A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

  • Jinsong Hu1,

    Affiliated with

    • Youcai Xu2Email author and

      Affiliated with

      • Bing Hu2

        Affiliated with

        Boundary Value Problems20102010:781750

        DOI: 10.1155/2010/781750

        Received: 24 August 2010

        Accepted: 14 November 2010

        Published: 30 November 2010

        Abstract

        We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear three-level implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient.

        1. Introduction

        A symmetric version of regularized long wave equation (SRLWE),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ1_HTML.gif
        (1.1)
        has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [1]. The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq1_HTML.gif solitary wave solutions are
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ2_HTML.gif
        (1.2)
        The four invariants and some numerical results have been obtained in [1], where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq2_HTML.gif is the velocity, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq3_HTML.gif . Obviously, eliminating http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq4_HTML.gif from (1.1), we get a class of SRLWE:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ3_HTML.gif
        (1.3)

        Equation (1.3) is explicitly symmetric in the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq6_HTML.gif derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2, 3]. The SRLW equation also arises in many other areas of mathematical physics [46]. Numerical investigation indicates that interactions of solitary waves are inelastic [7]; thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In [8], Guo studied the existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In [9], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initial-boundary value problem of SRLWEs (see [915]).

        In applications, the viscous damping effect is inevitable, and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ4_HTML.gif
        (1.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ5_HTML.gif
        (1.5)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq7_HTML.gif are positive constants, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq8_HTML.gif is the dissipative coefficient, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq9_HTML.gif is the damping coefficient. Equations (1.4)-(1.5) are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered. Existence, uniqueness, and wellposedness of global solutions to (1.4)-(1.5) are presented (see [1620]). But it is difficult to find the analytical solution to (1.4)-(1.5), which makes numerical solution important.

        To authors' knowledge, the finite difference method to dissipative SRLWEs with damping term (1.4)-(1.5) has not been studied till now. In this paper, we propose linear three level implicit finite difference scheme for (1.4)-(1.5) with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ6_HTML.gif
        (1.6)
        and the boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ7_HTML.gif
        (1.7)

        We show that this difference scheme is uniquely solvable, convergent, and stable in both theoretical and numerical senses.

        Lemma 1.1.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq11_HTML.gif , the solution of (1.4)–(1.7) satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq14_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq15_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq16_HTML.gif is a generic positive constant that varies in the context.

        Proof.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ8_HTML.gif
        (1.8)
        Multiplying (1.4) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq17_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq18_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ9_HTML.gif
        (1.9)
        According to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ10_HTML.gif
        (1.10)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ11_HTML.gif
        (1.11)
        Then, multiplying (1.5) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq19_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq20_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ12_HTML.gif
        (1.12)
        By
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ13_HTML.gif
        (1.13)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ14_HTML.gif
        (1.14)
        Adding (1.14) to (1.11), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ15_HTML.gif
        (1.15)

        So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq21_HTML.gif is decreasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq22_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq24_HTML.gif . Then, it indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq26_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq27_HTML.gif . It is followed from Sobolev inequality that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq28_HTML.gif .

        2. Finite Difference Scheme and Its Error Estimation

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq30_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%2F781750/MediaObjects/13661_2010_Article_954_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq35_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq36_HTML.gif . We define the difference operators as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ16_HTML.gif
        (2.1)
        Then, the average three-implicit finite difference scheme for the solution of (1.4)–(1.7) is as follow:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ17_HTML.gif
        (2.2)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ18_HTML.gif
        (2.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ19_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ20_HTML.gif
        (2.5)

        Lemma 2.1.

        Summation by parts follows [12, 21] that for any two discrete functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq37_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ21_HTML.gif
        (2.6)

        Lemma 2.2 (discrete Sobolev's inequality [12, 21]).

        There exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq39_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ22_HTML.gif
        (2.7)

        Lemma 2.3 (discrete Gronwall inequality [12, 21]).

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq41_HTML.gif are nonnegative functions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq42_HTML.gif is nondecreasing. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq43_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ23_HTML.gif
        (2.8)

        Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq44_HTML.gif .

        Theorem 2.4.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq46_HTML.gif , then the solution of (2.2)–(2.5) satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ24_HTML.gif
        (2.9)

        Proof.

        Taking an inner product of (2.2) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq47_HTML.gif   (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq48_HTML.gif ) and considering the boundary condition (2.5) and Lemma 2.1, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ25_HTML.gif
        (2.10)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq49_HTML.gif . Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ26_HTML.gif
        (2.11)
        we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ27_HTML.gif
        (2.12)
        Taking an inner product of (2.3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq50_HTML.gif  (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq51_HTML.gif ), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ28_HTML.gif
        (2.13)
        Adding (2.12) to (2.13), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ29_HTML.gif
        (2.14)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ30_HTML.gif
        (2.15)
        Equation (2.14) can be changed to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ31_HTML.gif
        (2.16)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq52_HTML.gif , and (2.16) is changed to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ32_HTML.gif
        (2.17)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq53_HTML.gif is sufficiently small which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq54_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ33_HTML.gif
        (2.18)
        Summing up (2.18) from 1 to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq55_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ34_HTML.gif
        (2.19)

        From Lemma 2.3, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq56_HTML.gif , which implies that, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq58_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq59_HTML.gif . By Lemma 2.2, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq60_HTML.gif .

        Theorem 2.5.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq62_HTML.gif , the solution of difference scheme (2.2)–(2.5) satisfies:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ35_HTML.gif
        (2.20)

        Proof.

        Differentiating backward (2.2)–(2.5) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq63_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ36_HTML.gif
        (2.21)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ37_HTML.gif
        (2.22)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ38_HTML.gif
        (2.23)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ39_HTML.gif
        (2.24)
        Computing the inner product of (2.21) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq64_HTML.gif   (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq65_HTML.gif ) and considering (2.24) and Lemma 2.1, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ40_HTML.gif
        (2.25)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq66_HTML.gif . It follows from Theorem 2.4 that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ41_HTML.gif
        (2.26)
        By the Schwarz inequality and Lemma 2.1, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ42_HTML.gif
        (2.27)
        Noting that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ43_HTML.gif
        (2.28)
        it follows from (2.25) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ44_HTML.gif
        (2.29)
        Computing the inner product of (2.22) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq67_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq68_HTML.gif ) and considering (2.24) and Lemma 2.1, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ45_HTML.gif
        (2.30)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ46_HTML.gif
        (2.31)
        then (2.30) is changed to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ47_HTML.gif
        (2.32)
        Adding (2.29) to (2.32), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ48_HTML.gif
        (2.33)
        Leting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq69_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq70_HTML.gif . Choosing suitable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq71_HTML.gif which is small enough to satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq72_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ49_HTML.gif
        (2.34)
        Summing up (2.34) from 1 to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq73_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ50_HTML.gif
        (2.35)

        By Lemma 2.3, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq74_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq76_HTML.gif . It follows from Theorem 2.4 and Lemma 2.2 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq78_HTML.gif .

        3. Solvability

        Theorem 3.1.

        The solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq79_HTML.gif of (2.2)–(2.5) is unique.

        Proof.

        Using the mathematical induction, clearly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq81_HTML.gif are uniquely determined by initial conditions (2.4). then select appropriate second-order methods (such as the C-N Schemes) and calculate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq83_HTML.gif (i.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq85_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq86_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq87_HTML.gif are uniquely determined). Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq88_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq89_HTML.gif are the only solution, now consider http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq91_HTML.gif in (2.2) and (2.3):
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ51_HTML.gif
        (3.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ52_HTML.gif
        (3.2)
        Taking an inner product of (3.1) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq92_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ53_HTML.gif
        (3.3)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ54_HTML.gif
        (3.4)
        then it holds
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ55_HTML.gif
        (3.5)
        Taking an inner product of (3.2) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq93_HTML.gif and adding to (3.5), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ56_HTML.gif
        (3.6)

        which implies that (3.1)-(3.2) have only zero solution. So the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq95_HTML.gif of (2.2)–(2.5) is unique.

        4. Convergence and Stability

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq97_HTML.gif be the solution of problem (1.4)–(1.7); that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq99_HTML.gif , then the truncation of the difference scheme (2.2)–(2.5) is
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ57_HTML.gif
        (4.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ58_HTML.gif
        (4.2)

        Making use of Taylor expansion, it holds http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq100_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq101_HTML.gif .

        Theorem 4.1.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq103_HTML.gif , then the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq105_HTML.gif in the senses of norms http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq107_HTML.gif , respectively, to the difference scheme (2.2)–(2.5) converges to the solution of problem (1.4)–(1.7) and the order of convergence is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq108_HTML.gif .

        Proof.

        Subtracting (2.2) from (4.1) subtracting (2.3) from (4.2), and letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq109_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq110_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ59_HTML.gif
        (4.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ60_HTML.gif
        (4.4)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ61_HTML.gif
        (4.5)
        Computing the inner product of (4.3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq111_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ62_HTML.gif
        (4.6)
        According to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ63_HTML.gif
        (4.7)
        it follow from Lemma 1.1, Theorems 2.4, and 2.5 that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ64_HTML.gif
        (4.8)
        By the Schwarz inequality, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ65_HTML.gif
        (4.9)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ66_HTML.gif
        (4.10)
        it follows from (4.9)–(4.10) and (4.6) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ67_HTML.gif
        (4.11)
        Computing the inner product of (4.4) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq112_HTML.gif , we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ68_HTML.gif
        (4.12)
        Adding (4.12) to (4.11), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ69_HTML.gif
        (4.13)
        Leting
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ70_HTML.gif
        (4.14)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ71_HTML.gif
        (4.15)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq113_HTML.gif is sufficiently small which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq114_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ72_HTML.gif
        (4.16)
        Summing up (4.16) from 1 to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq115_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ73_HTML.gif
        (4.17)
        Select appropriate second-order methods (such as the C-N Schemes), and calculate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq116_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq117_HTML.gif , which satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ74_HTML.gif
        (4.18)
        Noticing that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ75_HTML.gif
        (4.19)
        we then have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ76_HTML.gif
        (4.20)
        By Lemma 2.3, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ77_HTML.gif
        (4.21)
        This yields
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ78_HTML.gif
        (4.22)
        By Lemma 2.2, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ79_HTML.gif
        (4.23)

        Similarly to Theorem 4.1, we can prove the result as follows.

        Theorem 4.2.

        Under the conditions of Theorem 4.1, the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq119_HTML.gif of (2.2)–(2.5) is stable in the senses of norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq121_HTML.gif , respectively.

        5. Numerical Simulations

        Since the three-implicit finite difference scheme can not start by itself, we need to select other two-level schemes (such as the C-N Scheme) to get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq123_HTML.gif . Then, reusing initial value http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq125_HTML.gif , we can work out http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq126_HTML.gif . Iterative numerical calculation is not required, for this scheme is linear, so it saves computing time.

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq127_HTML.gif , the damping does not have an effect and the dissipative will not appear. So the initial conditions of (1.4)–(1.7) are same as those of (1.1):
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ80_HTML.gif
        (5.1)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq130_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq131_HTML.gif . Since we do not know the exact solution of (1.4)-(1.5), an error estimates method in [21] is used: a comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. We consider the solution on mesh http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq132_HTML.gif as the reference solution. In Table 1, we give the ratios in the sense of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq133_HTML.gif at various time steps.
        Table 1

        The error ratios in the sense of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq134_HTML.gif at various time steps.

          

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq135_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq136_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq137_HTML.gif

        μ

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq138_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq139_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq140_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq141_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq142_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq143_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq144_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq145_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq146_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq147_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq148_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq149_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq150_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq151_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq152_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq153_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq154_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq155_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq156_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq157_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq158_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq159_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq160_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq161_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq162_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq163_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq164_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq165_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq166_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq167_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq168_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq169_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq170_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq171_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq172_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq173_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq174_HTML.gif

         

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq175_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq176_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq177_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq178_HTML.gif

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq179_HTML.gif , a wave figure comparison of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq181_HTML.gif at various time steps is as in Figures 1 and 2.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Fig1_HTML.jpg
        Figure 1

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq182_HTML.gif , the wave graph of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq183_HTML.gif at various times.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Fig2_HTML.jpg
        Figure 2

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq184_HTML.gif , the wave graph of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq185_HTML.gif at various times.

        From Table 1, it is easy to find that the difference scheme in this paper is second-order convergent. Figures 1 and 2 show that the height of wave crest is more and more low with time elapsing due to the effect of damping and dissipativeness. It simulates that the continue energy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq186_HTML.gif of problem (1.4)–(1.7) in Lemma 1.1 is digressive. Numerical experiments show that the finite difference scheme is efficient.

        Declarations

        Acknowledgments

        The work of Jinsong Hu was supported by the research fund of key disciplinary of application mathematics of Xihua University (Grant no. XZD0910-09-1). The work of Youcai Xu was supported by the Youth Research Foundation of Sichuan University (no. 2009SCU11113).

        Authors’ Affiliations

        (1)
        School of Mathematics and Computer Engineering, Xihua University
        (2)
        School of Mathematics, Sichuan University

        References

        1. Seyler CE, Fenstermacher DL: A symmetric regularized-long-wave equation. Physics of Fluids 1984,27(1):4-7. 10.1063/1.864487View ArticleMATH
        2. Albert J: On the decay of solutions of the generalized Benjamin-Bona-Mahony equations. Journal of Mathematical Analysis and Applications 1989,141(2):527-537. 10.1016/0022-247X(89)90195-9MathSciNetView ArticleMATH
        3. Amick CJ, Bona JL, Schonbek ME: Decay of solutions of some nonlinear wave equations. Journal of Differential Equations 1989,81(1):1-49. 10.1016/0022-0396(89)90176-9MathSciNetView ArticleMATH
        4. Ogino T, Takeda S: Computer simulation and analysis for the spherical and cylindrical ion-acoustic solitons. Journal of the Physical Society of Japan 1976,41(1):257-264. 10.1143/JPSJ.41.257MathSciNetView Article
        5. Makhankov VG: Dynamics of classical solitons (in non-integrable systems). Physics Reports. Section C 1978,35(1):1-128.MathSciNetView Article
        6. Clarkson PA: New similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations. Journal of Physics A 1989,22(18):3821-3848. 10.1088/0305-4470/22/18/020MathSciNetView ArticleMATH
        7. Bogolubsky IL: Some examples of inelastic soliton interaction. Computer Physics Communications 1977,13(3):149-155. 10.1016/0010-4655(77)90009-1View Article
        8. Guo B: The spectral method for symmetric regularized wave equations. Journal of Computational Mathematics 1987,5(4):297-306.MathSciNetMATH
        9. Zheng JD, Zhang RF, Guo BY: The Fourier pseudo-spectral method for the SRLW equation. Applied Mathematics and Mechanics 1989,10(9):801-810.MathSciNet
        10. Zheng JD: Pseudospectral collocation methods for the generalized SRLW equations. Mathematica Numerica Sinica 1989,11(1):64-72.MATH
        11. Shang YD, Guo B: Legendre and Chebyshev pseudospectral methods for the generalized symmetric regularized long wave equations. Acta Mathematicae Applicatae Sinica 2003,26(4):590-604.MathSciNetMATH
        12. Bai Y, Zhang LM: A conservative finite difference scheme for symmetric regularized long wave equations. Acta Mathematicae Applicatae Sinica 2007,30(2):248-255.MathSciNetMATH
        13. Wang T, Zhang L, Chen F: Conservative schemes for the symmetric regularized long wave equations. Applied Mathematics and Computation 2007,190(2):1063-1080. 10.1016/j.amc.2007.01.105MathSciNetView ArticleMATH
        14. Wang TC, Zhang LM: Pseudo-compact conservative finite difference approximate solution for the symmetric regularized long wave equation. Acta Mathematica Scientia. Series A 2006,26(7):1039-1046.MathSciNetMATH
        15. Wang TC, Zhang LM, Chen FQ: Pseudo-compact conservative finite difference approximate solutions for symmetric regularized-long-wave equations. Chinese Journal of Engineering Mathematics 2008,25(1):169-172.MathSciNetMATH
        16. Shang Y, Guo B, Fang S: Long time behavior of the dissipative generalized symmetric regularized long wave equations. Journal of Partial Differential Equations 2002,15(1):35-45.MathSciNetMATH
        17. Shang YD, Guo B: Global attractors for a periodic initial value problem for dissipative generalized symmetric regularized long wave equations. Acta Mathematica Scientia. Series A 2003,23(6):745-757.MathSciNet
        18. Guo B, Shang Y: Approximate inertial manifolds to the generalized symmetric regularized long wave equations with damping term. Acta Mathematicae Applicatae Sinica 2003,19(2):191-204. 10.1007/s10255-003-0095-1MathSciNetView ArticleMATH
        19. Shang Y, Guo B: Exponential attractor for the generalized symmetric regularized long wave equation with damping term. Applied Mathematics and Mechanics 2005,26(3):259-266.MathSciNet
        20. Shaomei F, Boling G, Hua Q: The existence of global attractors for a system of multi-dimensional symmetric regularized wave equations. Communications in Nonlinear Science and Numerical Simulation 2009,14(1):61-68. 10.1016/j.cnsns.2007.07.001MathSciNetView ArticleMATH
        21. Hu B, Xu Y, Hu J: Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation. Applied Mathematics and Computation 2008,204(1):311-316. 10.1016/j.amc.2008.06.051MathSciNetView ArticleMATH

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