Research Article  Open Access
Two Energy Conserving Numerical Schemes for the KleinGordonZakharov Equations
Abstract
Two new difference schemes are proposed for an initialboundaryvalue problem of the KleinGordonZakharov (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 initialboundaryvalue 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 initialboundaryvalue problem (1)–(4) formally satisfy the following energy identity: where the potential functionis defined as
In [2] Ozawa et al. proved the wellposedness of the equations in threedimensional 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 VuQuoc 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 KleinGordon equation, KleinGordonSchödinger equations, SineGordon 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: 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 twoorder 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 For the two iterative schemes, we use an error restrictorto 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 energy. The curves of discrete energyobtained 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 variousand. Finally, errors produced by our two schemes and the schemes in [5, 6] are compared in Tables 3 and 4.
