A finite difference scheme for generalized regularized long-wave equation

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Abstract

In this paper, a finite difference method for a Cauchy problem of generalized regularized long-wave (GRLW) equation was considered. An energy conservative finite difference scheme was proposed. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

Introduction

The generalized regularized long-wave (GRLW) equationut+ux+α(up)x-βuxxt=0,where p is a positive integer, and α and β are positive constants, was first put forward as a model for small-amplitude long-waves on the surface of water in a channel by Peregrine [1], [2]. A special case of (1), i.e.,ut+ux+uux-βuxxt=0,which is usually called the regularized long-wave (RLW) equation, has been studied by many authors. Mathematical theory and numerical methods for (2) was considered in [3], [4], [5], [6], [7], [8]. However, a few study have been done for the Eq. (1). Recently, Dogˇan Kaya [9] gives the exact solitary-solutions of GRLW equation (1), and construct numerical solutions using ADM (Adomian decomposition method) without using any discretization technique.

In this paper, we consider the following initial value problem of generalized regularized long-wave equation,ut+ux+α(up)x-βuxxt=0,u|t=0=u0(x).It has the following conservation lawE(t)=uL22+βuxL22=const.Using a customary designation, we shall refer to the functional E(t) as the energy integral, although it is not necessarily identifiable with energy in the original physical problem. The single solitary wave solution of (3) isu(x,t)=A·sech2p-1[k(x+x0-σt)],whereA=(p+1)(σ-1)2α1p-1,k=p-12βσ-1σ,and σ, x0 are arbitrary constants and p  2.

The purpose of this paper is to present a conservative finite difference scheme for the initial value problem (3), (4), which simulates conservation law (5) that the differential Eq. (3) possesses, and proof convergence and stability of the scheme. An outline of the paper is as follows. In Section 2, a conservative finite difference scheme for the initial value problem (3), (4). In Section 3, convergence and stability of the scheme are proved. Numerical experiments are reported in Section 4.

Section snippets

Finite difference scheme and conservation law

As usual, the following notations will be used(ujn)x=uj+1n-ujnh,(ujn)x¯=ujn-uj-1nh,(ujn)xˆ=uj+1n-uj-1n2h,(ujn)tˆ=ujn+1-ujn-12τ,(un,vn)=hjujnvjn,un2=(un,un),un=supj|ujn|,where h and τ are the spatial and temporal step sizes, respectively, and xj = jh, tn = . Superscript n denotes a quantity associated with the time-level tn and subscript j denotes a quantity associated with space mesh point xj. In this paper, C denote general constant, which may have different value in different place.

Now we

Convergence and stability of finite difference scheme

First, we consider the truncation error of the finite difference scheme (7), (8). Suppose vjn=u(xj,tn). Then we haveErjn=(vjn)tˆ+1-θ2[(vjn+1)xˆ+(vjn-1)xˆ]+θ(vjn)xˆ-β(vjn)xx¯tˆ+pα2(p+1){(vjn)p-1((vjn+1)xˆ+(vjn-1)xˆ)+[(vjn)p-1(vjn+1+vjn-1)]xˆ}.According to Taylor’s expansion, it can be easily obtained that linear part of (13) at point (xj,tn) satisfies(vjn)tˆ+1-θ2[(vjn+1)xˆ+(vjn-1)xˆ]+θ(vjn)xˆ-β(vjn)xx¯tˆ=(vt+vx-βvxxt)|(xj,tn)+O(h2+τ2).The last term of (13) can be written asQ=pα2(p+1){(vjn)p-1((vj

Algorithm and numerical experiments

In the numerical experiments, we solve the problem (3), (4) in [−50,50]. According to solitary wave solution (6), we add boundary conditions u0n=ujn=0 or take u0n=u(-50,tn),uJn=u(50,tn). In addition, let x0 = 0 in (6), α=12,β=1,u0(x)=Asech2p-1(kx), and consider three cases: p = 4, p = 4 and p = 8. Thus, the system (7), (8) can be rewritten asAjnuj-1n+1+Bjnujn+1+Cjnuj+1n+1=Djn(j=1,,J-1,n=1,,N-1),u0n=uJn=0(n=0,1,,N),uj0=u0(jh)(j=1,,J),whereAjn=-1-θ2λ-1h2-p4(p+1)λ[(ujn)p-1+(uj-1n)p-1](j=1,,J-1),Bjn=1+

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This work is supported by NSF grant of China (No. 10471023).

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