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

  • Jin-Ming Zuo1Email author,

    Affiliated with

    • Yao-Ming Zhang1,

      Affiliated with

      • Tian-De Zhang2 and

        Affiliated with

        • Feng Chang2

          Affiliated with

          Boundary Value Problems20102010:516260

          DOI: 10.1155/2010/516260

          Received: 28 May 2010

          Accepted: 14 October 2010

          Published: 20 October 2010

          Abstract

          A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Numerical examples show the efficiency of the scheme.

          1. Introduction

          In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ1_HTML.gif
          (11)
          with an initial condition
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ2_HTML.gif
          (12)
          and boundary conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ3_HTML.gif
          (13)
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq1_HTML.gif is a integer and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq2_HTML.gif is a known smooth function. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq3_HTML.gif , (1.1) is called as usual Rosenau-RLW equation. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq4_HTML.gif , (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ4_HTML.gif
          (14)
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ5_HTML.gif
          (15)

          It is known the conservative scheme is better than the nonconservative ones. Zhang et al. [1] point out that the nonconservative scheme may easily show nonlinear blow up. In [2] Li and Vu-Quoc said " http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq5_HTML.gif in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation". In [311], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general Rosenau-RLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.

          2. A Nonlinear-Implicit Conservative Scheme

          In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1.1)–(1.3) and give its numerical analysis.

          2.1. The Nonlinear-Implicit Scheme and Its Conservative Law

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

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

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

          Lemma 2.1 (see [12]).

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

          Theorem 2.2.

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

          Proof.

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

          Then (2.8) is gotten from (2.10).

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

          Then (2.9) is gotten from (2.15). This completes the proof of Theorem 2.2.

          2.2. Existence and Prior Estimates of Difference Solution

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

          Lemma 2.3.

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

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

          Theorem 2.4.

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

          Proof.

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

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

          Hence, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq45_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq46_HTML.gif . It follows from Lemma 2.3 that exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq47_HTML.gif which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq48_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq49_HTML.gif , then it can be proved that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq50_HTML.gif is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.

          Next we will give some priori estimates of difference solutions. First the following two lemmas [14] are introduced:

          Lemma 2.5 (discrete Sobolev's estimate).

          For any discrete function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq51_HTML.gif on the finite interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq52_HTML.gif , there is the inequality
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ24_HTML.gif
          (219)

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

          Lemma 2.6 (discrete Gronwall's inequality).

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

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

          Theorem 2.7.

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

          hold.

          Proof.

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

          This completes the proof of Theorem 2.7.

          Remark 2.8.

          Theorem 2.7 implies that scheme (2.3)–(2.5) is unconditionally stable.

          2.3. Convergence and Uniqueness of Difference Solution

          First, we consider the convergence of scheme (2.3)–(2.5). We define the truncation error as follows:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ30_HTML.gif
          (225)

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

          Theorem 2.9.

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

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

          Theorem 2.10.

          Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme (2.3)–(2.5) converges to the solution of problem (1.1)–(1.3) with order http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq66_HTML.gif in the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq67_HTML.gif norm.

          Proof.

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

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

          Theorem 2.11.

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

          Proof.

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

          This completes the proof of Theorem 2.11.

          Remark 2.12.

          All results above in this paper are correct for initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary.

          3. Numerical Experiments

          In order to test the correction of the numerical analysis in this paper, we consider the following initial-boundary value problems of the general Rosenau-RLW equation:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ45_HTML.gif
          (31)
          with an initial condition
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ46_HTML.gif
          (32)
          and boundary conditions
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ47_HTML.gif
          (33)
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq76_HTML.gif . Then the exact solution of the initial value problem (3.1)-(3.2) is
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_Equ48_HTML.gif
          (34)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq77_HTML.gif is wave velocity.

          It follows from (3.4) that the initial-boundary value problem (3.1)–(3.3) is consistent to the boundary value problem (3.3) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq78_HTML.gif . In the following examples, we always choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq80_HTML.gif .

          Tables 1, 2, and 3 give the errors in the sense of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq81_HTML.gif -norm and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq82_HTML.gif -norm of the numerical solutions under various steps of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq84_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq85_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq87_HTML.gif . The three tables verify the second-order convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq88_HTML.gif and discrete energy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq89_HTML.gif computed by scheme (2.3)–(2.5) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F516260/MediaObjects/13661_2010_Article_932_IEq91_HTML.gif .
          Table 1

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

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

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

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

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

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

          0.4

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

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

            

          0.2

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

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

          3.953 296

          3.930 200

          0.1

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

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

          3.987 130

          3.980 050

          0.05

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

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

          3.997 412

          3.994 759

          0.025

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

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

          4.221 051

          4.151 348

          Table 2

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

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

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

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

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

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

          0.4

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

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

            

          0.2

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

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

          3.961 294

          3.934 592

          0.1

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

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

          3.996 350

          3.988 283

          0.05

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

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

          4.003 273

          4.000 165

          0.025

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

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

          4.290 286

          4.242 688

          Table 3

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

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

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

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

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

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

          0.4

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

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

            

          0.2

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

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

          3.885 945

          3.851 797

          0.1

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

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

          3.975 084

          3.965 980

          0.05

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

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

          4.000 294

          3.995 992

          0.025

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

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

          4.391 818

          4.381 199

          Table 4

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

           

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

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

          10

          1.897 658 262 960 01

          0.533 175 231 580 85

          20

          1.897 658 268 873 21

          0.533 175 231 872 51

          30

          1.897 658 262 993 93

          0.533 175 231 177 25

          40

          1.897 658 265 568 93

          0.533 175 231 478 09

          50

          1.897 658 260 975 87

          0.533 175 231 776 18

          60

          1.897 658 265 384 88

          0.533 175 231 074 05

          Table 5

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

           

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

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

          10

          2.672 608 675 265 30

          1.113 462 678 852 70

          20

          2.672 608 676 236 58

          1.113 462 678 465 22

          30

          2.672 608 674 147 13

          1.113 462 678 083 94

          40

          2.672 608 672 639 88

          1.113 462 678 711 58

          50

          2.672 608 672 874 71

          1.113 462 678 330 06

          60

          2.672 608 679 729 44

          1.113 462 678 958 71

          Table 6

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

           

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

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

          10

          3.988 663 320 390 89

          1.917 613 014 656 71

          20

          3.988 663 260 854 26

          1.917 613 014 739 89

          30

          3.988 663 167 685 49

          1.917 613 014 820 83

          40

          3.988 663 194 506 97

          1.917 613 014 927 44

          50

          3.988 663 973 359 89

          1.917 613 014 009 71

          60

          3.988 663 621 972 59

          1.917 613 014 679 17

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

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

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

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

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

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

          Declarations

          Acknowledgments

          The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (nos. 10871117 and 10571110).

          Authors’ Affiliations

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

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