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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 462018, 13 pages
Two Energy Conserving Numerical Schemes for the Klein-Gordon-Zakharov Equations
1Department of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
2Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Received 28 July 2013; Accepted 19 September 2013
Academic Editor: Orazio Descalzi
Copyright © 2013 Juan Chen and Luming Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
In this paper, we consider the following initial-boundary-value problem of the KGZ equations (see ): 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.
In  Ozawa et al. proved the well-posedness of the equations in three-dimensional space. Adomian discussed the existence of its nonperturbative solutions (see ). In  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  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 ). 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
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.
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 .
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
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
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
Then substituting (45) into (44) we have
Now, adding (42) to(46), we get
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 ). Thus. Hence, the following inequalities can be obtained by (52):
Then, applying Lemma 3, we get
So the proof of Theorem 7 is complete.
5. Another Conservative Difference Scheme
Scheme B. We consider the following:
Theorem 8. Scheme B admits the following invariant: where
Proof. Computing the inner product of (56) withand taking the real part, we have
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
Then, substituting (71) into (62) and choosing, we get the following inequality:
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 , 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.
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.
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 forinnorm and forinnorm. 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.
This paper is supported by the National Natural Science Foundation of China (11001034).
- R. O. Dendy, Plama Dynamics, Oxford University Press, Oxford, UK, 1990.
- T. Ozawa, K. Tsutaya, and Y. Tsutsumi, “Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions,” Mathematische Annalen, vol. 313, no. 1, pp. 127–140, 1999.
- G. Adomian, “Non-perturbative solution of the Klein-Gordon-Zakharov equation,” Applied Mathematics and Computation, vol. 81, no. 1, pp. 89–92, 1997.
- B. Guo and G. Yuan, “Global smooth solution for the Klein-Gordon-Zakharov equations,” Journal of Mathematical Physics, vol. 36, no. 8, pp. 4119–4124, 1995.
- T. Wang, J. Chen, and L. Zhang, “Conservative difference methods for the Klein-Gordon-Zakharov equations,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 430–452, 2007.
- J. Chen and L.-m. Zhang, “Numerical simulation for the initial-boundary value problem of the Klein-Gordon-Zakharov equations,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 2, pp. 325–336, 2012.
- F. Zhang, V. M. Pérez-García, and L. Vázquez, “Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme,” Applied Mathematics and Computation, vol. 71, no. 2-3, pp. 165–177, 1995.
- S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839–1875, 1995.
- Q. S. Chang and L. B. Xu, “A numerical method for a system of generalized nonlinear Schrödinger equations,” Journal of Computational Mathematics, vol. 4, no. 3, pp. 191–199, 1986.
- Q. Chang, E. Jia, and W. Sun, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” Journal of Computational Physics, vol. 148, no. 2, pp. 397–415, 1999.
- T. Wang and L. Zhang, “Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1780–1794, 2006.
- T. Wang, T. Nie, L. Zhang, and F. Chen, “Numerical simulation of a nonlinearly coupled Schrödinger system: a linearly uncoupled finite difference scheme,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 607–621, 2008.
- L. Zhang and Q. Chang, “A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 603–612, 2003.
- D. Furihata, “Finite-difference schemes for nonlinear wave equation that inherit energy conservation property,” Journal of Computational and Applied Mathematics, vol. 134, no. 1-2, pp. 37–57, 2001.
- Y. S. Wong, Q. Chang, and L. Gong, “An initial-boundary value problem of a nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 84, no. 1, pp. 77–93, 1997.
- S. Jiménez and L. Vázquez, “Analysis of four numerical schemes for a nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 35, no. 1, pp. 61–94, 1990.
- L. Zhang, “Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension,” Applied Mathematics and Computation, vol. 163, no. 1, pp. 343–355, 2005.
- L. M. Zhang and Q. S. Chang, “Convergence and stability of a conservative finite difference scheme for a class of equation system in interaction of complex Schrödinger field and real Klein-Gordon field,” Numerical Mathematics, vol. 22, no. 3, pp. 362–370, 2000.
- W. Bao and L. Yang, “Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations,” Journal of Computational Physics, vol. 225, no. 2, pp. 1863–1893, 2007.
- F. Zhang and L. Vázquez, “Two energy conserving numerical schemes for the Sine-Gordon equation,” Applied Mathematics and Computation, vol. 45, no. 1, pp. 17–30, 1991.
- G. L. Payne, D. R. Nicholson, and R. M. Downie, “Numerical solution of the Zakharov equations,” Journal of Computational Physics, vol. 50, no. 3, pp. 482–498, 1983.
- R. T. Glassey, “Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension,” Mathematics of Computation, vol. 58, no. 197, pp. 83–102, 1992.
- R. T. Glassey, “Approximate solutions to the Zakharov equations via finite differences,” Journal of Computational Physics, vol. 100, no. 2, pp. 377–383, 1992.
- Q. S. Chang and H. Jiang, “A conservative difference scheme for the Zakharov equations,” Journal of Computational Physics, vol. 113, no. 2, pp. 309–319, 1994.
- Q. S. Chang, B. L. Guo, and H. Jiang, “Finite difference method for generalized Zakharov equations,” Mathematics of Computation, vol. 64, no. 210, pp. 537–553, 1995.
- Y. L. Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, 1991.
- S. Liu, Z. Fu, S. Liu, and Z. Wang, “The periodic solutions for a class of coupled nonlinear Klein-Gordon equations,” Physics Letters A, vol. 323, no. 5-6, pp. 415–420, 2004.