Boundary Value Problems volume 2010, Article number: 516260 (2010)
A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW 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 second-order convergent. Numerical examples show the efficiency of the scheme.
In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW 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 Rosenau-RLW equation. When 
, (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) 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 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 [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, Sine-Gordon equation, Klein-Gordon 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 Rosenau-RLW 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.
In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1.1)–(1.3) and give its numerical analysis.
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.
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 finite-dimensional 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.
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 second-order 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 initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary.
In order to test the correction of the numerical analysis in this paper, we consider the following initial-boundary value problems of the general Rosenau-RLW 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 initial-boundary 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 second-order 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 
.
with 
for 
.
with 
for 
.
with 
for 
.
and discrete energy 
with 
at various 
for 
.
and discrete energy 
with 
at various 
for 
.
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).
Exact solutions of 
at 
and numerical solutions computed by scheme (2. 3)–(2.5) at 
for 
.
Exact solutions of 
at 
and numerical solutions computed by scheme (2. 3)–(2.5) at 
for 
.
Exact solutions of 
at 
and numerical solutions computed by scheme (2. 3)–(2.5) at 
for 
.
Zhang F, Pérez-García VM, Vázquez L: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Applied Mathematics and Computation 1995,71(2-3):165-177. 10.1016/0096-3003(94)00152-T
Li S, Vu-Quoc L: Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM Journal on Numerical Analysis 1995,32(6):1839-1875. 10.1137/0732083
Chang Q, Xu L: A numerical method for a system of generalized nonlinear Schrödinger equations. Journal of Computational Mathematics 1986,4(3):191-199.
Chang Q, Jia E, Sun W: Difference schemes for solving the generalized nonlinear Schrödinger equation. Journal of Computational Physics 1999,148(2):397-415. 10.1006/jcph.1998.6120
Wang T-C, Zhang L-M: Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator. Applied Mathematics and Computation 2006,182(2):1780-1794. 10.1016/j.amc.2006.06.015
Wang T, Guo B, Zhang L: New conservative difference schemes for a coupled nonlinear Schrödinger system. Applied Mathematics and Computation 2010,217(4):1604-1619. 10.1016/j.amc.2009.07.040
Zhang L: A finite difference scheme for generalized regularized long-wave equation. Applied Mathematics and Computation 2005,168(2):962-972. 10.1016/j.amc.2004.09.027
Zhang F, Vázquez L: Two energy conserving numerical schemes for the sine-Gordon equation. Applied Mathematics and Computation 1991,45(1):17-30. 10.1016/0096-3003(91)90087-4
Wong YS, Chang Q, Gong L: An initial-boundary value problem of a nonlinear Klein-Gordon equation. Applied Mathematics and Computation 1997,84(1):77-93. 10.1016/S0096-3003(96)00065-3
Chang QS, Guo BL, Jiang H: Finite difference method for generalized Zakharov equations. Mathematics of Computation 1995,64(210):537-553. 10.1090/S0025-5718-1995-1284664-5
Hu JS, Zheng KL: Two conservative difference schemes for the generalized Rosenau equation. Boundary Value Problems 2010, 2010:-18.
Hu B, Xu Y, Hu J: Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation. Applied Mathematics and Computation 2008,204(1):311-316. 10.1016/j.amc.2008.06.051
Browder FE: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In Proceedings of Symposia in Applied Mathematics, vol. 17. American Mathematical Society, Providence, RI, USA; 1965:24-49.
Zhou Y: Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, Beijing, China; 1991:vi+260.
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zuo, JM., Zhang, YM., Zhang, TD. et al. A New Conservative Difference Scheme for the General Rosenau-RLW Equation. Bound Value Probl 2010, 516260 (2010). https://doi.org/10.1155/2010/516260
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DOI: https://doi.org/10.1155/2010/516260