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Boundary Value Problems

, 2010:543503 | Cite as

Two Conservative Difference Schemes for the Generalized Rosenau Equation

  • Jinsong Hu
  • Kelong ZhengEmail author
Open Access
Research Article

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.

Keywords

Difference Scheme Fixed Point Theorem Finite Difference Scheme Finite Difference Method Discontinuous Galerkin Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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

with an initial condition

and boundary conditions

where Open image in new window is a integer.

When Open image in new window , (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 Open image in new window -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:

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 Open image in new window and Open image in new window be the uniform step size in the spatial and temporal direction, respectively. Denote Open image in new window , Open image in new window , and Open image in new window . Define

and in the paper, Open image in new window denotes a general positive constant which may have different values in different occurrences.

Since Open image in new window , then the following finite difference scheme is considered:

Lemma 2.1 (see [8]).

For any two mesh functions, Open image in new window , one has

Theorem 2.2.

Suppose Open image in new window , then the scheme (2.2)–(2.4) is conservative for discrete energy, that is,

Proof.

Computing the inner product of (2.2) with Open image in new window , according to boundary condition (2.4) and Lemma 2.1, we have

According to

we obtain

By the definition of Open image in new window , (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 Open image in new window be a finite dimensional inner product space. Suppose that Open image in new window is continuous and there exists an Open image in new window such that Open image in new window for all Open image in new window with Open image in new window . Then there exists Open image in new window such that Open image in new window and Open image in new window .

Theorem 2.4.

There exists Open image in new window satisfying the difference scheme (2.2)–(2.4).

Proof.

By the mathematical induction, for Open image in new window , assume that Open image in new window satisfy (2.2)–(2.4). Next we prove that there exists Open image in new window satisfying (2.2)–(2.4).

Define a operator Open image in new window on Open image in new window as follows:

Taking the inner product of (2.13) with Open image in new window , we get

Obviously, for all Open image in new window , Open image in new window with Open image in new window . It follows from Lemma 2.3 that there exists Open image in new window which satisfies Open image in new window . Let Open image in new window , it can be proved that Open image in new window 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 Open image in new window be the solution of problem (1.1)–(1.3), Open image in new window , then the truncation of the scheme (2.2)–(2.4) is

Using Taylor expansion, we know that Open image in new window holds if Open image in new window .

Lemma 2.5.

Suppose that Open image in new window , then the solution of the initial-boundary value problem (1.1)–(1.3) satisfies

Proof.

It follows from (1.4) that
Using Hölder inequality and Schwartz inequality, we get

Hence, Open image in new window . According to Sobolev inequality, we have Open image in new window .

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

There exist two constant Open image in new window and Open image in new window such that

Lemma 2.7 (Discrete Gronwall inequality [10]).

Suppose Open image in new window are nonnegative mesh functions and Open image in new window is nondecreasing. If Open image in new window and

Theorem 2.8.

Suppose Open image in new window , then the solution Open image in new window of (2.2) satisfies Open image in new window , which yield Open image in new window .

Proof.

It follows from (2.7) that
Using Lemma 2.1 and Schwartz inequality, we get

According to Lemma 2.6, we have Open image in new window .

Theorem 2.9.

Suppose Open image in new window , then the solution Open image in new window of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is Open image in new window .

Proof.

Subtracting (2.15) from (2.2) and letting Open image in new window , we have
Computing the inner product of (2.24) with Open image in new window , and using Open image in new window , we get
According to Lemma 2.5, Theorem 2.8, and Schwartz inequality, we have
Furthermore,
Substituting (2.27)–(2.29) into (2.25), we get
Similarly to the proof of (2.23), we have
and (2.30) can be rewritten as
Let Open image in new window , then (2.32) is written as follows:
If Open image in new window is sufficiently small which satisfies Open image in new window , then
Noticing
According to Lemma 2.7, we get Open image in new window , that is,
It follows from (2.31) that
By using Lemma 2.6, we have

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 Open image in new window .

3. Linearized Finite Difference Scheme

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

Theorem 3.1.

Suppose Open image in new window , then the scheme (3.1), (2.3), and (2.4) are conservative for discrete energy, that is,

Proof.

Computing the inner product of (3.1) with Open image in new window , we have
According to Lemma 2.1, we get
Adding (3.3) and (3.5) to (3.6), we obtain

By the definition of Open image in new window , (3.2) holds.

Theorem 3.2.

The difference scheme (3.1) is uniquely solvable.

Proof.

we use the mathematical induction. Obviously, Open image in new window is determined by (2.3) and we can choose a two-order method to compute Open image in new window (e.g., by scheme (2.2)). Assuming that Open image in new window are uniquely solvable, consider Open image in new window in (3.1) which satisfies
Taking the inner product of (3.8) with Open image in new window , we get
Notice that
It follows from (3.8) that

That is, there uniquely exists trivial solution satisfying (3.8). Hence, Open image in new window 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):

Using Taylor expansion, we know that Open image in new window holds if Open image in new window .

Theorem 3.3.

Suppose Open image in new window , then the solution of (3.1) satisfies Open image in new window , which yield Open image in new window .

Proof.

It follows from (3.2) that
According to (2.23), we have
that is,
If Open image in new window is sufficiently small which satisfies Open image in new window , we get
which yields Open image in new window . According to (2.23), we get
Using Lemma 2.6, we obtain

Theorem 3.4.

Suppose Open image in new window , then the solution Open image in new window 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 Open image in new window .

Proof.

Subtracting (3.12) from (3.1) and letting Open image in new window , we have
Computing the inner product of (3.19) with Open image in new window , we get
Notice that
and similarly
Furthermore, we get
Substituting (3.22)–(3.24) into (3.20), we get
Similarly to the proof of (2.31), (3.25) can be written as
Let Open image in new window , then (3.26) is written as follows:
that is,
If Open image in new window is sufficiently small which satisfies Open image in new window , then
Choosing a two-order method to compute Open image in new window (e.g., by scheme (2.2)) and noticing
According to Lemma 2.7, we get Open image in new window , that is,
According to (2.31), we get
By using Lemma 2.6, we have

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 Open image in new window .

4. Numerical Experiments

Consider the generalized Rosenau equation:

with an initial condition

and boundary conditions

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 Open image in new window as reference solution and obtain the error estimates on mesh Open image in new window , respectively, for different choices of Open image in new window , where we take Open image in new window . To verify the stability of schemes, we take Open image in new window . The maximal errors Open image in new window are listed on Tables 1, 2, and 3.
We have shown in Theorems 2.2 and 3.1 that the numerical solutions Open image in new window of Scheme (2.2) and Scheme (3.1) satisfy the conservation of energy, respectively. In Figure 1, we give the values of Open image in new window for Open image in new window with fixed Open image in new window for Scheme (2.2). In Figure 2, the values of Open image in new window for Scheme (3.1) are presented. We can see that scheme (2.2) preserves the discrete energy better than scheme (3.1).
Figure 1

Energy of scheme (2. 2) when Open image in new window and Open image in new window .

Figure 2

Energy of scheme (3. 1) when Open image in new window and Open image in new window .

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

Notes

Acknowledgment

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

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Copyright information

© J. Hu and K. Zheng. 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.

Open AccessThis 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.

Authors and Affiliations

  1. 1.School of Mathematics and Computer EngineeringXihua UniversityChengduChina
  2. 2.School of ScienceSouthwest University of Science and TechnologyMianyangChina

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