorithm not only gives highly accurate results and preserves conservation laws of charge and energy, but also can save the CPU time.
The remainder of this paper is organized as follows. In Section 2, notations and some lemmas are given. Section 3 devotes to the establishment of the time two-mesh compact difference scheme. The convergence of the TT-MCD scheme is analyzed in Section 4. In Section 5, two numerical examples are given to verify the feasibility and effectiveness. The article ends with a brief conclusions section. Throughout this paper, the symbol M is used to denote a generic positive constant.

Notations and Some Lemmas
First, for the temporal approximation on the fine mesh T F , we define τ F = T N , t n = nτ F (n = 0, 1, · · · , N ). Similarly, for the coarse mesh T C , denote τ C = T N where N = N s , (2 ≤ s ∈ Z + ), t ks = kτ C (k = 0, 1, · · · , N). For the spatial approximation, let h = b−a J for positive integer J, x j = a + jh, j = 0, 1, · · · , J. Let u n j = u(x j , t n ) and I h = {x 0 , x 1 , · · · , x J } denote the set of nodes of the interval [a, b]. We use the following notations for simplicity: where τ denotes the time step length τ C or τ F . Let H h,0 denote the set of mesh functions u defined on I h with boundary conditions u 0 = u j = 0. We define the discrete inner products and norms via And for any complex-valued function u = v + iw, let Next, we give some auxiliary lemmas, which will be used later.

The Time Two-Mesh Compact Difference Scheme
In order to construct the compact difference scheme [7], we first split the NLS and ν n+ 1 From Equations (2) and (3), we obtain i(u j ) n t + q(|u| 2 ) Then substituting Equations (2) into (4), we have i 12 (u j−1 ) n t + 10(u j ) n t + (u j+1 ) n t + η(u j ) Next, based on Equation (5), a time two-mesh CD scheme for problem Equation (1) is constructed as follows.
Step 1: Letting U ks C,j = V ks C,j + iW ks C,j , we find V ks+1 C,j , W ks+1 where Step 2: Based on the solutions U ks C,j obtained from Step 1, we use the Lagrange's linear interpolation formula to compute U n C,j (n = 1, 2, · · · , N ), that is, at time levels t ks−l (l = 0, 1, · · · , k, and k = 1, 2, · · · , N, ks − l = n), we have Step 3: Taking the mean value of the solution U n+1 C obtained from Step 2 and the former time level fine mesh solution U n F as the initial value, we construct a linear system on the time fine mesh as follows to solve the solutions V n+1 F,j , W n+1 where

Remark 1.
A modification, analogous to the Gauss-Seidel method for linear systems, of our algorithm is that the fine mesh solution U n F is also used in calculation of the fine mesh solution U n+1 F , by which one can improve the calculation accuracy of fine mesh solutions.

The Convergence Analysis of the TT-MCD Scheme
In this section, we first consider error analysis of the nonlinear system on the time coarse mesh. Denote e ks where R ks For simplification, we further denote φ = g(v, w) − g(V, W) and ψ = g(w, v) − g(W, V). Then Equations (13) and (14) can be written as Based on the above set of error equations, we obtain the following error estimation of the coarse mesh solution.
Theorem 1. Suppose that the exact solution u n = v n + iw n to the initial boundary value problem Equation (1) is sufficiently smooth and let U n C = V n C + iW n C be the numerical solution on the time coarse mesh. Then, there exist a positive constant M independent of h, τ C such that Proof. (I) The proof contains two cases. First, we consider the case of n = ks, (k = 1, · · · , N). For simplification, we will omit the subindex j and the mark C of coarse mesh in Equations (15) and (16). Taking the inner product (·, ·) on both sides of Equation (15) with E n+1 + E n , we obtain Using Lemma 1 and Cauchy-Schwarz inequality, we obtain Using two order Taylor expansion at a point (V Combining Equation (17) with Equation (18), we then have Similarly, taking the inner product (·, ·) on both sides of Equation (16) with e n+1 + e n , and then using Lemma 1, Cauchy-Schwarz inequality and Taylor expansion for function g, we can obtain summing from 0 to n − 1 in Equations (19) and (20), respectively, and then adding the two inequalities, we have By taking τ C small enough so that τ C < 2 3M and applying Lemma 2, we obtain u n − U n C L 2 ≤ M(τ 2 C + h 4 ). Further, noticing that n = ks, then we have (II) The second case is n = ks − l, (l = 1, · · · , k − 1, and k = 1, 2, · · · , N). Based on the Lagrange's interpolation formula, we obtain Subtracting Equations (9) and (22), we can have Using (21) and triangle inequality, we obtain u ks−l − U ks−l C L 2 ≤ M(τ 2 C + h 4 ). In addition to synthesizing the above two cases, we then obtain the result of Theorem 1.
Next, we give the convergence result on the time fine mesh. Letting e n F,j = v n j − V n F,j , E n F,j = w n j − W n F,j , (1 ≤ j ≤ J − 1, 0 ≤ n ≤ N ), then from Equations (5), (10) and (11), we obtain where and v n+ 1 Based on the above set of error equations, we then obtain the following error estimation of the fine mesh solution.
Theorem 2. Suppose that the exact solution u n = v n + iw n to the initial boundary value problem Equation (1) is sufficiently smooth and let U n F = V n F + iW n F be the numerical solution on the time fine mesh. Then, there exists a positive constant M independent of h, τ C , τ F such that Proof. Omitting the subindex j and taking the inner product (·, ·) on both sides of Equation (23) For each term on the right side of Equation (25), we estimate them as follows: where the assumption max Then, from Equations (26)-(28), we obtain Taking the inner product on both sides of Equation (24) with e n+1 F + e n F , and then in an entirely analogous manner, a similar estimate may be obtained as follows.
Adding Equations (29) and (30) and summing from n = 0 to m, we obtain Here we have used Lemma 1 and the assumption that τ F is small enough. Furthermore, we use the technique shown in [48,49] and Lemma 2 to obtain the conclusion.

Numerical Results
In this section, we present two numerical examples to illustrate the efficiency of the algorithm discussed in Section 4.

Single Soliton Solution to the NLS Equation
In order to test the accuracy of the algorithm presented in Section 4, Equation (1) with the coefficient η = 1, q = 2 and the exact solution [24] u( is solved by time two-mesh (TT-M) CD scheme Equations (6)- (12) and standard nonlinear implicit (SNI) CD scheme Equations (6)-(8) at the domain of (t, x) ∈ (0, 1] × [−30, 30], respectively. The NLS Equation (1) has an infinite number of conservation laws including, e.g., Further, the numerical invariants are evaluated as Let U F and U S be the numerical solutions of the TT-MCD scheme and the SNI-CD scheme, respectively. Here, we also used some notations to denote the convergence rates of the TT-MCD scheme in time and space, respectively. Similarly, notations Rate t SN I and Rate x SN I are defined in the SNI-CD scheme. In Table 1, the discrete L 2 -norm errors, convergence rates in time and the time cost of the TT-MCD scheme and the SNI-CD scheme are given, respectively. These data are obtained by taking fixed the space step h = 0.02 and changed the time step τ C = 5τ F = 1/4, 1/8, 1/16, 1/32. Further, Table 2 presents the discrete L 2 -norm errors, convergence rates in space and the time cost of the two schemes by taking fixed τ C = 1/50, τ F = 1/2500 and changed h = 1/2, 1/4, 1/8, 1/16. From Tables 1 and 2, we observe that the errors of the two schemes are almost identical and the temporal and spatial convergence rates of the both schemes are nearly approach two and four, respectively, which are in agreement with our theoretical results. Furthermore, from Tables 1 and 2, one can also clearly that our TT-MCD scheme has much lower time cost than the SNI-CD scheme.
The curves of real and imaginary parts of the TT-MCD and exact solutions at t = 1.0 are shown in Figure 1, which indicates that our numerical solution and the exact solution are in good agreement. The numerical invariant values are plotted in Figure 2, which shows that our method preserves the two conservation laws. A comparison of L 2 norm errors of the original TT-M method and its of the proposed TT-M method at t = 1 with τ C = 0.1, τ F = 0.01, h = 0.1 is shown in Figure 3, which implies that our method can improve the calculation accuracy of fine grid solutions.
This initial condition yields a two-soliton solution. As a result of solitons, after the collision, double solitons preserve their properties such as shape, velocity, and amplitudes, which will be observed at the simulations of double solitons are given in Figure 4. The numerical experiment is performed from t = 0 to t = 6 with τ C = 0.03, τ F = 0.01, h = 0.1. As time progresses, the two solitons are traveling in opposite directions. The faster soliton eventually catches up with the slower one. Then, the two solitons collide and separate, but recover their shapes afterward despite a strongly nonlinear interaction. These results are in qualitative agreement with the behavior predicted by the soliton theory [6]. The evolution of the two invariants is given in Figure 5, in which the charge Q seems to remain constant all the time. The energy E also keeps constant except for a slight change that occurs during the interaction of two solitons. The results given in Figure 5 show that our method preserves the conservation laws for charge and energy.

Birth of Standing Soliton with the Maxwellian Initial Condition
Theory predict that if a soliton u(x, t) will appear over time, otherwise the soliton decay away [24]. Here, we consider the birth of soliton with the Maxwellian initial condition [24]: The values of all parameters are chosen to be h = 0.08, τ C = 0.025, τ F = 0.005 and q = 2 to exhibit the behaviors of solutions for A = 1 and A = 1.78 and time running up from t = 0 to t = 6 are given in Figures 6 and 7, respectively. It is seen from Figures 6 and 7 that the approximate solution of |u| decay by time increases for A = 1 unless for A = 1.78 soliton's amplitude, shape, and speed are protected, and the locations of solitons do not change for both cases. In Figure 8, the numerical invariants Q and E for A = 1.78 are plotted. As it is seen undoubtedly from Figure 8 that TT-MCD method produces charge Q and energy E are almost constant.

Conclusions
In this article, a TT-MCD algorithm is presented for the one-dimensional nonlinear Shördinger equation. This new TT-MCD algorithm contains three steps of a nonlinear implicit CD system on the time coarse mesh of size τ C , some useful values on the time fine mesh of size τ F , and a linear system on the time fine mesh. Especially, the fine mesh solution, which becomes available, is also used as the initial guess in the third step to improve the calculation accuracy of fine grid solutions. The discrete L 2 -norm convergence result with O(τ 4 C + τ 2 F + h 4 ) is proved in detail. Three different test problems have been investigated. The performance and accuracy of the algorithm have been shown by investigating calculating the error discrete L 2 -norm and two conservation laws of charge and energy and their relative changes. The obtained results show that the new TT-MCD algorithm can produce numerical solutions of the NLS equation with high accuracy and preserve the conservation laws of charge and energy. Moreover, the computing cost of the TT-MCD method is less than that of the standard nonlinear implicit (SNI) CD scheme. In the near future, we will focus on the extension of the time two-mesh idea to multidimensional nonlinear Shördinger equations combining with alternating direction implicit (ADI) or locally one-dimensional (LOD) techniques. Data Availability Statement: All the data were computed using our algorithm.