Two Conservative Difference Schemes for the Generalized Rosenau Equation

  • Jinsong Hu1 and

    Affiliated with

    • Kelong Zheng2Email author

      Affiliated with

      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:

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

      with an initial condition

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

      and boundary conditions

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

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

      When http://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 http://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:

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq4_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq7_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq8_HTML.gif . Define

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

      and in the paper, http://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 http://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:

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ6_HTML.gif
      (2.2)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ7_HTML.gif
      (2.3)
      http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq11_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ9_HTML.gif
      (2.5)
      Furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq12_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ10_HTML.gif
      (2.6)

      Theorem 2.2.

      Suppose http://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,
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ12_HTML.gif
      (2.8)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ13_HTML.gif
      (2.9)

      According to

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

      By the definition of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq17_HTML.gif is continuous and there exists an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq18_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq19_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq20_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq21_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq22_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq24_HTML.gif .

      Theorem 2.4.

      There exists http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq26_HTML.gif , assume that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq29_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq30_HTML.gif as follows:

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq31_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ18_HTML.gif
      (2.14)

      Obviously, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq33_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq35_HTML.gif which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq36_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq37_HTML.gif , it can be proved that http://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 http://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), http://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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq41_HTML.gif holds if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq42_HTML.gif .

      Lemma 2.5.

      Suppose that http://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
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ22_HTML.gif
      (2.18)

      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq44_HTML.gif . According to Sobolev inequality, we have http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq47_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq48_HTML.gif are nonnegative mesh functions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq49_HTML.gif is nondecreasing. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq50_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ24_HTML.gif
      (2.20)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ25_HTML.gif
      (2.21)

      Theorem 2.8.

      Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq51_HTML.gif , then the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq52_HTML.gif of (2.2) satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq53_HTML.gif , which yield http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq55_HTML.gif .

      Theorem 2.9.

      Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq56_HTML.gif , then the solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq59_HTML.gif , we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq60_HTML.gif , and using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq61_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ29_HTML.gif
      (2.25)
      where
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ31_HTML.gif
      (2.27)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ32_HTML.gif
      (2.28)
      Furthermore,
      http://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
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ36_HTML.gif
      (2.32)
      Let http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ37_HTML.gif
      (2.33)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq63_HTML.gif is sufficiently small which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq64_HTML.gif , then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq65_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq66_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ39_HTML.gif
      (2.35)
      Noticing
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ40_HTML.gif
      (2.36)
      and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq67_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq68_HTML.gif . Hence
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq69_HTML.gif , that is,
      http://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
      http://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
      http://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 http://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:

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

      Theorem 3.1.

      Suppose http://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,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq72_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ47_HTML.gif
      (3.3)
      where
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ49_HTML.gif
      (3.5)
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ51_HTML.gif
      (3.7)

      By the definition of http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq76_HTML.gif are uniquely solvable, consider http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq77_HTML.gif in (3.1) which satisfies
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq78_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ53_HTML.gif
      (3.9)
      Notice that
      http://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
      http://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, http://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):

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq80_HTML.gif holds if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq81_HTML.gif .

      Theorem 3.3.

      Suppose http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq83_HTML.gif , which yield http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ58_HTML.gif
      (3.14)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ59_HTML.gif
      (3.15)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq85_HTML.gif is sufficiently small which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq86_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ60_HTML.gif
      (3.16)
      which yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq87_HTML.gif . According to (2.23), we get
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ62_HTML.gif
      (3.18)

      Theorem 3.4.

      Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq88_HTML.gif , then the solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq91_HTML.gif , we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq92_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ64_HTML.gif
      (3.20)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ65_HTML.gif
      (3.21)
      Notice that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ66_HTML.gif
      (3.22)
      and similarly
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ67_HTML.gif
      (3.23)
      Furthermore, we get
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ70_HTML.gif
      (3.26)
      Let http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ71_HTML.gif
      (3.27)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ72_HTML.gif
      (3.28)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq94_HTML.gif is sufficiently small which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq95_HTML.gif , then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq96_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq97_HTML.gif , we have
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_Equ75_HTML.gif
      (3.31)
      we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq99_HTML.gif , that is,
      http://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
      http://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
      http://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 http://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:

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

      with an initial condition

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

      and boundary conditions

      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq102_HTML.gif , respectively, for different choices of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq103_HTML.gif , where we take http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq105_HTML.gif . The maximal errors http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq107_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq109_HTML.gif .

       

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

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

      http://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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq148_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq150_HTML.gif .

       

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

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

      http://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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq189_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq191_HTML.gif .

       

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

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

      http://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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq231_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq232_HTML.gif with fixed http://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 http://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).
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq235_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq236_HTML.gif .

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F543503/MediaObjects/13661_2009_Article_937_IEq237_HTML.gif and http://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

      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.