Open Access

A New Conservative Difference Scheme for the General Rosenau-RLW Equation

Boundary Value Problems20102010:516260

DOI: 10.1155/2010/516260

Received: 28 May 2010

Accepted: 14 October 2010

Published: 20 October 2010

Abstract

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.

1. Introduction

In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ1_HTML.gif
(11)
with an initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ2_HTML.gif
(12)
and boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ3_HTML.gif
(13)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq1_HTML.gif is a integer and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq2_HTML.gif is a known smooth function. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq3_HTML.gif , (1.1) is called as usual Rosenau-RLW equation. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq4_HTML.gif , (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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ4_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ5_HTML.gif
(15)

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 " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq5_HTML.gif 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 [311], 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.

2. A Nonlinear-Implicit Conservative Scheme

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.

2.1. The Nonlinear-Implicit Scheme and Its Conservative Law

For convenience, we introduce the following notations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ6_HTML.gif
(21)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq7_HTML.gif denote the spatial and temporal mesh sizes, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq9_HTML.gif , respectively,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ7_HTML.gif
(22)

and in the paper, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq10_HTML.gif denotes a general positive constant, which may have different values in different occurrences.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq11_HTML.gif , then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ8_HTML.gif
(23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ9_HTML.gif
(24)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ10_HTML.gif
(25)

Lemma 2.1 (see [12]).

For any two mesh functions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq12_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ11_HTML.gif
(26)
Furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq13_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ12_HTML.gif
(27)

Theorem 2.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq14_HTML.gif , then scheme (2.3)–(2.5) is conservative in the senses:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ13_HTML.gif
(28)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ14_HTML.gif
(29)

Proof.

Multiplying (2.3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq15_HTML.gif , according to boundary condition (2.5), and then summing up for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq16_HTML.gif from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq17_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ15_HTML.gif
(210)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ16_HTML.gif
(211)

Then (2.8) is gotten from (2.10).

Computing the inner product of (2.3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq18_HTML.gif , according to boundary condition (2.5) and Lemma 2.1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ17_HTML.gif
(212)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ18_HTML.gif
(213)
According to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ19_HTML.gif
(214)
we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq19_HTML.gif . It follows from (2.12) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ20_HTML.gif
(215)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ21_HTML.gif
(216)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq20_HTML.gif for scheme (2.3)–(2.5), we introduce the following Brouwer fixed point theorem [13].

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq21_HTML.gif be a finite-dimensional inner product space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq22_HTML.gif be the associated norm, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq23_HTML.gif be continuous. Assume, moreover, that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq24_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq27_HTML.gif . Then, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq28_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq30_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq32_HTML.gif , then have the following.

Theorem 2.4.

There exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq33_HTML.gif which satisfies scheme (2.3)–(2.5).

Proof.

It follows from the original problem (1.1)–(1.3) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq34_HTML.gif satisfies scheme (2.3)–(2.5). Assume there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq35_HTML.gif which satisfy scheme (2.3)–(2.5), as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq36_HTML.gif , now we try to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq37_HTML.gif , satisfy scheme (2.3)–(2.5).

We define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq38_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq39_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ22_HTML.gif
(217)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq40_HTML.gif . Computing the inner product of (2.17) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq41_HTML.gif and considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq43_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ23_HTML.gif
(218)

Hence, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq45_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq46_HTML.gif . It follows from Lemma 2.3 that exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq47_HTML.gif which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq48_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq49_HTML.gif , then it can be proved that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq50_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq51_HTML.gif on the finite interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq52_HTML.gif , there is the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ24_HTML.gif
(219)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq53_HTML.gif are two constants independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq54_HTML.gif and step length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq55_HTML.gif .

Lemma 2.6 (discrete Gronwall's inequality).

Suppose that the discrete function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq56_HTML.gif satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ25_HTML.gif
(220)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq58_HTML.gif are nonnegative constants. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ26_HTML.gif
(221)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq59_HTML.gif is sufficiently small, such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq60_HTML.gif .

Theorem 2.7.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq61_HTML.gif , then the following inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ27_HTML.gif
(222)

hold.

Proof.

It is follows from (2.9) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ28_HTML.gif
(223)
According to Lemma 2.5, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ29_HTML.gif
(224)

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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ30_HTML.gif
(225)

then from Taylor's expansion, we obtain the following.

Theorem 2.9.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq63_HTML.gif , then the truncation errors of scheme (2.3)–(2.5) satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ31_HTML.gif
(226)

as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq65_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq66_HTML.gif in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq67_HTML.gif norm.

Proof.

Subtracting (2.3) from (2.25) letting
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ32_HTML.gif
(227)
we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ33_HTML.gif
(228)
Computing the inner product of (2.28) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq68_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ34_HTML.gif
(229)
From the conservative property (1.5), it can be proved by Lemma 2.5 that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq69_HTML.gif . Then by Theorem 2.7 we can estimate (2.29) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ35_HTML.gif
(230)
According to the following inequality [11]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ36_HTML.gif
(231)
Substituting (2.30)–(2.31) into (2.29), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ37_HTML.gif
(232)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ38_HTML.gif
(233)
then (2.32) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ39_HTML.gif
(234)
Choosing suitable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq70_HTML.gif which is small enough, we obtain by Lemma 2.6 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ40_HTML.gif
(235)
From the discrete initial conditions, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq71_HTML.gif is of second-order accuracy, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ41_HTML.gif
(236)
Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ42_HTML.gif
(237)

It follows from Lemma 2.5, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq72_HTML.gif . This completes the proof of Theorem 2.10.

Theorem 2.11.

Scheme (2.3)–(2.5) is uniquely solvable.

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq74_HTML.gif both satisfy scheme (2.3)–(2.5), let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq75_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ43_HTML.gif
(238)
Similarly to the proof of Theorem 2.10, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ44_HTML.gif
(239)

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.

3. Numerical Experiments

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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ45_HTML.gif
(31)
with an initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ46_HTML.gif
(32)
and boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ47_HTML.gif
(33)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq76_HTML.gif . Then the exact solution of the initial value problem (3.1)-(3.2) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ48_HTML.gif
(34)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq77_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq78_HTML.gif . In the following examples, we always choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq80_HTML.gif .

Tables 1, 2, and 3 give the errors in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq81_HTML.gif -norm and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq82_HTML.gif -norm of the numerical solutions under various steps of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq84_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq85_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq87_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq88_HTML.gif and discrete energy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq89_HTML.gif computed by scheme (2.3)–(2.5) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq91_HTML.gif .
Table 1

The errors of numerical solutions at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq92_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq93_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq94_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq95_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq96_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq97_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq98_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq99_HTML.gif

0.4

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq100_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq101_HTML.gif

  

0.2

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq102_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq103_HTML.gif

3.953 296

3.930 200

0.1

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq104_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq105_HTML.gif

3.987 130

3.980 050

0.05

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq106_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq107_HTML.gif

3.997 412

3.994 759

0.025

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq108_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq109_HTML.gif

4.221 051

4.151 348

Table 2

The errors of numerical solutions at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq110_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq111_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq112_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq113_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq114_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq115_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq116_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq117_HTML.gif

0.4

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq118_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq119_HTML.gif

  

0.2

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq120_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq121_HTML.gif

3.961 294

3.934 592

0.1

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq122_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq123_HTML.gif

3.996 350

3.988 283

0.05

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq124_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq125_HTML.gif

4.003 273

4.000 165

0.025

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq126_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq127_HTML.gif

4.290 286

4.242 688

Table 3

The errors of numerical solutions at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq128_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq130_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq131_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq132_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq133_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq134_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq135_HTML.gif

0.4

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq136_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq137_HTML.gif

  

0.2

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq138_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq139_HTML.gif

3.885 945

3.851 797

0.1

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq140_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq141_HTML.gif

3.975 084

3.965 980

0.05

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq143_HTML.gif

4.000 294

3.995 992

0.025

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq144_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq145_HTML.gif

4.391 818

4.381 199

Table 4

Discrete mass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq146_HTML.gif and discrete energy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq147_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq148_HTML.gif at various https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq149_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq150_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq151_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq152_HTML.gif

10

1.897 658 262 960 01

0.533 175 231 580 85

20

1.897 658 268 873 21

0.533 175 231 872 51

30

1.897 658 262 993 93

0.533 175 231 177 25

40

1.897 658 265 568 93

0.533 175 231 478 09

50

1.897 658 260 975 87

0.533 175 231 776 18

60

1.897 658 265 384 88

0.533 175 231 074 05

Table 5

Discrete mass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq153_HTML.gif and discrete energy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq155_HTML.gif at various https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq156_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq157_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq158_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq159_HTML.gif

10

2.672 608 675 265 30

1.113 462 678 852 70

20

2.672 608 676 236 58

1.113 462 678 465 22

30

2.672 608 674 147 13

1.113 462 678 083 94

40

2.672 608 672 639 88

1.113 462 678 711 58

50

2.672 608 672 874 71

1.113 462 678 330 06

60

2.672 608 679 729 44

1.113 462 678 958 71

Table 6

Discrete mass https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq160_HTML.gif and discrete energy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq161_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq162_HTML.gif at various https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq163_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq164_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq165_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq166_HTML.gif

10

3.988 663 320 390 89

1.917 613 014 656 71

20

3.988 663 260 854 26

1.917 613 014 739 89

30

3.988 663 167 685 49

1.917 613 014 820 83

40

3.988 663 194 506 97

1.917 613 014 927 44

50

3.988 663 973 359 89

1.917 613 014 009 71

60

3.988 663 621 972 59

1.917 613 014 679 17

Figures 1, 2, and 3 plot the exact solutions at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq167_HTML.gif and the numerical solutions computed by scheme (2.3)–(2.5) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq168_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq169_HTML.gif , which also show the accuracy of scheme (2.3)–(2.5).
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Fig1_HTML.jpg
Figure 1

Exact solutions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq170_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq171_HTML.gif and numerical solutions computed by scheme (2. 3)–(2.5) at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq172_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq173_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Fig2_HTML.jpg
Figure 2

Exact solutions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq174_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq175_HTML.gif and numerical solutions computed by scheme (2. 3)–(2.5) at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq176_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq177_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Fig3_HTML.jpg
Figure 3

Exact solutions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq178_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq179_HTML.gif and numerical solutions computed by scheme (2. 3)–(2.5) at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq180_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq181_HTML.gif .

Declarations

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).

Authors’ Affiliations

(1)
School of Science, Shandong University of Technology
(2)
School of Mathematics, Shandong University

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Copyright

© Jin-Ming Zuo et al. 2010

This article is published under license to BioMed Central Ltd. 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.