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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 517818, 15 pages
Numerical Simulation for General Rosenau-RLW Equation: An Average Linearized Conservative Scheme
1School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China
2Department 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.
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.
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.
As already pointed out by Fei et al. , 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.  have proposed a Crank-Nicolson difference scheme for the Rosenau-RLW equation. The difference scheme in  is nonlinear implicit, so it requires heavy iterative calculations and is not suitable for parallel computation. In a recent work , 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 ). For any two mesh functions: , one has Furthermore, if , then
Proof. Multiplying (2.2) with , according to the boundary conditions (2.4), then summing up for from 1 to , we obtain
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.
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 ). Assuming that are uniquely solvable, consider in (2.2) which satisfies
Taking the inner product of (3.1) with , we obtain
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
This is that.
Lemma 4.2 (discrete Sobolev’s inequality ). There exist two constants and such that
Lemma 4.3 (discrete Gronwall inequality ). 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. 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.
5. Numerical Experiments
In this section, we conduct some numerical experiments to verify our theoretical results obtained in the previous sections.
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.
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.
We make a comparison between C-N scheme  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  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.
From the numerical results, the scheme of this paper is accurate and efficient.
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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