Open Access

Two Conservative Difference Schemes for the Generalized Rosenau Equation

Boundary Value Problems20102010:543503

DOI: 10.1155/2010/543503

Received: 31 October 2009

Accepted: 26 January 2010

Published: 8 March 2010

Abstract

Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.

1. Introduction

Consider the following initial-boundary value problem for generalized Rosenau equation:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ1_HTML.gif
(1.1)

with an initial condition

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ2_HTML.gif
(1.2)

and boundary conditions

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq1_HTML.gif is a integer.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq2_HTML.gif , (1.1) is called as usual Rosenau equation proposed by Rosenau [1] for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park [2]. Many numerical schemes have been proposed, such as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq3_HTML.gif -conforming finite element method by Chung and Pani [3], discontinuous Galerkin method by Choo et al. [4], orthogonal cubic spline collocation method by Manickam [5], and finite difference method by Chung [6] and Omrani et al. [7]. As for the generalized case, however, there are few studies on theoretical analysis and numerical methods.

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ4_HTML.gif
(1.4)

Hence, we propose two conservative difference schemes which simulate conservative law (1.4). 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, a linearized difference scheme is proposed and theoretical results are obtained. In Section 4, some numerical experiments are shown.

2. Nonlinear Finite Difference Scheme

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq5_HTML.gif be the uniform step size in the spatial and temporal direction, respectively. Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq8_HTML.gif . Define

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ5_HTML.gif
(2.1)

and in the paper, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq9_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%2F543503/MediaObjects/13661_2009_Article_937_IEq10_HTML.gif , then the following finite difference scheme is considered:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ6_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ7_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ8_HTML.gif
(2.4)

Lemma 2.1 (see [8]).

For any two mesh functions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq11_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ9_HTML.gif
(2.5)
Furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq12_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ10_HTML.gif
(2.6)

Theorem 2.2.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq13_HTML.gif , then the scheme (2.2)–(2.4) is conservative for discrete energy, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ11_HTML.gif
(2.7)

Proof.

Computing the inner product of (2.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq14_HTML.gif , according to boundary condition (2.4) and Lemma 2.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ12_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ13_HTML.gif
(2.9)

According to

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ14_HTML.gif
(2.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ15_HTML.gif
(2.11)
we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ16_HTML.gif
(2.12)

By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq15_HTML.gif , (2.7) holds.

To prove the existence of solution for scheme (2.2)–(2.4), the following Browder fixed point Theorem should be introduced. For the proof, see [9].

Lemma 2.3 (Browder fixed point Theorem).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq16_HTML.gif be a finite dimensional inner product space. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq17_HTML.gif is continuous and there exists an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq18_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq19_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq20_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq21_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq22_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq24_HTML.gif .

Theorem 2.4.

There exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq25_HTML.gif satisfying the difference scheme (2.2)–(2.4).

Proof.

By the mathematical induction, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq26_HTML.gif , assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq27_HTML.gif satisfy (2.2)–(2.4). Next we prove that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq28_HTML.gif satisfying (2.2)–(2.4).

Define a operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq29_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq30_HTML.gif as follows:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ17_HTML.gif
(2.13)
Taking the inner product of (2.13) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq31_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ18_HTML.gif
(2.14)

Obviously, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq33_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq34_HTML.gif . It follows from Lemma 2.3 that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq35_HTML.gif which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq36_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq37_HTML.gif , it can be proved that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq38_HTML.gif is the solution of the scheme (2.2)–(2.4).

Next, we discuss the convergence and stability of the scheme (2.2)–(2.4). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq39_HTML.gif be the solution of problem (1.1)–(1.3), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq40_HTML.gif , then the truncation of the scheme (2.2)–(2.4) is

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ19_HTML.gif
(2.15)

Using Taylor expansion, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq41_HTML.gif holds if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq42_HTML.gif .

Lemma 2.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq43_HTML.gif , then the solution of the initial-boundary value problem (1.1)–(1.3) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ20_HTML.gif
(2.16)

Proof.

It follows from (1.4) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ21_HTML.gif
(2.17)
Using Hölder inequality and Schwartz inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ22_HTML.gif
(2.18)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq44_HTML.gif . According to Sobolev inequality, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq45_HTML.gif .

Lemma 2.6 (Discrete Sobolev's inequality [10]).

There exist two constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq47_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ23_HTML.gif
(2.19)

Lemma 2.7 (Discrete Gronwall inequality [10]).

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq48_HTML.gif are nonnegative mesh functions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq49_HTML.gif is nondecreasing. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq50_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ24_HTML.gif
(2.20)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ25_HTML.gif
(2.21)

Theorem 2.8.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq51_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq52_HTML.gif of (2.2) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq53_HTML.gif , which yield https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq54_HTML.gif .

Proof.

It follows from (2.7) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ26_HTML.gif
(2.22)
Using Lemma 2.1 and Schwartz inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ27_HTML.gif
(2.23)

According to Lemma 2.6, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq55_HTML.gif .

Theorem 2.9.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq56_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq57_HTML.gif of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq58_HTML.gif .

Proof.

Subtracting (2.15) from (2.2) and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq59_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ28_HTML.gif
(2.24)
Computing the inner product of (2.24) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq60_HTML.gif , and using https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq61_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ29_HTML.gif
(2.25)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ30_HTML.gif
(2.26)
According to Lemma 2.5, Theorem 2.8, and Schwartz inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ31_HTML.gif
(2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ32_HTML.gif
(2.28)
Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ33_HTML.gif
(2.29)
Substituting (2.27)–(2.29) into (2.25), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ34_HTML.gif
(2.30)
Similarly to the proof of (2.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ35_HTML.gif
(2.31)
and (2.30) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ36_HTML.gif
(2.32)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq62_HTML.gif , then (2.32) is written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ37_HTML.gif
(2.33)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq63_HTML.gif is sufficiently small which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq64_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ38_HTML.gif
(2.34)
Summing up (2.34) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq65_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq66_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ39_HTML.gif
(2.35)
Noticing
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ40_HTML.gif
(2.36)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq67_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq68_HTML.gif . Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ41_HTML.gif
(2.37)
According to Lemma 2.7, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq69_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ42_HTML.gif
(2.38)
It follows from (2.31) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ43_HTML.gif
(2.39)
By using Lemma 2.6, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ44_HTML.gif
(2.40)

This completes the proof of Theorem 2.9.

Similarly, the following theorem can be proved.

Theorem 2.10.

Under the conditions of Theorem 2.9, the solution of the scheme (2.2)–(2.4) is stable by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq70_HTML.gif .

3. Linearized Finite Difference Scheme

In this section, we propose a linear-implicit finite difference scheme as follows:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ45_HTML.gif
(3.1)

Theorem 3.1.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq71_HTML.gif , then the scheme (3.1), (2.3), and (2.4) are conservative for discrete energy, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ46_HTML.gif
(3.2)

Proof.

Computing the inner product of (3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq72_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ47_HTML.gif
(3.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ48_HTML.gif
(3.4)
According to Lemma 2.1, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ49_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ50_HTML.gif
(3.6)
Adding (3.3) and (3.5) to (3.6), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ51_HTML.gif
(3.7)

By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq73_HTML.gif , (3.2) holds.

Theorem 3.2.

The difference scheme (3.1) is uniquely solvable.

Proof.

we use the mathematical induction. Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq74_HTML.gif is determined by (2.3) and we can choose a two-order method to compute https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq75_HTML.gif (e.g., by scheme (2.2)). Assuming that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq76_HTML.gif are uniquely solvable, consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq77_HTML.gif in (3.1) which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ52_HTML.gif
(3.8)
Taking the inner product of (3.8) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq78_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ53_HTML.gif
(3.9)
Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ54_HTML.gif
(3.10)
It follows from (3.8) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ55_HTML.gif
(3.11)

That is, there uniquely exists trivial solution satisfying (3.8). Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq79_HTML.gif in (3.1) is uniquely solvable.

To discuss the convergence and stability of the scheme (3.1), we denote the truncation of the scheme (3.1):

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ56_HTML.gif
(3.12)

Using Taylor expansion, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq80_HTML.gif holds if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq81_HTML.gif .

Theorem 3.3.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq82_HTML.gif , then the solution of (3.1) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq83_HTML.gif , which yield https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq84_HTML.gif .

Proof.

It follows from (3.2) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ57_HTML.gif
(3.13)
According to (2.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ58_HTML.gif
(3.14)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ59_HTML.gif
(3.15)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq85_HTML.gif is sufficiently small which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq86_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ60_HTML.gif
(3.16)
which yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq87_HTML.gif . According to (2.23), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ61_HTML.gif
(3.17)
Using Lemma 2.6, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ62_HTML.gif
(3.18)

Theorem 3.4.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq88_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq89_HTML.gif of the schemes (3.1), (2.3), and (2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq90_HTML.gif .

Proof.

Subtracting (3.12) from (3.1) and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq91_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ63_HTML.gif
(3.19)
Computing the inner product of (3.19) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq92_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ64_HTML.gif
(3.20)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ65_HTML.gif
(3.21)
Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ66_HTML.gif
(3.22)
and similarly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ67_HTML.gif
(3.23)
Furthermore, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ68_HTML.gif
(3.24)
Substituting (3.22)–(3.24) into (3.20), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ69_HTML.gif
(3.25)
Similarly to the proof of (2.31), (3.25) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ70_HTML.gif
(3.26)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq93_HTML.gif , then (3.26) is written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ71_HTML.gif
(3.27)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ72_HTML.gif
(3.28)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq94_HTML.gif is sufficiently small which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq95_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ73_HTML.gif
(3.29)
Summing up (3.29) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq96_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq97_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ74_HTML.gif
(3.30)
Choosing a two-order method to compute https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq98_HTML.gif (e.g., by scheme (2.2)) and noticing
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ75_HTML.gif
(3.31)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ76_HTML.gif
(3.32)
According to Lemma 2.7, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq99_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ77_HTML.gif
(3.33)
According to (2.31), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ78_HTML.gif
(3.34)
By using Lemma 2.6, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ79_HTML.gif
(3.35)

This completes the proof of Theorem 3.4.

Similarly, the following theorem can be proved that.

Theorem 3.5.

Under the conditions of Theorem 3.4, the solution of the schemes (3.1), (2.3), and (2.4) are stable by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq100_HTML.gif .

4. Numerical Experiments

Consider the generalized Rosenau equation:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ80_HTML.gif
(4.1)

with an initial condition

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ81_HTML.gif
(4.2)

and boundary conditions

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ82_HTML.gif
(4.3)
We construct two schemes to (4.1)–(4.3) as nonlinear scheme (2.2) and linearized scheme (3.1). Since we do not know the exact solution of (4.1)–(4.3), we consider the solution on mesh https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq101_HTML.gif as reference solution and obtain the error estimates on mesh https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq102_HTML.gif , respectively, for different choices of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq103_HTML.gif , where we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq104_HTML.gif . To verify the stability of schemes, we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq105_HTML.gif . The maximal errors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq106_HTML.gif are listed on Tables 1, 2, and 3.
Table 1

The errors estimates in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq107_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq109_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq110_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq111_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq112_HTML.gif

 

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq113_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq114_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq115_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq116_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq117_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq118_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq119_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq120_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq121_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq122_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq123_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq124_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq125_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq126_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq127_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq128_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq129_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq130_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq131_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq132_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq133_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq134_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq135_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq136_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq137_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq138_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq139_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq140_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq141_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq143_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq144_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq145_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq146_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq147_HTML.gif

Table 2

The errors estimates in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq148_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq150_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq151_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq152_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq153_HTML.gif

 

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq154_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq155_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq156_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq157_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq158_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq159_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq160_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq161_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq162_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq163_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq164_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq165_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq166_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq167_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq168_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq169_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq170_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq172_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq173_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq174_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq175_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq176_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq177_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq178_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq179_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq180_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq181_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq182_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq183_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq184_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq185_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq186_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq187_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq188_HTML.gif

Table 3

The errors estimates in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq189_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq190_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq191_HTML.gif .

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq192_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq193_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq194_HTML.gif

 

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

Scheme (2.2)

Scheme (3.1)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq195_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq196_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq197_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq198_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq199_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq200_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq201_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq202_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq203_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq204_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq205_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq206_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq207_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq208_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq209_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq210_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq211_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq212_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq213_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq214_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq215_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq216_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq217_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq218_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq219_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq220_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq221_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq222_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq223_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq224_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq225_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq226_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq227_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq228_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq229_HTML.gif

We have shown in Theorems 2.2 and 3.1 that the numerical solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq230_HTML.gif of Scheme (2.2) and Scheme (3.1) satisfy the conservation of energy, respectively. In Figure 1, we give the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq231_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq232_HTML.gif with fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq233_HTML.gif for Scheme (2.2). In Figure 2, the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq234_HTML.gif for Scheme (3.1) are presented. We can see that scheme (2.2) preserves the discrete energy better than scheme (3.1).
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Fig1_HTML.jpg
Figure 1

Energy of scheme (2. 2) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq235_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq236_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Fig2_HTML.jpg
Figure 2

Energy of scheme (3. 1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq238_HTML.gif .

From the numerical results, two finite difference schemes of this paper are efficient.

Declarations

Acknowledgment

This work was supported by the Youth Research Foundation of SWUST (no. 08zx3125).

Authors’ Affiliations

(1)
School of Mathematics and Computer Engineering, Xihua University
(2)
School of Science, Southwest University of Science and Technology

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Copyright

© J. Hu and K. Zheng. 2010

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