Mathematical Problems in EngineeringVolume 2012 (2012), Article ID 517818, 15 pagesdoi:10.1155/2012/517818

Research Article

Numerical Simulation for General Rosenau-RLW Equation: An Average Linearized Conservative Scheme

Xintian Pan¹ and Luming Zhang²

¹School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China²Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China

Received 31 October 2011; Revised 6 February 2012; Accepted 9 March 2012

Academic Editor: John Burns

Copyright © 2012 Xintian Pan 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.

Abstract.

Numerical solutions for the general Rosenau-RLW equation are considered and an energy conservative linearized finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and a priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that the scheme is efficient and reliable.

1. Introduction

In this paper, we examine the use of the finite difference method for the general Rosenau-RLW equation with an initial condition and boundary conditions where is a integer and is a known smooth function. When , the equation (1.1) is called usual Rosenau-RLW equation. When , (1.1) is called modified Rosenau-RLW equation.

It can be proved easily that the problem (1.1)–(1.3) possesses the following conservative laws:

As already pointed out by Fei et al. [1], the nonconservative difference schemes may easily show nonlinear blow-up, and the conservative difference schemes perform better than the non-conservative ones. In [2–15], some conservative finite difference schemes were used for Sine-Gordon equation, Cahn-Hilliard equation, Klein-Gordon equation, a system of Schrödinger equation, Zakharov equations, Rosenau equation, GRLW equation, Klein-Gordon-Schrödinger equation, respectively. Numerical results of all the schemes are very good.

As far as computational studies are concerned, Zuo et al. [16] have proposed a Crank-Nicolson difference scheme for the Rosenau-RLW equation. The difference scheme in [16] is nonlinear implicit, so it requires heavy iterative calculations and is not suitable for parallel computation. In a recent work [14], we have made some preliminary computation by proposing a conservative linearized difference scheme for GRLW equation which is unconditionally stable and reduces the computational work, and the numerical results are encouraging. In this paper, we continue our work and propose a conservative linearized difference scheme for the general Rosenau-RLW equation which is unconditionally stable and second-order convergent and simulates conservative laws (1.4)-(1.5) at the same time.

The remainder of this paper is organized as follows. In Section 2, an energy conservative linearized difference scheme for the general Rosenau-RLW equation is described and the discrete conservative laws of the difference scheme are discussed. In Section 3, we show that the scheme is uniquely solvable. In Section 4, convergence and stability of the scheme are proved. In Section 5, numerical experiments are reported.

2. An Average Linearized Conservative Scheme and Its Discrete Conservative Law

In this section, we describe a new conservative difference scheme for the problems of (1.1)–(1.3). Let and be the uniform step size in the spatial and temporal direction, respectively. Denote , , and . Define

and in the paper, denotes a general positive constant which may have different values in different occurrences.

Notice that . We consider the following three-level average linearized conservative scheme for the IBV problems (1.1)–(1.3): where is a real constant. The scheme (2.2)–(2.4) is three level and linear implicit, so it can be easily implemented. It should be pointed out that we need another suitable two-level scheme (such as C-N scheme) to compute . For convenience, the last term of (2.2) is defined by

Lemma 2.1 (see [17]). For any two mesh functions: , one has Furthermore, if , then

Theorem 2.2. Suppose and . Then the scheme (2.2)–(2.4) admits the following invariant:

Proof. Multiplying (2.2) with , according to the boundary conditions (2.4), then summing up for from 1 to , we obtain Let Then we obtain (2.8) from (2.10).Taking the inner product of (2.2) with , according to Lemma 2.1, we have Now, computing the last term of the left-hand side in (2.12), we have Substitute (2.13) into (2.12), and we let By the definition of , (2.9) holds.

3. Solvability

In this section, we will prove the solvability of the difference scheme (2.2).

Theorem 3.1. The difference scheme (2.2) is uniquely solvable.

Proof. By the mathematical induction. It is obvious that is uniquely determined by (2.3). We can choose a second-order method to compute (such as C-N scheme [16]). Assuming that are uniquely solvable, consider in (2.2) which satisfies Taking the inner product of (3.1) with , we obtain where .Notice that It follows from (3.2) that That is, there uniquely exists trivial solution satisfying (3.1). Hence, in (2.2) is uniquely solvable. This completes the proof of Theorem 3.1.

Remark 3.2. All results above in this paper are correct for IBV problem of the general Rosenau-RLW equation with finite or infinite boundary.

4. Convergence and Stability of Finite Difference Scheme

First we will consider the truncation error of the difference scheme of (2.2)–(2.4). Denote . We define the truncation error as follows: Using Taylor expansion, we obtain that holds if .

This is that.

Lemma 4.1. Assume is smooth enough, then the local truncation error of difference scheme (2.2)–(2.4) is .

Next, we will discuss the convergence and stability of finite difference scheme (2.2)–(2.4). The following two lemmas are introduced.

Lemma 4.2 (discrete Sobolev’s inequality [18]). There exist two constants and such that

Lemma 4.3 (discrete Gronwall inequality [18]). Suppose are nonnegative mesh functions and is nondecreasing. If and then

Lemma 4.4. Suppose , then the solution of (2.2) satisfies , which yield .

Proof. It follows from (2.9) that Thus This implies for small which satisfies , we get Using Lemma 4.2, we obtain

Remark 4.5. Lemma 4.4 implies that scheme (2.2)–(2.4) is unconditionally stable.

Theorem 4.6. Assume that and . Then the solution of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is by the norm.

Proof. Subtracting (4.1) from (2.2) and letting , we have Taking the inner product in (4.9) with , we obtain where According to Lemma 4.4, the fifth term of right-hand side of (4.10) is estimated as follows: and similarly we can prove In addition, it is obvious that Substituting (4.12)–(4.15) into (4.10), we get Let , then (4.16) can be written as follows: Thus Hence, for sufficiently small, such that , we obtain Summing up (4.19) from 1 to yields Choose a second-order method to compute (such as C-N scheme) and notice that From the discrete initial conditions, we know that is of second-order accuracy, then Then we obtain An application of Lemma 4.3 yields Thus It follows from Lemma 4.2 that This completes the proof of Theorem 4.6.

Similarly, we can prove stability of the difference solution.

Theorem 4.7. Under the conditions of Theorem 4.6, the solution of the scheme (3.1)–(2.4) is unconditionally stable by the norm.

5. Numerical Experiments

In this section, we conduct some numerical experiments to verify our theoretical results obtained in the previous sections.

Consider the general Rosenau-RLW equation with an initial condition and boundary conditions The exact solution of the system (5.1)-(5.2) has the following form: where is a integer and .

It follows from (5.4) that the initial-boundary value problem (5.1)–(5.3) is consistent to the initial value problem (5.1)-(5.2) for , . In the numerical experiments, we take , and consider three cases , respectively. The errors in the sense of -norm and -norm of the numerical solutions are listed on Tables 1, 2, and 3 for three cases with . Tables 1, 2, and 3 verify the second-order convergence and good stability of the numerical solutions.

Table 1: The errors of numerical solutions at with and .

0.25

—
—

0.125

3.981465
3.909609

0.0625

4.025718
4.002124

0.03125

4.123894
4.118692

Table 2: The errors of numerical solutions at with and .

0.25

—
—

0.125

3.981732
3.870541

0.0625

4.025923
4.022161

0.03125

4.124036
4.108427

Table 3: The errors of numerical solutions at with and .

0.25

—
—

0.125

3.980476
3.906148

0.0625

4.025511
4.003231

0.03125

4.123924
4.118027

We have shown in Theorem 2.2 that the numerical solution of the scheme (2.2) satisfies the conservation of discrete mass and energy, respectively. In Tables 4, 5, and 6, the values of and( for the scheme (2.2) are presented for three cases under steps with and 1, respectively. It is easy to see from Tables 4, 5, and 6 that the scheme (2.2) preserves the discrete mass and discrete energy very well; thus it can be used to computing for a long time.

Table 4: Discrete mass and energy of scheme (2.2) for a few of values at different time with and .

2
3.7953131
1.0663504
3.7953130
1.0661275
3.7953129
1.0659045

4
3.7953035
1.0663504
3.7953024
1.0661275
3.7953024
1.0659045

6
3.7952587
1.0663504
3.7952598
1.0661275
3.7952591
1.0659045

8
3.7950800
1.0663504
3.7950814
1.0661275
3.7950829
1.0659045

10
3.7943581
1.0663504
3.7943658
1.0661275
3.7943715
1.0659045

Table 5: Discrete mass and energy of scheme (2.2) for a few of values at different time with and .

2
6.2655606
2.8676742
6.2655603
2.8670879
6.2655600
2.8665016

4
6.2653440
2.8676742
6.2653437
2.8670879
6.2653435
2.8665016

6
6.2648079
2.8676742
6.2648083
2.8670879
6.2648086
2.8665016

8
6.2635545
2.8676742
6.2635570
2.8670879
6.2635595
2.8665016

10
6.2606441
2.8676742
6.2606529
2.8670879
6.2606616
2.8665016

Table 6: Discrete mass and energy of scheme (2.2) for a few of values at different time with and .

2
9.7202265
4.7351269
9.7202105
4.7345960
9.7201954
4.7340650

4
9.7140074
4.7351270
9.7139777
4.7345960
9.7139486
4.7340650

6
9.7024346
4.7351270
9.7023976
4.7345960
9.7023610
4.7340650

8
9.6834925
4.7351270
9.6834566
4.7345960
9.6834209
4.7340650

10
9.6536097
4.7351270
9.6535850
4.7345960
9.6535602
4.7340650

We make a comparison between C-N scheme [16] and our scheme with under the meshes in Figures 1 and 2 when . It is obvious from Figures 1 and 2 that our scheme performs better than C-N scheme [16] in the numerical precision when and 1. Figures 1 and 2 also show that numerical precision of the scheme (2.2) depends on the choice of parameter . The curves of the solitary waves with time computed by the scheme (2.2) with for and for under mesh sizes of are given in Figures 3 and 4, respectively; the waves at agree with the ones at quite well, which also demonstrate the accuracy of the scheme in present paper.

Figure 1: Errors in the sense of computed by the scheme (2.2) when and .

Figure 2: Errors in the sense of computed by the scheme (2.2) when and .

Figure 3: Exact solutions of at and numerical solutions computed by the scheme (2.2) at with for .

Figure 4: Exact solutions of at and numerical solutions computed by the scheme (2.2) at with for .

From the numerical results, the scheme of this paper is accurate and efficient.

Acknowledgments

This work is supported by the Youth Research Foundation of WFU (no. 2011Z17). The authors would like to thank the editor and the reviewers for their valuable comments and suggestions.

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