Two Energy Conserving Numerical Schemes for the Klein-Gordon-Zakharov Equations

Abstract

Two new difference schemes are proposed for an initial-boundary-value problem of the Klein-Gordon-Zakharov (KGZ) equations. They have the advantage that there is a discrete energy which is conserved. Their stability and convergence of difference solutions are proved in order O(ℎ²+𝜏²) on the basis of the prior estimates. Results of numerical experiments demonstrate the efficiency of the new schemes.

1. Introduction

In this paper, we consider the following initial-boundary-value problem of the KGZ equations (see [1]): 𝑈_𝑡𝑡−𝑈_𝑥𝑥+𝑈+𝑁𝑈+|𝑈|²𝑈=0,−𝑥_𝐿<𝑥<𝑥_𝑅𝑁,0≤𝑡≤𝑇,_𝑡𝑡−𝑁_𝑥𝑥=|𝑈|²_𝑥𝑥,−𝑥_𝐿<𝑥<𝑥_𝑅,≤𝑡≤𝑇,𝑈|_𝑡=0=𝑈₀(𝑥),𝑈_𝑡|_𝑡=0=𝑈₁(𝑥),𝑁|_𝑡=0=𝑁₀(𝑥),𝑁_𝑡|_𝑡=0=𝑁₁(𝑥),𝑈|_{𝑥=𝑥𝐿}=𝑈|_{𝑥=𝑥𝑅}=0,𝑁|_{𝑥=𝑥𝐿}=𝑁|_{𝑥=𝑥𝑅}=0, (1)(2)(3)(4) where a complex unknown function𝑈(𝑥,𝑡)denotes the fast time scale component of electric field raised by electrons and a real unknown function𝑁(𝑥,𝑡)denotes the deviation of ion density from its equilibrium;𝑈₀(𝑥),𝑈₁(𝑥),𝑁₀(𝑥), and𝑁₁(𝑥)are known smooth functions.

The solutions𝑈(𝑥,𝑡)and𝑁(𝑥,𝑡)of the initial-boundary-value problem (1)–(4) formally satisfy the following energy identity: 𝐸=^𝑥𝑅_𝑥𝐿𝑈_𝑡²+𝑈_𝑥²+|𝑈|²+𝑁|𝑈|²+12|𝑉|²+12|𝑁|²+12|𝑈|⁴=const, (5) where the potential function𝑉is defined as 𝑉=−𝑓_𝑥,𝑓_𝑥𝑥=𝑁_𝑡. (6)

In [2] Ozawa et al. proved the well-posedness of the equations in three-dimensional space. Adomian discussed the existence of its nonperturbative solutions (see [3]). In [4] Guo and Yuan studied the global smooth solutions for the Cauchy problem of these equations. Furthermore, in [5, 6] the authors proposed three difference schemes for the KGZ equations. It is well known that a conservative scheme performs better than a nonconservative one; for example, Zhang et al. in [7] pointed out that the nonconservative schemes may easily show nonlinear blowup and Li and Vu-Quoc also said, “in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation" (see [8]). Up to now, many conservative finite difference schemes have been studied for the Klein-Gordon equation, Klein-Gordon-Schödinger equations, Sine-Gordon equation, Zakharov equations, and so on (see [9–25]). Numerical results of all the schemes are very good. Therefore, in this paper we will generalize the technique of these methods to propose two new conservative difference schemes which are unconditionally stable and more accurate for the KGZ equations.

The paper is organized as follows. In Section 2, a new difference scheme (i.e., Scheme A) is proposed, and its discrete conservative law is discussed. In Section 3, some prior estimates for difference solutions are made. In Section 4, convergence and stability for the new scheme are proved using discrete energy method. In Section 5, another conservative scheme (i.e., Scheme B) is constructed, and its discrete conservative law is discussed. In Section 6, some prior estimates of Scheme B are obtained by induction, then convergence of the scheme is analyzed. Finally, in Section 7, some numerical results are provided to demonstrate the theoretical results.

2. Finite Difference Scheme and Its Conservative Law

Before we propose the new difference scheme for the KGZ (1)–(4), we give some notations as follows: 𝑥_𝑗=𝑥_𝐿𝑥+𝑗ℎ,0≤𝑗≤𝐽=^𝑅−𝑥_𝐿ℎ,𝑡^𝑛𝑇=𝑛𝜏,𝑛=0,1,2,…,𝜏,𝑤^𝑛_𝑗𝑥=𝑤^𝑛_𝑗+1−𝑤^𝑛_𝑗ℎ,𝑤^𝑛_𝑗𝑥=𝑤^𝑛_𝑗−𝑤^𝑛_𝑗−1ℎ,𝑤^𝑛_𝑗𝑡=𝑤_𝑗^𝑛+1−𝑤^𝑛_𝑗𝜏,𝑤^𝑛_𝑗𝑡=𝑤^𝑛_𝑗−𝑤_𝑗^𝑛−1𝜏, (7) whereℎand𝜏are step size of space and time, respectively.

Also we define the following inner product and norms: 𝑤^𝑛,𝑢^𝑛=ℎ^𝐽_𝑗=0𝑤^𝑛_𝑗𝑢^𝑛_𝑗,𝑤^𝑛𝑝_𝑝=ℎ^𝐽_𝑗=0𝑤^𝑛_𝑗^𝑝,𝑤^𝑛_∞=sup_{0≤𝑗≤𝐽}𝑤^𝑛_𝑗. (8)

In this paper,𝐶stands for a general positive constant which may take different values on different occasions. For briefness, we omit the subscript2of‖𝑤^𝑛‖₂.

Lemma 1. For any two mesh functions{𝑤_𝑗}and{𝑣_𝑗},𝑗=0,1,…,𝐽, there is the identity ℎ^𝐽−1_𝑗=1𝑤_𝑗(𝑣_𝑗)_𝑥𝑥=−ℎ^𝐽−1_𝑗=0𝑤_𝑗𝑥𝑣_𝑗𝑥−𝑤₀𝑣_0𝑥+𝑤_𝐽𝑣_𝐽𝑥. (9)

It is easy to prove this lemma directly.

Now, we consider the following difference scheme for the KGZ equations (1)–(4).

Scheme A. We consider the following: 𝑈^𝑛_𝑗𝑡𝑡+ℎ²𝑈12^{𝑛𝑗𝑥𝑥𝑡𝑡}−12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1_𝑥𝑥+12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1+12𝑁^𝑛_𝑗𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1+14𝑈_𝑗^𝑛+12+𝑈_𝑗^𝑛−12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1𝑁=0,_{𝑛𝑗𝑡𝑡}+ℎ²𝑁12^{𝑛𝑗𝑥𝑥𝑡𝑡}−12𝑁_𝑗^𝑛+1+𝑁_𝑗^𝑛−1_𝑥𝑥=𝑈^𝑛_𝑗²_𝑥𝑥,𝑈⁰_𝑗=𝑈₀𝑥_𝑗,𝑁⁰_𝑗=𝑁₀𝑥_𝑗,𝑈^𝑛₀=𝑈^𝑛_𝐽=0,𝑁^𝑛₀=𝑁^𝑛_𝐽𝑈=0,_1𝑗−𝑈_𝑗⁻¹=2𝜏𝑈₁𝑥_𝑗,𝑁¹_𝑗−𝑁_𝑗⁻¹=2𝜏𝑁₁𝑥_𝑗. (10)(11)(12)(13) By (10), (12), (11), and (13), we obtain the following: 2𝜏²𝑈¹_𝑗−𝑈⁰_𝑗−𝜏𝑈₁𝑥_𝑗+ℎ²6𝜏²𝑈¹_𝑗−𝑈⁰_𝑗−𝜏𝑈₁𝑥_𝑗𝑥𝑥−𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗𝑥𝑥+𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗+𝑁⁰_𝑗𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗+12𝑈¹_𝑗²+𝑈¹_𝑗−2𝜏𝑈₁𝑥_𝑗²×𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗2=0,𝜏²𝑁¹_𝑗−𝑁⁰_𝑗−𝜏𝑁₁𝑥_𝑗+ℎ²6𝜏²𝑁¹_𝑗−𝑁⁰_𝑗−𝜏𝑁₁𝑥_𝑗𝑥𝑥−𝑁¹_𝑗−𝜏𝑁₁𝑥_𝑗𝑥𝑥=𝑈⁰_𝑗²_𝑥𝑥. (14) Here, we also define the potential function{𝑓^𝑛_𝑗}to be such that 𝑓^𝑛_𝑗𝑥𝑥=𝑁^𝑛_𝑗𝑡,𝑗=1,2,…,𝐽−1,𝑓^𝑛₀=𝑓^𝑛_𝐽=0. (15)

Theorem 2. The difference scheme (10)–(15) possesses the following invariant: 𝐸^𝑛=𝐸^𝑛−1=⋯=𝐸⁰=𝑐𝑜𝑛𝑠𝑡, (16) where 𝐸^𝑛=𝑈^𝑛_𝑡²−ℎ²𝑈12^{𝑛𝑥𝑡2}+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12+𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²+14𝑈^𝑛+14₄+𝑈^𝑛4₄+12𝑓^𝑛_𝑥²−ℎ²𝑁24^𝑛𝑡2+14𝑁^𝑛+12+𝑁^𝑛2. (17)

Proof. Computing the inner product of (10) with𝑈^𝑛+1−𝑈^𝑛−1and taking the real part, we have 𝑈^𝑛_𝑡²−𝑈_𝑡^𝑛−12−ℎ²𝑈12^{𝑛𝑥𝑡2}−𝑈^𝑛−1_𝑥𝑡²+12𝑈_𝑥^𝑛+12−𝑈_𝑥^𝑛−12+12𝑈^𝑛+12−𝑈^𝑛−12+12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12−𝑈_𝑗^𝑛−12+14𝑈^𝑛+14₄−𝑈^𝑛−14₄=0. (18) Next, computing the inner product of (11) with(1/2)(𝑓^𝑛+𝑓^𝑛−1)and using (15), we obtain 12𝑓^𝑛_𝑥²−𝑓_𝑥^𝑛−12−ℎ²𝑁24^𝑛𝑡2−𝑁_𝑡^𝑛−12+14𝑁^𝑛+12−𝑁^𝑛−12+12ℎ^𝐽−1_𝑗=1𝑈^𝑛_𝑗²𝑁_𝑗^𝑛+1−𝑁_𝑗^𝑛−1=0. (19) In the computation of (18) and (19), we have used the boundary conditions and Lemma 1. Then, result (16) follows from (18) and (19).

3. Some Prior Estimates for Difference Solutions

In this section, we will estimate the difference solutions of Scheme A after introducing two important lemmas proved in [26].

Lemma 3 (discrete Sobolev's inequality). For any discrete function𝑢_ℎ={𝑢_𝑗∣𝑗=0,1,…,𝐽}on the finite interval[0,𝑙]and for any given𝜀>0, there exists a constant 𝐶, depending only on𝜀, such that 𝛿^𝑘𝑢_ℎ𝑝≤𝜀𝛿^𝑛𝑢_ℎ2+𝐶𝑢_ℎ2, (20) where2≤𝑝≤∞,0≤𝑘<𝑛.

Lemma 4 (Gronwall's inequality). Suppose that the nonnegative mesh functions{𝑤(𝑛),𝜌(𝑛),𝑛=1,2,…,𝑁,𝑁𝜏=𝑇}satisfy the inequality 𝑤(𝑛)≤𝜌(𝑛)+𝜏^𝑛_𝑙=1𝐵_𝑙𝑤(𝑙), (21) where𝐵_𝑙(𝑙=1,2,…,𝑁)are nonnegative constant. Then, for any0≤𝑛≤𝑁, there is 𝑤(𝑛)≤𝜌(𝑛)exp𝑛𝜏^𝑛_𝑙=1𝐵_𝑙. (22)

Theorem 5. Assume that𝑈₀(𝑥)∈𝐻¹,𝑈₁(𝑥)∈𝐿²,𝑁₀(𝑥)∈𝐻¹, and 𝑁₁(𝑥)∈𝐿²;then the following estimates hold: 𝑈^𝑛_𝑡≤𝐶,𝑈^𝑛_𝑥≤𝐶,𝑈^𝑛𝑈≤𝐶,_𝑛∞≤𝐶,𝑓^𝑛_𝑥≤𝐶,𝑁^𝑛≤𝐶,‖𝑈^𝑛‖₄≤𝐶. (23)

Proof. Applying Young's inequality, it is easy to see that −ℎ²𝑈12^{𝑛𝑥𝑡2}ℎ=−³12^𝐽−1_𝑗=1𝑈^𝑛_𝑗𝑥𝑡²ℎ=−³12^𝐽−1_𝑗=1𝑈^𝑛_𝑗𝑥𝑡𝑈^𝑛_𝑗𝑥𝑡ℎ=−12^𝐽−1_𝑗=1𝑈^𝑛_𝑗+1−𝑈^𝑛_𝑗𝑡𝑈^𝑛_𝑗+1−𝑈^𝑛_𝑗𝑡ℎ≥−6^𝐽−1_𝑗=1𝑈^𝑛_𝑗+1𝑡²+𝑈^𝑛_𝑗𝑡²1=−3𝑈^𝑛_𝑡², (24) and by (15), we have −ℎ²𝑁24^𝑛𝑡2ℎ=−³24^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑡²ℎ=−³24^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑥²ℎ=−24^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥−𝑓^𝑛_𝑗−1𝑥²ℎ≥−12^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥²+𝑓^𝑛_𝑗−1𝑥²1=−6𝑓^𝑛_𝑥²; (25) then from (16) we get 23𝑈^𝑛_𝑡²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12+𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²+14𝑈^𝑛+14₄+𝑈^𝑛4₄+13𝑓^𝑛_𝑥²+14𝑁^𝑛+12+𝑁^𝑛2≤𝐸^𝑛=𝐶. (26) Since 12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12≤14ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗²+𝑈_𝑗^𝑛+14≤14𝑁^𝑛2+𝑈^𝑛+14₄,12ℎ^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²≤14𝑁^𝑛+12+𝑈^𝑛4₄, (27) it follows from (26) that 23𝑈^𝑛_𝑡²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+13𝑓^𝑛_𝑥²≤𝐶. (28) Therefore 𝑈^𝑛_𝑡≤𝐶,𝑈^𝑛_𝑥≤𝐶,𝑈^𝑛≤𝐶,𝑓^𝑛_𝑥≤𝐶. (29) Besides, we can obtain the following estimates by Lemma 3: 𝑈^𝑛_∞≤𝐶,𝑈^𝑛₄≤𝐶. (30) On the other hand, by inequality𝑎𝑏≤(1/4)𝑎²+𝑏², we have ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12≤ℎ^𝐽−1_𝑗=114𝑁^𝑛_𝑗²+𝑈_𝑗^𝑛+14=14𝑁^𝑛2+𝑈^𝑛+14₄,ℎ^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²≤ℎ^𝐽−1_𝑗=114𝑁_𝑗^𝑛+12+𝑈^𝑛_𝑗⁴=14𝑁^𝑛+12+𝑈^𝑛4₄. (31)(32) Thus, it follows from (26) that 𝑁^𝑛≤𝐶. (33) This completes the proof.

4. Convergence and Stability of the Difference Scheme

In this section, we will discuss the convergence and stability of the difference scheme (10)–(15). First, we define the truncation errors by 𝑟^𝑛_𝑗=𝑈𝑥_𝑗,𝑡^𝑛_𝑡𝑡+ℎ²𝑈𝑥12^𝑗,𝑡^𝑛_{𝑥𝑥𝑡𝑡}−12𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1_𝑥𝑥+12𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1+12𝑁𝑥_𝑗,𝑡^𝑛𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1+14𝑈𝑥_𝑗,𝑡^𝑛+12+𝑈𝑥_𝑗,𝑡^𝑛−12×𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1,𝜎^𝑛_𝑗=𝑁𝑥_𝑗,𝑡^𝑛_𝑡𝑡+ℎ²𝑁𝑥12^𝑗,𝑡^𝑛_{𝑥𝑥𝑡𝑡}−12𝑁𝑥_𝑗,𝑡^𝑛+1+𝑁𝑥_𝑗,𝑡^𝑛−1_𝑥𝑥−𝑈𝑥_𝑗,𝑡^𝑛2_𝑥𝑥. (34)(35)

Lemma 6. Assume that the conditions of Theorem 5 are satisfied and𝑈(𝑥.𝑡)∈𝐶^4,4,𝑁(𝑥,𝑡)∈𝐶^4,4;then the truncation errors of the difference scheme (10)–(15) satisfy|𝑟^𝑛_𝑗|+|𝜎^𝑛_𝑗|=𝑂(𝜏²+ℎ²)as𝜏→0,ℎ→0.

By Taylor's expansion, Lemma 6 can be proved directly. Besides, we note that the approximations of the initial conditions (13) have truncation errors of order𝑂(𝜏²), which are consistent with the scheme.

Now, we are going to analyze the convergence of the difference scheme (10)–(15).

Set the following: 𝑒^𝑛_𝑗=𝑈𝑥_𝑗,𝑡^𝑛−𝑈^𝑛_𝑗,𝜂^𝑛_𝑗=𝑁𝑥_𝑗,𝑡^𝑛−𝑁^𝑛_𝑗,1ℎ²𝐹^𝑛_𝑗+1−2𝐹^𝑛_𝑗+𝐹^𝑛_𝑗−1=1𝜏𝜂_𝑗^𝑛+1−𝜂^𝑛_𝑗,𝐹𝑗=1,2,…,𝐽−1,_𝑛0=𝐹^𝑛_𝐽=0. (36)

Theorem 7. Assume that the conditions of Lemma 6 are satisfied; then the solutions of the difference scheme (10)–(15) converge to the solutions of the problem stated in (1)–(4) with order𝑂(𝜏²+ℎ²)in the𝐿_∞norm for𝑈^𝑛and in the𝐿₂norm for𝑁^𝑛.

Proof. Subtracting (10) from (34), we obtain 𝑟^𝑛_𝑗=𝑒^𝑛_𝑗𝑡𝑡+ℎ²𝑒12^{𝑛𝑗𝑥𝑥𝑡𝑡}−12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1_𝑥𝑥+12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+12𝜂^𝑛_𝑗𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1+12𝑁^𝑛_𝑗𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛+1+𝐿₁; (37) that is, 𝑒^𝑛_𝑗𝑡𝑡+ℎ²𝑒12^{𝑛𝑗𝑥𝑥𝑡𝑡}−12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1_𝑥𝑥+12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1=𝑟^𝑛_𝑗−12𝜂^𝑛_𝑗𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1−12𝑁^𝑛_𝑗𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛+1−𝐿₁, (38) where 𝐿₁=14𝑈𝑥_𝑗,𝑡^𝑛+12+𝑈𝑥_𝑗,𝑡^𝑛−12×𝑈𝑥_𝑗,𝑡^𝑛+1+𝑈𝑥_𝑗,𝑡^𝑛−1−14𝑈_𝑗^𝑛+12+𝑈_𝑗^𝑛−12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1=14𝑈𝑥_𝑗,𝑡^𝑛+12+𝑈𝑥_𝑗,𝑡^𝑛−12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+14𝑈𝑥_𝑗,𝑡^𝑛+12+𝑈𝑥_𝑗,𝑡^𝑛−12−𝑈_𝑗^𝑛+12−𝑈_𝑗^𝑛−12⋅𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1=14𝑈𝑥_𝑗,𝑡^𝑛+12+𝑈𝑥_𝑗,𝑡^𝑛−12𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛−1+14𝑈𝑥_𝑗,𝑡^𝑛+1𝑒_𝑗^𝑛+1+𝑒_𝑗^𝑛+1𝑈_𝑗^𝑛+1+𝑒_𝑗^𝑛−1𝑈_𝑗^𝑛−1+𝑈𝑥_𝑗,𝑡^𝑛−1𝑒_𝑗^𝑛−1⋅𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1. (39) Then computing the inner product of (38) with𝑒^𝑛+1−𝑒^𝑛−1and taking the real part, we have ‖𝑒^𝑛_𝑡‖²−‖𝑒_𝑡^𝑛−1‖²−ℎ²12‖𝑒^𝑛_𝑥𝑡‖²−‖𝑒^𝑛−1_𝑥𝑡‖²+12‖𝑒_𝑥^𝑛+1‖²−‖𝑒_𝑥^𝑛−1‖²+12‖𝑒^𝑛+1‖²−‖𝑒^𝑛−1‖²≤𝐶𝜏‖𝑟^𝑛‖²+‖𝜂^𝑛‖²+‖𝑁^𝑛𝑒^𝑛+1+𝑒^𝑛−1‖²+‖𝑒^𝑛_𝑡‖²+‖𝑒_𝑡^𝑛−1‖²+‖𝑒^𝑛+1‖²+‖𝑒^𝑛−1‖². (40) From Lemma 3 and Theorem 5 it follows that 𝑁^𝑛𝑒^𝑛+1+𝑒^𝑛−1≤𝐶𝑁^𝑛⋅𝑒^𝑛+1+𝑒^𝑛−1_∞≤𝐶𝑒_𝑥^𝑛+1+𝑒_𝑥^𝑛−1+𝐶𝑒^𝑛+1+𝑒^𝑛−1≤𝐶𝑒_𝑥^𝑛+1+𝑒_𝑥^𝑛−1+𝑒^𝑛+1+𝑒^𝑛−1. (41) So, substituting (41) into (40), we have 𝑒^𝑛_𝑡²−𝑒_𝑡^𝑛−12−ℎ²𝑒12^{𝑛𝑥𝑡2}−𝑒^𝑛−1_𝑥𝑡²+12𝑒_𝑥^𝑛+12−𝑒_𝑥^𝑛−12+12𝑒^𝑛+12−𝑒^𝑛−12≤𝐶𝜏𝑟^𝑛2+𝜂^𝑛2+𝑒_𝑥^𝑛+12+𝑒_𝑥^𝑛−12+𝑒^𝑛+12+𝑒^𝑛−12+𝑒^𝑛_𝑡²+𝑒_𝑡^𝑛−12. (42) Next, subtracting (11) from (35), we obtain 𝜂^𝑛_𝑗𝑡𝑡+ℎ²𝜂12^{𝑛𝑗𝑥𝑥𝑡𝑡}−12𝜂_𝑗^𝑛+1+𝜂_𝑗^𝑛−1_𝑥𝑥=𝜎^𝑛_𝑗+𝑈𝑥_𝑗,𝑡^𝑛2−𝑈^𝑛_𝑗²_𝑥𝑥. (43) Computing the inner product of (43) with(1/2)(𝐹^𝑛+𝐹^𝑛−1), we get −1𝐹2𝜏^𝑛𝑥2−𝐹_𝑥^𝑛−12+ℎ²𝜂24𝜏^𝑛𝑡2−𝜂_𝑡^𝑛−12−1𝜂4𝜏^𝑛+12−𝜂^𝑛−12=𝜎^𝑛_𝑗,12𝐹^𝑛+𝐹^𝑛−1+𝑈𝑥_𝑗,𝑡^𝑛2−𝑈^𝑛_𝑗²_𝑥𝑥,12𝐹^𝑛+𝐹^𝑛−1. (44) Note that 𝑈𝑥_𝑗,𝑡^𝑛2−𝑈^𝑛_𝑗²_𝑥𝑥,12𝐹^𝑛+𝐹^𝑛−1≤𝐶𝐹^𝑛_𝑥²+𝐹_𝑥^𝑛−12+𝑒^𝑛_𝑥²+𝑒^𝑛2,𝜎^𝑛,12𝐹^𝑛+𝐹^𝑛−1=12ℎ^𝐽−1_𝑗=1𝜎^𝑛_𝑗𝐹^𝑛_𝑗+𝐹_𝑗^𝑛−1≤𝐶𝜎^𝑛2+𝐹^𝑛_𝑥²+𝐹_𝑥^𝑛−12. (45) Then substituting (45) into (44) we have 𝐹^𝑛_𝑥²−𝐹_𝑥^𝑛−12−ℎ²𝜂12^𝑛𝑡2−𝜂_𝑡^𝑛−12+12𝜂^𝑛+12−𝜂^𝑛−12≤𝐶𝜏𝜎^𝑛2+𝐹^𝑛_𝑥²+𝐹_𝑥^𝑛−12+𝑒^𝑛_𝑥²+𝑒^𝑛2. (46) Now, adding (42) to(46), we get 𝑒^𝑛_𝑡²−𝑒_𝑡^𝑛−12−ℎ²𝑒12^{𝑛𝑥𝑡2}−𝑒^𝑛−1_𝑥𝑡²+12𝑒_𝑥^𝑛+12−𝑒_𝑥^𝑛−12+12𝑒^𝑛+12−𝑒^𝑛−12+𝐹^𝑛_𝑥²−𝐹_𝑥^𝑛−12−ℎ²𝜂12^𝑛𝑡2−𝜂_𝑡^𝑛−12+12𝜂^𝑛+12−𝜂^𝑛−12≤𝐶𝜏𝑟^𝑛2+𝜂^𝑛2+𝜎^𝑛2+𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥²+𝑒_𝑥^𝑛−12+𝑒^𝑛+12+𝑒^𝑛2+𝑒^𝑛−12+𝑒^𝑛_𝑡²+𝑒_𝑡^𝑛−12+𝐹^𝑛_𝑥²+𝐹_𝑥^𝑛−12. (47) Let 𝐵^𝑛=𝑒^𝑛_𝑡²−ℎ²𝑒12^{𝑛𝑥𝑡2}+12𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥²+12𝑒^𝑛+12+𝑒^𝑛2+𝐹^𝑛_𝑥²−ℎ²𝜂12^𝑛𝑡2+12𝜂^𝑛+12+𝜂^𝑛2. (48) It is easy to see that 𝐵^𝑛≥23𝑒^𝑛_𝑡²+12𝑒_𝑥^𝑛+12+𝑒^𝑛_𝑥²+12𝑒^𝑛+12+𝑒^𝑛2+23𝐹^𝑛_𝑥²+12𝜂^𝑛+12+𝜂^𝑛2. (49) Then by (47) and Lemma 6 we have 𝐵^𝑛−𝐵^𝑛−1≤𝐶𝜏𝑟^𝑛2+𝜎^𝑛2+𝐶𝜏𝐵^𝑛+𝐵^𝑛−1≤𝐶𝜏ℎ²+𝜏²²+𝐶𝜏𝐵^𝑛+𝐵^𝑛−1. (50) Summing (50) up for𝑛and applying Lemma 4, we get 𝐵^𝑁≤𝐵⁰+𝐶ℎ²+𝜏²²()𝐵exp𝐶𝑁𝜏≤𝐶⁰+ℎ²+𝜏²². (51) Therefore, it follows from (49) that 23𝑒^𝑁_𝑡²+12𝑒_𝑥^𝑁+12+𝑒^𝑁_𝑥²+12𝑒^𝑁+12+𝑒^𝑁2+23𝐹^𝑛_𝑥²+12𝜂^𝑁+12+𝜂^𝑁2≤𝐶𝐵⁰+ℎ²+𝜏²². (52) Note that𝑒⁰,𝑒¹,𝜂⁰, and 𝜂¹are two-order precision and‖𝐹⁰_𝑥‖=𝑂(ℎ²+𝜏²)(see [22]). Thus𝐵⁰=𝑂(ℎ²+𝜏²)². Hence, the following inequalities can be obtained by (52): 𝑒^𝑁_𝑥≤𝑂ℎ²+𝜏²,𝑒^𝑁≤𝑂ℎ²+𝜏²𝑒_𝑁𝑡,≤𝑂ℎ²+𝜏²,𝜂^𝑁≤𝑂ℎ²+𝜏². (53)(54) Then, applying Lemma 3, we get 𝑒^𝑁_𝑥∞≤𝑂ℎ²+𝜏². (55)
So the proof of Theorem 7 is complete.

In the same way, we can prove that the solutions of the difference schemes (10)–(15) are unconditionally stable for initial data.

5. Another Conservative Difference Scheme

In this section, we will propose another conservative difference scheme for the problem given in (1)–(4) and discuss the discrete conservative law of this scheme.

Now, we consider the finite difference simulations for (1) and (2) as follows.

Scheme B. We consider the following: 𝑈^𝑛_𝑗𝑡𝑡−12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1_𝑥𝑥+12𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1+12𝑁^𝑛_𝑗𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1+18𝑈_𝑗^𝑛+12+2𝑈^𝑛_𝑗²+𝑈_𝑗^𝑛−12×𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−11=0,2𝑁^𝑛_𝑗𝑡𝑡+14𝑁^𝑛_𝑗+1+𝑁^𝑛_{𝑗−1𝑡𝑡}−12𝑁_𝑗^𝑛+1+𝑁_𝑗^𝑛−1_𝑥𝑥=𝑈^𝑛_𝑗²_𝑥𝑥. (56)(57)

In addition, the initial and boundary conditions (3) and (4) are also, respectively, approximated as 𝑈⁰_𝑗=𝑈₀𝑥_𝑗,𝑁⁰_𝑗=𝑁₀𝑥_𝑗,𝑈^𝑛₀=𝑈^𝑛_𝐽=0,𝑁^𝑛₀=𝑁^𝑛_𝐽𝑈=0,_1𝑗−𝑈_𝑗⁻¹=2𝜏𝑈₁𝑥_𝑗,𝑁¹_𝑗−𝑁_𝑗⁻¹=2𝜏𝑁₁𝑥_𝑗. (58)(59)

We also define the function{𝑓^𝑛_𝑗}by 𝑓^𝑛_𝑗𝑥𝑥=𝑁^𝑛_𝑗𝑡,𝑗=1,2,…,𝐽−1,𝑓^𝑛₀=𝑓^𝑛_𝐽=0. (60) In (56) and (57), let𝑛=0. Then eliminating𝑈⁻¹and𝑁⁻¹from (58) and (59), we get 2𝜏²𝑈¹_𝑗−𝑈⁰_𝑗−𝜏𝑈₁𝑥_𝑗−𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗𝑥𝑥+𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗+𝑁⁰_𝑗𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗+14𝑈¹_𝑗²+2𝑈⁰_𝑗²+𝑈¹_𝑗−2𝜏𝑈₁𝑥_𝑗²×𝑈¹_𝑗−𝜏𝑈₁𝑥_𝑗1=0,𝜏²𝑁¹_𝑗−𝑁⁰_𝑗−𝜏𝑁₁𝑥_𝑗+12𝜏²𝑁¹_𝑗+1−𝑁⁰_𝑗+1−𝜏𝑁₁𝑥_𝑗+1+𝑁¹_𝑗−1−𝑁⁰_𝑗−1−𝜏𝑁₁𝑥_𝑗−1−𝑁¹_𝑗−𝜏𝑁₁𝑥_𝑗𝑥𝑥=𝑈⁰_𝑗²_𝑥𝑥. (61)

Theorem 8. Scheme B admits the following invariant: 𝐸^𝑛=𝐸^𝑛−1=⋯=𝐸⁰=𝑐𝑜𝑛𝑠𝑡, (62) where 𝐸^𝑛=𝑈^𝑛_𝑡²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+14𝑓^𝑛_𝑥²+12ℎ^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²+𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12+14ℎ^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥+14𝑁^𝑛+12+𝑁^𝑛2+18𝑈^𝑛+14₄+𝑈^𝑛4₄+2ℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗². (63)

Proof. Computing the inner product of (56) with𝑈^𝑛+1−𝑈^𝑛−1and taking the real part, we have 𝑈^𝑛_𝑡²−𝑈_𝑡^𝑛−12+12𝑈_𝑥^𝑛+12−𝑈_𝑥^𝑛−12+12𝑈^𝑛+12−𝑈^𝑛−12+12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12−𝑈_𝑗^𝑛−12+𝐿₁=0, (64) where 𝐿₁=18Reℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12+2𝑈^𝑛_𝑗²+𝑈_𝑗^𝑛−12×𝑈_𝑗^𝑛+1+𝑈_𝑗^𝑛−1𝑈_𝑗^𝑛+1−𝑈_𝑗^𝑛−1=18ℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12+2𝑈^𝑛_𝑗²+𝑈_𝑗^𝑛−12×𝑈_𝑗^𝑛+12−𝑈_𝑗^𝑛−12=18ℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+14+2𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗²−2𝑈^𝑛_𝑗²𝑈_𝑗^𝑛−12−𝑈_𝑗^𝑛−14=18𝑈^𝑛+14₄+2ℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗²−2ℎ^𝐽−1_𝑗=1𝑈^𝑛_𝑗²𝑈_𝑗^𝑛−12−𝑈^𝑛−14₄. (65)
Next, computing the inner product of (57) with(1/2)(𝑓^𝑛+𝑓^𝑛−1)and by (60), we obtain −1𝑓4𝜏^𝑛𝑥2−𝑓_𝑥^𝑛−12+𝐿₂−1𝑁4𝜏^𝑛+12−𝑁^𝑛−12=ℎ2𝜏^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²−𝑁_𝑗^𝑛−1𝑈^𝑛_𝑗², (66) where 𝐿₂=14ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗+1+𝑁^𝑛_{𝑗−1𝑡𝑡}⋅12𝑓^𝑛_𝑗+𝑓_𝑗^𝑛−1=1ℎ4𝜏^𝐽−1_𝑗=1𝑓^𝑛_𝑗+1+𝑓^𝑛_{𝑗−1𝑥𝑥}−𝑓^𝑛−1_𝑗+1+𝑓^𝑛−1_{𝑗−1𝑥𝑥}⋅12𝑓^𝑛_𝑗+𝑓_𝑗^𝑛−11=−ℎ4𝜏^𝐽−1_𝑗=1𝑓^𝑛_𝑗+1+𝑓^𝑛_𝑗−1𝑥−𝑓^𝑛−1_𝑗+1+𝑓^𝑛−1_𝑗−1𝑥⋅12𝑓^𝑛_𝑗+𝑓_𝑗^𝑛−1_𝑥1=−ℎ4𝜏^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥−𝑓_𝑗^𝑛−1_𝑥𝑓^𝑛−1_𝑗−1𝑥. (67) Then 14𝑓^𝑛_𝑥²−𝑓_𝑥^𝑛−12+14ℎ^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥−𝑓_𝑗^𝑛−1_𝑥𝑓^𝑛−1_𝑗−1𝑥+14𝑁^𝑛+12−𝑁^𝑛−12+ℎ2𝜏^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²−𝑁_𝑗^𝑛−1𝑈^𝑛_𝑗²=0. (68) Hence, result (62) is obtained by adding (68) to (64). This completes the proof.

6. Convergence and Stability of the Scheme

Before we prove the convergence of Scheme B, we estimate the difference solutions of this scheme.

Theorem 9. Assume that the conditions of Theorem 5 are satisfied; then the following estimates hold: 𝑈^𝑛_𝑡≤𝐶,𝑈^𝑛_𝑥≤𝐶,𝑈^𝑛𝑈≤𝐶,_𝑛∞≤𝐶,𝑁^𝑛≤𝐶,𝑈^𝑛₄≤𝐶. (69)

Proof (by induction). First, because of the inequality 1𝑎⋅𝑏≤𝑝(𝜀𝑎)^𝑝+1𝑞1𝜀𝑏^𝑞,1𝑝+1𝑞=1,𝑝,𝑞,𝜀>0, (70) we have 12ℎ^𝐽−1_𝑗=1𝑁^𝑛_𝑗𝑈_𝑗^𝑛+12≤𝜀₁2𝑁^𝑛2+18𝜀₁‖𝑈^𝑛+1‖⁴₄,12ℎ^𝐽−1_𝑗=1𝑁_𝑗^𝑛+1𝑈^𝑛_𝑗²≤18𝑁^𝑛+12+12𝑈^𝑛4₄. (71) Then, substituting (71) into (62) and choosing𝜀₁=√2𝜀>1, we get the following inequality: ‖𝑈^𝑛_𝑡‖²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+14𝑓^𝑛_𝑥²+14ℎ^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥+14𝑁^𝑛+12+𝑁^𝑛2+18𝑈^𝑛+14₄+𝑈^𝑛4₄+2ℎ^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗²1≤𝐶+8𝑁^𝑛+12+12𝑈^𝑛4₄+𝜀₁2𝑁^𝑛2+18𝜀₁𝑈^𝑛+14₄. (72) That is, 𝑈^𝑛_𝑡²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+14𝑓^𝑛_𝑥²+14ℎ^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥+18𝑁^𝑛+12+14𝑁^𝑛2+18−18𝜀₁𝑈^𝑛+14₄+18𝑈^𝑛4₄+2ℎ8^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗²1≤𝐶+2𝑈^𝑛4₄+𝜀₁2𝑁^𝑛2. (73) Note that 14ℎ^𝐽−1_𝑗=1𝑓^𝑛_𝑗𝑥𝑓^𝑛_𝑗−1𝑥≤14ℎ^𝐽−1_𝑗=112𝑓^𝑛_𝑗²_𝑥+𝑓^𝑛_𝑗−1²_𝑥=14𝑓^𝑛_𝑥², (74) so we have ‖𝑈^𝑛_𝑡‖²+12𝑈_𝑥^𝑛+12+𝑈^𝑛_𝑥²+12𝑈^𝑛+12+𝑈^𝑛2+18𝑁^𝑛+12+14𝑁^𝑛2+18−18𝜀₁𝑈^𝑛+14₄+18𝑈^𝑛4₄+2ℎ8^𝐽−1_𝑗=1𝑈_𝑗^𝑛+12𝑈^𝑛_𝑗²1≤𝐶+2𝑈^𝑛4₄+𝜀₁2𝑁^𝑛2. (75)
Obviously, by (58), (59), and the conditions of Theorem 5, the following inequalities hold: 𝑈_𝑡⁻¹≤𝐶,𝑈⁰_𝑡𝑈≤𝐶,_0𝑥≤𝐶,𝑈⁰𝑈≤𝐶,_0∞≤𝐶,𝑁⁰≤𝐶,𝑈⁰₄≤𝐶. (76)
Assume that Theorem 9 holds when𝑛=𝑘; that is, 𝑈_𝑡^𝑘−1≤𝐶,𝑈^𝑘_𝑥≤𝐶,𝑈^𝑘𝑈≤𝐶,_𝑘∞≤𝐶,𝑁^𝑘≤𝐶,𝑈^𝑘₄≤𝐶. (77) By (75) and (77), we get 𝑈^𝑘_𝑡²+12𝑈_𝑥^𝑘+12+𝑈^𝑘_𝑥²+12𝑈^𝑘+12+𝑈^𝑘2+18𝑁^𝑘+12+14𝑁^𝑘2+18−18𝜀₁𝑈^𝑘+14₄+18𝑈^𝑘4₄+2ℎ8^𝐽−1_𝑗=1𝑈_𝑗^𝑘+12𝑈^𝑘_𝑗²1≤𝐶+2𝑈^𝑘4₄+𝜀₁2𝑁^𝑘2≤𝐶, (78) from which the following inequalities are obtained, 𝑈^𝑘_𝑡≤𝐶,𝑈_𝑥^𝑘+1≤𝐶,𝑈^𝑘+1𝑁≤𝐶,_𝑘+1≤𝐶,𝑈^𝑘+1₄≤𝐶, (79) and applying Lemma 3, we have 𝑈^𝑘+1_∞≤𝐶. (80) Then, for any𝑛∈{0,1,2,…,𝑁}, the following estimates are obtained: 𝑈^𝑛_𝑡≤𝐶,𝑈^𝑛_𝑥≤𝐶,𝑈^𝑛𝑈≤𝐶,_𝑛∞≤𝐶,𝑁^𝑛≤𝐶,𝑈^𝑛₄≤𝐶. (81) This completes the proof.

Theorem 10. Assume that the conditions of Lemma 6 are satisfied; then the solutions of the difference scheme (56)–(61) converge to the solutions of the problem given in (1)–(4) with order𝑂(𝜏²+ℎ²)in the𝐿_∞norm for𝑈^𝑛and in the𝐿₂norm for𝑁^𝑛.

Here, we omit details of the proof of this theorem because it can be proved in the same way as that used to prove Theorem 7.

7. Numerical Experiments

In this section, we compute the following numerical example to demonstrate the effectiveness of our two difference schemes: 𝑈_𝑡𝑡−𝑈_𝑥𝑥+𝑈+𝑁𝑈+|𝑈|²𝑁𝑈=0,−20<𝑥<20,0≤𝑡≤𝑇,_𝑡𝑡−𝑁_𝑥𝑥=|𝑈|²_𝑥𝑥𝑈()()()()𝑈()=√,−20<𝑥<20,0≤𝑡≤𝑇,−20,𝑡=𝑈20,𝑡=0,𝑁−20,𝑡=𝑁20,𝑡=0,0≤𝑡≤𝑇,𝑥,010−√22sech1+√52𝑥⋅exp𝑖21+√5𝑥𝑁(),−20≤𝑥≤20,𝑥,0=−2sech²1+√52𝑥,−20≤𝑥≤20. (82)

The analytic solution of KGZ equations, which is derived in [27], will be used in our computation for comparison. The solution can be written as 𝑈()=√𝑥,𝑡10−√22sech1+√52𝑖𝑥−𝑡⋅exp21+√5,𝑁()𝑥−𝑡𝑥,𝑡=−2sech²1+√52𝑥−𝑡. (83)(84)

In order to quantify the numerical results, we define the “error” functions and “rate of convergence” as 𝑒(ℎ,𝜏)=𝑒^𝑛_∞=sup_{0≤𝑗≤𝐽}𝑈𝑥_𝑗,𝑡^𝑛−𝑈^𝑛_𝑗,rate₁=log₂𝑒(ℎ,𝜏)𝑒(),𝜂()=𝜂ℎ/2,𝜏/2ℎ,𝜏^𝑛=ℎ^𝐽_𝑗=0𝑈𝑥_𝑗,𝑡^𝑛−𝑁^𝑛_𝑗²,rate₂=log₂𝜂(ℎ,𝜏)𝜂().ℎ/2,𝜏/2 (85)(86) For the two iterative schemes, we use an error restrictor𝜀≤0.000001to control the iterative procedures.

Firstly, in Figures 1 and 2, the solitary waves computed by Scheme A and Scheme B are compared with the waves of analytic solution, respectively. From (12) and (58), we will see that the the boundary conditions discretization produces no error in computation, so it is harmless to discrete energy𝐸^𝑛. The curves of discrete energy𝐸^𝑛obtained by the two schemes are plotted in Figure 3. Secondly, Tables 1 and 2 give the errors and the rates of convergence for Scheme A and Scheme B with variousℎand𝜏. Finally, errors produced by our two schemes and the schemes in [5, 6] are compared in Tables 3 and 4.

Table 1 

Errors and rates of convergence for Scheme A with different ℎ and 𝜏 at time 𝑇=1.

Table 2 

Errors and rates of convergence for Scheme B with different ℎ and 𝜏 at time 𝑇=5.

Table 3 

Error comparison for Scheme A and scheme in [6] with 𝜃=0.5 at time 𝑇=5.

Table 4 

Error comparison for Scheme B and Scheme I in [5] at time 𝑇=10.

(a)

(b)

Figure 1 

|𝑈|and𝑁computed by Scheme A withℎ=𝜏=0.1.Comparison between analytic solution and numerical solution at time𝑇=1.

(a)

(b)

Figure 2 

|𝑈|and𝑁computed by Scheme B withℎ=0.1,𝜏=0.05.Comparison between analytic solution and numerical solution at time𝑇=5.

(a)

(b)

Figure 3 

Discrete energy𝐸𝑛computed by Scheme A and Scheme B withℎ=0.05,𝜏=0.05.

In Figures 1 and 2, it is obvious that the solitary waves computed by Scheme A and Scheme B agree with the ones computed by exact solutions quite well. Figure 3 shows that both Schemes A and B possess satisfactory conservative property. Tables 1 and 2 verify the second-order convergence and good stability for the two schemes. Furthermore, Tables 3 and 4 show that our two schemes are more accurate than schemes in [5, 6]. Therefore, it is clear that our two new difference schemes are efficient and accurate for the studied problem.

8. Conclusion

In this paper, we study the finite difference method for the KGZ equations. We propose two difference schemes, both of them are conservative on discrete energy law. The two schemes are shown to possess second-order accuracy for𝑈in𝐿_∞norm and for𝑁in𝐿₂norm. Numerical results demonstrate that the two schemes are accurate and efficient. It is worth mentioning that our methods can be directly extended to two dimensions and/or three dimensions, but some new techniques are required to be used to deal with the prior estimates.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (11001034).