A new conservative finite difference scheme is presented for an initialboundary value problem of the general RosenauRLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and secondorder convergent. Numerical examples show the efficiency of the scheme.
1. Introduction
In this paper, we consider the following initialboundary value problem of the general RosenauRLW equation:
with an initial condition
and boundary conditions
where is a integer and is a known smooth function. When , (1.1) is called as usual RosenauRLW equation. When , (1.1) is called as modified RosenauRLW (MRosenauRLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:
It is known the conservative scheme is better than the nonconservative ones. Zhang et al. [1] point out that the nonconservative scheme may easily show nonlinear blow up. In [2] Li and VuQuoc 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 [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, SineGordon equation, KleinGordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general RosenauRLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
2. A NonlinearImplicit Conservative Scheme
In this section, we propose a nonlinearimplicit conservative scheme for the initialboundary value problem (1.1)–(1.3) and give its numerical analysis.
2.1. The NonlinearImplicit Scheme and Its Conservative Law
For convenience, we introduce the following notations
where and denote the spatial and temporal mesh sizes, , , respectively,
and in the paper, denotes a general positive constant, which may have different values in different occurrences.
Since , then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:
Lemma 2.1 (see [12]).
For any two mesh functions, , one has
Furthermore, if , then
Theorem 2.2.
Suppose that , then scheme (2.3)–(2.5) is conservative in the senses:
Proof.
Multiplying (2.3) with , according to boundary condition (2.5), and then summing up for from 1 to , we have
Let
Then (2.8) is gotten from (2.10).
Computing the inner product of (2.3) with , according to boundary condition (2.5) and Lemma 2.1, we obtain
where
According to
we have . It follows from (2.12) that
Let
Then (2.9) is gotten from (2.15). This completes the proof of Theorem 2.2.
2.2. Existence and Prior Estimates of Difference Solution
To show the existence of the approximations for scheme (2.3)–(2.5), we introduce the following Brouwer fixed point theorem [13].
Lemma 2.3.
Let be a finitedimensional inner product space, be the associated norm, and be continuous. Assume, moreover, that there exist , for all , , . Then, there exists a such that and .
Let , , then have the following.
Theorem 2.4.
There exists which satisfies scheme (2.3)–(2.5).
Proof.
It follows from the original problem (1.1)–(1.3) that satisfies scheme (2.3)–(2.5). Assume there exists which satisfy scheme (2.3)–(2.5), as , now we try to prove that , satisfy scheme (2.3)–(2.5).
We define on as follows:
where . Computing the inner product of (2.17) with and considering and , we obtain
Hence, for all , there exists . It follows from Lemma 2.3 that exists which satisfies . Let , then it can be proved that is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.
Next we will give some priori estimates of difference solutions. First the following two lemmas [14] are introduced:
Lemma 2.5 (discrete Sobolev's estimate).
For any discrete function on the finite interval , there is the inequality
where are two constants independent of and step length .
Lemma 2.6 (discrete Gronwall's inequality).
Suppose that the discrete function satisfies the inequality
where and are nonnegative constants. Then
where is sufficiently small, such that .
Theorem 2.7.
Suppose that , then the following inequalities
hold.
Proof.
It is follows from (2.9) that
According to Lemma 2.5, we obtain
This completes the proof of Theorem 2.7.
Remark 2.8.
Theorem 2.7 implies that scheme (2.3)–(2.5) is unconditionally stable.
2.3. Convergence and Uniqueness of Difference Solution
First, we consider the convergence of scheme (2.3)–(2.5). We define the truncation error as follows:
then from Taylor's expansion, we obtain the following.
Theorem 2.9.
Suppose that and , then the truncation errors of scheme (2.3)–(2.5) satisfy
as ,
Theorem 2.10.
Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme (2.3)–(2.5) converges to the solution of problem (1.1)–(1.3) with order in the norm.
Proof.
Subtracting (2.3) from (2.25) letting
we obtain
Computing the inner product of (2.28) with , we obtain
From the conservative property (1.5), it can be proved by Lemma 2.5 that . Then by Theorem 2.7 we can estimate (2.29) as follows:
According to the following inequality [11]
Substituting (2.30)–(2.31) into (2.29), we obtain
Let
then (2.32) can be rewritten as
Choosing suitable which is small enough, we obtain by Lemma 2.6 that
From the discrete initial conditions, we know that is of secondorder accuracy, then
Then we have
It follows from Lemma 2.5, we have . This completes the proof of Theorem 2.10.
Theorem 2.11.
Scheme (2.3)–(2.5) is uniquely solvable.
Proof.
Assume that and both satisfy scheme (2.3)–(2.5), let , we obtain
Similarly to the proof of Theorem 2.10, we have
This completes the proof of Theorem 2.11.
Remark 2.12.
All results above in this paper are correct for initialboundary value problem of the general RosenauRLW equation with finite or infinite boundary.
3. Numerical Experiments
In order to test the correction of the numerical analysis in this paper, we consider the following initialboundary value problems of the general RosenauRLW equation:
with an initial condition
and boundary conditions
where . Then the exact solution of the initial value problem (3.1)(3.2) is
where is wave velocity.
It follows from (3.4) that the initialboundary value problem (3.1)–(3.3) is consistent to the boundary value problem (3.3) for . In the following examples, we always choose , .
Tables 1, 2, and 3 give the errors in the sense of norm and norm of the numerical solutions under various steps of and at for and . The three tables verify the secondorder convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass and discrete energy computed by scheme (2.3)–(2.5) for and .
Table 1. The errors of numerical solutions at with for .
Table 2. The errors of numerical solutions at with for .
Table 3. The errors of numerical solutions at with for .
Table 4. Discrete mass and discrete energy with at various for .
Table 5. Discrete mass and discrete energy with at various for .
Table 6. Discrete mass and discrete energy with at various for .
Figures 1, 2, and 3 plot the exact solutions at and the numerical solutions computed by scheme (2.3)–(2.5) with at , which also show the accuracy of scheme (2.3)–(2.5).
Acknowledgments
The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (nos. 10871117 and 10571110).
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