Advances in Mathematical Physics

Volume 2014 (2014), Article ID 217393, 11 pages

http://dx.doi.org/10.1155/2014/217393

Research Article

Two Conservative Difference Schemes for Rosenau-Kawahara Equation

Jinsong Hu,1,2 Youcai Xu,¹ Bing Hu,¹ and Xiaoping Xie¹

¹School of Mathematics, Sichuan University, Chengdu 610064, China

²School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China

Received 14 September 2013; Revised 22 December 2013; Accepted 7 January 2014; Published 18 March 2014

Academic Editor: Ricardo Weder

Copyright © 2014 Jinsong Hu et al. 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.

Abstract.

Two conservative finite difference schemes for the numerical solution of the initialboundary value problem of Rosenau-Kawahara equation are proposed. The difference schemes simulate two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

1. Introduction

In the study of compact discrete systems, the wave-wave and wave-wall interactions cannot be described by the well-known KdV equation. To overcome this shortcoming of KdV equation, Rosenau proposed the following so-called Rosenau equation in [1, 2]:

which is usually used to describe the dense discrete system and simulate the long-chain transmission model through an L-C flow in radio and computer fields. Park proved the existence and uniqueness of solution to (1) in [3]. However, it is difficult to find its analytical solution. Therefore, the numerical study of (1) is very significant and attract many scholars (see, e.g., [4–10]).

As the further consideration of nonlinear wave, Zuo obtained Rosenau-Kawahara equation by adding viscous term and to Rosenau equation (1) and studied the solitary solution and periodic solution of this equation in [10]. In [11], Labidi and Biswas got the integral of Rosenau-Kawahara equation by using He’s principle. Then the solitary solution and two invariance of a generalized Rosenau-Kawahara equation are investigated in [12]. To our best knowledge, there is no study on the numerical method of Rosenau-Kawahara equation. Therefore, we will study the difference approximate solution of Rosenau-Kawahara equation with the initial data and boundary conditions as follows:

where is a known constant and is a smooth function. When , the solitary wave solution of (2) is (see [10])

The initial boundary value problem (2)–(4) is in accordance with the Cauchy problem of (2) when , . Hence, the boundary condition (4) is reasonable. It is easy to verify that (2)-(4) satisfy the following conservative quantities (see [12]):

where , are both constants only depending on initial data.

It is well known that a reasonable difference scheme has not only high-accuracy but also can maintain some physical properties of original problem. Lots of numerical experiments show that conservative difference scheme can simulate the conservative law of initial problem well since it could avoid the nonlinear blow-up [13–24]. Moreover, it is suitable to compute for long-time. Li and Vu-Quoc pointed in [14] that 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. Therefore, constructing the conservative difference scheme is a significant job.

The rest of this paper is organized as follows. We respectively propose a two-level nonlinear Crank-Nicolson difference scheme and three-level linear difference scheme for initial boundary value problems (2)–(4) in Sections 2 and 3. We analyze its two discrete conservative laws. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference schemes are of second-order convergence and unconditionally stable. In Section 4, we verify our theoretical analysis by numerical experiments.

2. Nonlinear Crank-Nicolson Conservative Difference Scheme

In this section, we propose a two-level nonlinear Crank-Nicolson difference scheme and give the theoretical analysis.

In the rest of this paper, denotes a general positive constant which may denote different value in different occurrence.

2.1. Nonlinear Difference Scheme and Its Conservative Law

Let and be the uniform step size in the spatial and temporal direction, respectively. The interval is divided into equal parts, where is a fixed positive integer. Denote , , . Let be the difference approximation of at , that is, . Denote . We define the difference operators, inner product, and norms that will be used in this paper as follows:

Consider the following finite difference scheme for problems (2)–(4):

The discrete boundary condition (11) is reasonable from the boundary condition (4).

Lemma 1. For any two discrete functions , we have

from summation by parts (see [22]). Thus,

And if , then

The following theorem shows how the difference schemes (9)–(11) simulate the conservative law numerically.

Theorem 2. Suppose that ; then the difference schemes (9)–(11) are conservative for discrete energy; that is,

Proof. Multiplying in the two sides of (9) and summing up for from to , from boundary (11) and Lemma 1, we obtain

From the definition of , (15) is deduced from (17).

Taking the inner of (9) with (i.e., ), from boundary (11) and Lemma 1, we get

where . On the other hand,

Substituting (19)–(22) into (18), we get

From the definition of , we get (16) by deducing (23).

2.2. Solvability of the Difference Scheme

In order to prove the solvability of difference scheme, we present the following Brouwer fixed point theorem [25].

Lemma 3 (Brouwer fixed point theorem). Let be a finite dimensional inner product space; suppose that is continuous and there exists an such that for all with . Then there exists such that and .

Theorem 4. There exists that satisfies difference schemes (9)–(11).

Proof. We use the mathematical induction to prove our theorem.

Suppose that there exist that satisfy difference scheme (9) for . Next we prove that there exists that satisfies difference scheme.

Let be an operator on defined by

Taking the inner product of (24) with , and noticing that

we have

Therefore, for any , if , then . From Lemma 3, there exists such that . Choose . It is easy to verify that satisfies difference scheme (9).

2.3. Convergence, Stability, and Uniqueness of Solution

Suppose that is the solution to (2)–(4). Denote . Then the truncation error of difference schemes (9)–(11) is

By Taylor expansion, we know that as .

Lemma 5. Suppose that . Then the solution of problems (2)–(4) satisfies

Proof. It follows from (7) that

Using Hölder inequality and Cauchy-Schwarz inequality, we can get

Thus,

which also yields , from Sobolev inequality.

Lemma 6 (discrete Sobolev inequality [22]). There exist constants and such that

Lemma 7 (discrete Gronwall inequality [22]). Assume that , are nonnegative gridding functions, and is increasing. If , and for any

then for any

Theorem 8. Suppose that . Then the solution of difference schemes (9)–(11) satisfies , , thus , .

Proof. From (16), we have

From (13) and by Cauchy-Schwarz inequality

Thus,

Therefore, from Lemma 6, we obtain , .

Remark 9. Theorem 8 implies that the solution of difference schemes (9)–(11) is unconditionally stable in the sense of norm .

Theorem 10. Suppose that . Then the solution of difference schemes (9)–(11) converges to the solution of (2)–(4) in the sense of norm , and the convergent rate is .

Proof. Subtracting (9)–(11) from (27)–(29) and denoting , we have

Taking the inner product of (40) with , we get

Similar to (19)–(21), we have

So, (43) can be rewritten as

As

from Lemma 5 and Theorem 8, we obtain

Using Cauchy-Schwarz inequality, we have

On the other hand,

Substituting (48) and (49) into (45), we get

Similar to the deduction process of (38), one can get

Therefore, (50) is changed into

Denoting , (52) is equivalent to

that is,

Choosing sufficiently small such that , we get

Summing up (55) from to , we get

Noticing that

and . we have

From Lemma 7, we get

that is,

Noticing equality (51), one can obtain

From Lemma 6, we have

Similar to Theorem 10, we have the following theorems.

Theorem 11. The solution to difference schemes (9)–(11) is unique.

3. A Linear Conservative Difference Scheme

In this section, we propose a three-level linear conservative difference scheme for (2)–(4) and give the theoretical analysis.

3.1. Linear Difference Scheme and Its Conservative Law

Consider the following finite difference scheme for problems (2)–(4):

The discrete boundary condition (65) is reasonable from the boundary condition (4).

The following theorem shows how the difference schemes (63)–(65) simulate the conservative law as follows.

Theorem 12. Suppose that ; then the difference schemes (63)–(65) are conservative for discrete energy; that is,

Proof. Multiplying in both sides of (63) and summing up for , from boundary (65) and Lemma 1, we obtain

Notice that

From the definition of , (66) is deduced from (68) and (69).

Taking the inner product of (63) with , from boundary (65) and Lemma 1, we obtain

where . On the other hand,

Substituting (71)–(74) into (70), we obtain

From the definition of , we get (67) by deducing (75).

3.2. Solvability of the Difference Scheme

Theorem 13. There exists unique solution for difference schemes (63)–(65).

Proof. Use the mathematical induction to prove it. It is obvious that is uniquely determined by the initial condition (64). We also can get by (9)–(11). Now suppose is solved uniquely. Consider the equation of (63) for

Taking the inner product of (76) with , from boundary condition (65) and Lemma 1, we get

Similar to (71)–(73), we have

Notice that

Substituting (78)–(81) into (77), we get

that is, (76) only admits zero solution. Therefore, there exists a unique that satisfies (63).

3.3. Convergence and Stability of the Difference Scheme

Suppose that is the solution to (2)–(4). Denote . Then the truncation error of difference schemes (63)–(65) is as follows:

By Taylor expansion, we know that as .

Theorem 14. Suppose that . Then the solution of difference schemes (63)–(65) satisfies , , ; thus , .

Proof. From (67), we know that

By (38), one can get

Then, from Lemma 6, we get , .

Remark 15. Theorem 14 implies that the solution of difference schemes (63)–(65) is unconditionally stable in the sense of norm .

Theorem 16. Suppose that . Then the solution of difference schemes (63)–(65) converges to the solution of (2)–(4) in the sense of norm , and the convergent order is .

Proof. Subtracting (63)–(65) from (83)–(85) and letting , we have

Taking the inner product of the two sides of (88) with , and using boundary condition (90) and Lemma 1, we obtain

Similar to (71)–(73), we have

Then (91) is changed into

By Lemma 5 and Theorem 14, it is shown that

Using Cauchy-Schwarz inequality, we have

On the other hand,

Substituting (95)-(96) into (93), we could obtain

Using the deduce process which is similar to (38), we have

Then (97) is changed into

Let , (99) can be rewritten as follows:

that is,

By choosing that is small enough such that , then

Summing up (102) from to , we obtain

Notice that

Firstly by selecting other two-level scheme with second-order convergence (such as the scheme (9)–(11)) to get , we have

Therefore

Using Lemma 7,we obtain

that is,

Combining with (98), we get

Then by Lemma 6, it is shown that

4. Numerical Simulations

In this section, let , , , , and

We denote the nonlinear Crank-Nicolson conservative difference schemes (9)–(11) as Scheme I and the linear three-level conservative difference schemes (63)–(65) as Scheme II. For some different values of and , we list errors of Scheme I and Scheme II at several different time in Tables 1 and 2, respectively. We verify the two-order accuracy of the difference scheme in Tables 3 and 4 by using the method of [26, 27]. The numerical simulation of two conservative quantities (6) and (7) is listed in Tables 5 and 6.

Table 1: The errors of Scheme I at various time steps with various and .

Table 2: The errors of Scheme II at various time steps with various and .

Table 3: The verification of the second convergence of Scheme I.

—
3.998318
3.988313
—
3.998969
3.999234

—
3.998548
3.996548
—
3.998630
3.999538

—
3.998359
3.998072
—
3.998171
3.999489

—
3.998109
3.998457
—
3.997769
3.999324

Table 4: The verification of the second convergence of Scheme II.

—
3.995835
3.997182
—
3.995455
3.993009

—
3.995103
3.999826
—
3.994473
3.994248

—
3.994599
4.001208
—
3.994140
3.995232

—
3.994217
4.002444
—
3.993690
3.996400

Table 5: When , numerical simulations on the conservation invariant .

Scheme I
Scheme II
Scheme I
Scheme II
Scheme I
Scheme II

4.120893241330
4.120918187462
4.120893214999
4.120899452450
4.120893201770
4.120894761190

4.120892312503
4.120918835632
4.120892200166
4.120898816174
4.120892129735
4.120894227434

4.120893031520
4.120921893266
4.120892362862
4.120899447408
4.120892167548
4.120894662524

4.120893107802
4.120925120669
4.120892370454
4.120900136940
4.120892149140
4.120895106600

4.120892997828
4.120924996792
4.120892342248
4.120900073781
4.120892155018
4.120895399389

Table 6: When , numerical simulations on the conservation invariant .

Scheme I
Scheme II
Scheme I
Scheme II
Scheme I
Scheme II

0.836201094485
0.836201094485
0.836201181724
0.836201181724
0.836201203535
0.836201203535

0.836201095943
0.836201096547
0.836201194492
0.836201202559
0.836201310312
0.836201469358

0.836201095943
0.836201096662
0.836201194483
0.836201206568
0.836201308626
0.836201594118

0.836201095952
0.836201096780
0.836201194484
0.836201210387
0.836201309421
0.836201718674

0.836201095955
0.836201096906
0.836201194369
0.836201214174
0.836201309604
0.836201843455

Finally, a numerical simulation figure comparison of at various time steps with is as follows. (see Figures 1 and 2).

Figure 1: When and , a numerical simulation figure on of Scheme I at various times.

Figure 2: When and , a numerical simulation figure on of Scheme II at various times.

Numerical simulations show that the finite difference Schemes I and II in this paper are efficient. The calculation results of Scheme I are slightly better than Scheme II. But iterative numerical calculation is not required, Scheme II can save computing time.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11171239), Open Fund of Key Laboratory of Mountain Hazards and Earth Surface Processes, CAS and Scientific Research Found of Sichuan Provincial Education Department (no. 11ZB009).

References

P. Rosenau, “A quasi-continuous description of a nonlinear transmission line,” Physica Scripta, vol. 34, pp. 827–829, 1986.

P. Rosenau, “Dynamics of dense discrete systems,” Progress of Theoretical Physics, vol. 79, pp. 1028–1042, 1988.

M. A. Park, “On the Rosenau equation,” Matemática Aplicada e Computacional, vol. 9, no. 2, pp. 145–152, 1990.

S. K. Chung and S. N. Ha, “Finite element Galerkin solutions for the Rosenau equation,” Applicable Analysis, vol. 54, no. 1-2, pp. 39–56, 1994.

K. Omrani, F. Abidi, T. Achouri, and N. Khiari, “A new conservative finite difference scheme for the Rosenau equation,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 35–43, 2008.

S. K. Chung, “Finite difference approximate solutions for the Rosenau equation,” Applicable Analysis, vol. 69, no. 1-2, pp. 149–156, 1998.

S. K. Chung and A. K. Pani, “Numerical methods for the Rosenau equation,” Applicable Analysis, vol. 77, no. 3-4, pp. 351–369, 2001.

S. A. V. Manickam, A. K. Pani, and S. K. Chung, “A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation,” Numerical Methods for Partial Differential Equations, vol. 14, no. 6, pp. 695–716, 1998.

Y. D. Kim and H. Y. Lee, “The convergence of finite element Galerkin solution for the Roseneau equation,” The Korean Journal of Computational & Applied Mathematics, vol. 5, no. 1, pp. 171–180, 1998.

J.-M. Zuo, “Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 835–840, 2009.

M. Labidi and A. Biswas, “Application of He’s principles to Rosenau-Kawahara equation,” Mathematics in Engineering, Science and Aerospace MESA, vol. 2, no. 2, pp. 183–197, 2011.

A. Biswas, H. Triki, and M. Labidi, “Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity,” Physics of Wave Phenomena, vol. 19, no. 1, pp. 24–29, 2011.

T. Wang, L. Zhang, and F. Chen, “Conservative schemes for the symmetric regularized long wave equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1063–1080, 2007.

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. 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.-C. Wang and L.-M. 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, B. Guo, and L. Zhang, “New conservative difference schemes for a coupled nonlinear Schrödinger system,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1604–1619, 2010.

L. Zhang, “A finite difference scheme for generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962–972, 2005.

Z. Fei 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.

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.

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.

J. Hu and K. Zheng, “Two conservative difference schemes for the generalized Rosenau equation,” Boundary Value Problems, vol. 2010, Article ID 543503, 18 pages, 2010.

X. Pan and L. Zhang, “On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3371–3378, 2012.

X. Pan and L. Zhang, “Numerical simulation for general Rosenau-RLW equation: an average linearized conservative scheme,” Mathematical Problems in Engineering, vol. 2012, Article ID 517818, 15 pages, 2012.

F. E. Browder, “Existence and uniqueness theorems for solutions of nonlinear boundary value problems,” in Proceedings of Symposia in Applied Mathematics, vol. 17, pp. 24–49, 1965.

T. Wang and B. Guo, “A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 878–888, 2009.

J. Hu, B. Hu, and Y. Xu, “C-N difference schemes for dissipative symmetric regularized long wave equations with damping term,” Mathematical Problems in Engineering, vol. 2011, Article ID 651642, 16 pages, 2011.