Journal of Applied Mathematics

Journal of Applied Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 462018 | https://doi.org/10.1155/2013/462018

Juan Chen, Luming Zhang, "Two Energy Conserving Numerical Schemes for the Klein-Gordon-Zakharov Equations", Journal of Applied Mathematics, vol. 2013, Article ID 462018, 13 pages, 2013. https://doi.org/10.1155/2013/462018

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

Academic Editor: Orazio Descalzi
Received28 Jul 2013
Accepted19 Sep 2013
Published30 Nov 2013

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() 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]): where a complex unknown functiondenotes the fast time scale component of electric field raised by electrons and a real unknown functiondenotes the deviation of ion density from its equilibrium;,,, andare known smooth functions.

The solutionsandof the initial-boundary-value problem (1)–(4) formally satisfy the following energy identity: where the potential functionis defined as

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 [925]). 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: whereandare step size of space and time, respectively.

Also we define the following inner product and norms:

In this paper,stands for a general positive constant which may take different values on different occasions. For briefness, we omit the subscriptof.

Lemma 1. For any two mesh functionsand,, there is the identity

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: By (10), (12), (11), and (13), we obtain the following: Here, we also define the potential functionto be such that

Theorem 2. The difference scheme (10)–(15) possesses the following invariant: where

Proof. Computing the inner product of (10) withand taking the real part, we have Next, computing the inner product of (11) withand using (15), we obtain 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 functionon the finite intervaland for any given, there exists a constant , depending only on, such that where.

Lemma 4 (Gronwall's inequality). Suppose that the nonnegative mesh functionssatisfy the inequality whereare nonnegative constant. Then, for any, there is

Theorem 5. Assume that, and then the following estimates hold:

Proof. Applying Young's inequality, it is easy to see that and by (15), we have then from (16) we get Since it follows from (26) that Therefore Besides, we can obtain the following estimates by Lemma 3: On the other hand, by inequality, we have Thus, it follows from (26) that 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

Lemma 6. Assume that the conditions of Theorem 5 are satisfied andthen the truncation errors of the difference scheme (10)–(15) satisfyas.

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 order, which are consistent with the scheme.

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

Set the following:

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 orderin thenorm forand in thenorm for.

Proof. Subtracting (10) from (34), we obtain that is, where Then computing the inner product of (38) withand taking the real part, we have From Lemma 3 and Theorem 5 it follows that So, substituting (41) into (40), we have Next, subtracting (11) from (35), we obtain Computing the inner product of (43) with, we get Note that Then substituting (45) into (44) we have Now, adding (42) to(46), we get Let It is easy to see that Then by (47) and Lemma 6 we have Summing (50) up forand applying Lemma 4, we get Therefore, it follows from (49) that Note that, and are two-order precision and(see [22]). Thus. Hence, the following inequalities can be obtained by (52): Then, applying Lemma 3, we get
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:

In addition, the initial and boundary conditions (3) and (4) are also, respectively, approximated as

We also define the functionby In (56) and (57), let. Then eliminatingandfrom (58) and (59), we get

Theorem 8. Scheme B admits the following invariant: where

Proof. Computing the inner product of (56) withand taking the real part, we have where
Next, computing the inner product of (57) withand by (60), we obtain where Then 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:

Proof (by induction). First, because of the inequality we have Then, substituting (71) into (62) and choosing, we get the following inequality: That is, Note that so we have
Obviously, by (58), (59), and the conditions of Theorem 5, the following inequalities hold:
Assume that Theorem 9 holds when; that is, By (75) and (77), we get from which the following inequalities are obtained, and applying Lemma 3, we have Then, for any, the following estimates are obtained: 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 orderin thenorm forand in thenorm for.

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:

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

In order to quantify the numerical results, we define the “error” functions and “rate of convergence” as