Advances in Mathematical Physics

Volume 2013 (2013), Article ID 423718, 7 pages

http://dx.doi.org/10.1155/2013/423718

Research Article

Conservative Linear Difference Scheme for Rosenau-KdV Equation

Jinsong Hu,¹ Youcai Xu,² and Bing Hu²

¹School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China

²School of Mathematics, Sichuan University, Chengdu 610064, China

Received 5 February 2013; Accepted 22 March 2013

Academic Editor: Hagen Neidhardt

Copyright © 2013 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.

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

1. Introduction

KdV equation has been used in very wide applications and undergone research which can be used to describe wave propagation and spread interaction as follows [1–4]:

In the study of the dynamics of dense discrete systems, the case of wave-wave and wave-wall interactions cannot be described using the well-known KdV equation. To overcome this shortcoming of the KdV equation, Rosenau [5, 6] proposed the so-called Rosenau equation:

The existence and the uniqueness of the solution for (2) were proved by Park [7], but it is difficult to find the analytical solution for (2). Since then, much work has been done on the numerical method for (2) ([8–13] and also the references therein). On the other hand, for the further consideration of the nonlinear wave, the viscous term needs to be included [14]

This equation is usually called the Rosenau-KdV equation. Zuo [14] discussed the solitary wave solutions and periodic solutions for (2). Recently, [15–17] discussed the solitary solutions for the generalized Rosenau-KdV equation with usual power law nonlinearity. In [15, 16], the authors also gave the two invariants for the generalized Rosenau-KdV equation. In particular, [16] not only derived the singular 1-solition solution by the ansatz method but also used perturbation theory to obtain the adiabatic parameter dynamics of the water waves. In [17], The ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation. The / method as well as the exp-function method are also applied to extract a few more solutions to this equation. But the numerical method to the initial-boundary value problem of Rosenau-KdV equation has not been studied till now. In this paper, we propose a conservative three-level finite difference scheme for the Rosenau-KdV equation (3) with the boundary conditions

and an initial condition

The initial boundary value problem (3)–(5) possesses the following conservative properties [15]:

The solitary wave solution for (3) is [14, 15]

When , the initial-boundary value problem (3)–(5) and the Cauchy problem (3) are consistent, so the boundary condition (4) is reasonable.

It is known the conservative scheme is better than the nonconservative ones. The nonconservative scheme may easily show nonlinear blow up. A lot of numerical experiments show that the conservative scheme can possesses some invariant properties of the original differential equation [18–29]. The conservative scheme is more suitable for long-time calculations. In [19], Li and Vu-Quoc 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.” In this paper, we propose a three-level linear finite difference scheme for the Rosenau-KdV equation (3)–(5). The difference scheme is conservative which simulates conservative properties (6) and (7) at the same time.

The rest of this paper is organized as follows. In Section 2, we propose a three-level linear finite difference scheme for the Rosenau-KdV equation and discuss the discrete conservative properties. In Section 3, we show that the scheme is uniquely solvable. Then, in Section 4, we prove that the finite difference scheme is of second-order convergence, unconditionally stable. Finally, some numerical tests are given in Section 5 to verify our theoretical analysis.

2. Finite Difference Scheme and Conservation Properties

Let and be the uniform step size in the spatial and temporal direction, respectively. Denote , , , and , . Throughout this paper, we denote as a generic positive constant independent of and , which may have different values in different occurrences. We introduce the following notations:

We propose a three-level linear finite difference scheme for the solution of (3)–(5) as follows:

From the boundary conditions (4), we know that (13) is reasonable.

Lemma 1. It follows from summation by parts that for any two mesh functions ,

Then one has

Furthermore, if , then

The difference scheme (10)–(13) simulates two conservative properties of the problems (6) and (7) as follows.

Theorem 2. Suppose that , then the difference scheme (10)–(13) is conservative:

Proof. Multiplying (10) with , summing up for from to , and considering the boundary condition (13) and Lemma 1, we get

Then, (17) is gotten from (19).

Taking an inner product of (10) with (i.e., ), considering the boundary condition (13) and Lemma 1, we obtain

where . According to

we have

Then, (18) is gotten from (23).

3. Solvability

Theorem 3. There exists which satisfies the difference scheme (10)–(13), .

Proof. Use mathematical induction to prove it. It is obvious that is uniquely determined by the initial condition (12). We also can get in order by two-level scheme (i.e., and are uniquely determined). Now suppose is solved uniquely. Consider the equation of (10) for

Taking an inner product of (24) with , we get

Similar to the proof of (21), we have

By

and from (25)–(27), we have

That is, (24) has only a trivial solution. Therefore, (10) determines uniquely. This completes the proof.

4. Convergence and Stability

Let be the solution of problem (3)–(5), , then the truncation error of the difference scheme (10)–(13) is as follows:

Making use of Taylor expansion, we know that holds if .

Lemma 4. Suppose that , then the solution of (3)–(5) satisfies

Proof. It is follows from (7) that

By Holder inequality and Schwarz inequality, we get

which implies that

Using Sobolev inequality, we get that .

Lemma 5 (discrete Sobolev's inequality [27]). There exist two constants and such that

Lemma 6 (discrete Gronwall inequality [27]). Suppose that  and   are nonnegative function and is nondecreasing. If , and

then

Theorem 7. Suppose , then the solution of (10)–(13) satisfies: , which yield .

Proof. It is follows from (18) that

According to (15) and Schwarz inequality, we get

Using Lemma 5, we have .

Theorem 8. Suppose , , then the solution of the difference scheme (10)–(13) converges to the solution of the problem (3)–(5) with order in norm .

Proof. Subtracting (10) from (29) and letting , we have

where . Computing the inner product of (39) with , we obtain

Similar to the proof of (21), we have

Then, (40) can be rewritten as follows:

Using Lemma 4 and Theorem 7, we get

According to the Schwarz inequality, we obtain

Noting that

and from (42)–(45), we have

Similar to the proof of (38), we have

Then, (46) can be rewritten as

Let . Then, (48) can be rewritten as follows:

which yields

If is sufficiently small, which satisfies , then

Summing up (51) from to , we have

First, we can get in order that satisfies by two-level scheme. Since

then we obtain

From Lemma 6 we get

which implies that

From (47) we have

By Lemma 5 we obtain

Finally, we can similarly prove results as follows.

Theorem 9. Under the conditions of Theorem 8, the solution of (10)–(13) is stable in norm .

5. Numerical Simulations

Since the three-level implicit finite difference scheme cannot start by itself, we need to select other two-level schemes (such as the Scheme) to get . Then, be reusing initial value , we can work out . Iterative numerical calculation is not required, for this scheme is linear, so it saves computing time. Let ,

In Table 1, we give the error at various time steps. Using the method in [30, 31], we verified the second convergence of the difference scheme in Table 2. Numerical simulations on two conservation invariants and are given in Table 3.

Table 1: The error at various time steps.

Table 2: The verification of the second convergence.

—
3.991564
4.000797
—
3.989657
3.990655

—
3.990757
4.004916
—
3.989749
3.995343

—
3.990539
4.009713
—
3.990136
3.999764

—
3.990427
4.014970
—
3.990030
4.004314

Table 3: Numerical simulations on the two conservation invariants and .

5.497722548019
1.984553365290
5.498060684522
1.984390175264
5.498145418391
1.984349335263

5.497724936513
1.984595075859
5.498060837192
1.984401029470
5.498145479109
1.984352109750

5.497728744900
1.984645964099
5.498061080542
1.984414367496
5.498145545374
1.984355520610

5.497731963790
1.984679827211
5.498061287046
1.984423270337
5.498145609535
1.984357811266

5.497734235191
1.984701501262
5.498061398506
1.984428974030
5.498145659050
1.984359292230

The wave graph comparison of between and at various times is given in Figures 1 and 2.

Figure 1: When , the wave graph of at various times.

Figure 2: When , the wave graph of at various times.

Numerical simulations show that the finite difference scheme is efficient.

Acknowledgments

The work was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB009) and the fund of Key Disciplinary of Computer Software and Theory, Sichuan, Grant no. SZD0802-09-1.

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