A finite difference scheme for generalized regularized long-wave equation☆
Introduction
The generalized regularized long-wave (GRLW) equationwhere 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.,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,It has the following conservation lawUsing 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) iswhereand 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 usedwhere h and τ are the spatial and temporal step sizes, respectively, and xj = jh, tn = nτ. 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 . Then we haveAccording to Taylor’s expansion, it can be easily obtained that linear part of (13) at point (xj,tn) satisfiesThe last term of (13) can be written as
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 or take . In addition, let x0 = 0 in (6), , and consider three cases: p = 4, p = 4 and p = 8. Thus, the system (7), (8) can be rewritten aswhere
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This work is supported by NSF grant of China (No. 10471023).