A symmetric version of regularized long wave equation (SRLWE),

has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [

1]. The

solitary wave solutions are

The four invariants and some numerical results have been obtained in [

1], where

is the velocity,

. Obviously, eliminating

from (1.1), we get a class of SRLWE:

Equation (1.3) is explicitly symmetric in the
and
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 [4–6]. 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 [9–15]).

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

where
are positive constants,
is the dissipative coefficient, and
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 [16–20]). 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

and the boundary conditions

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

Lemma 1.1.

Suppose that
,
, the solution of (1.4)–(1.7) satisfies
,
,
, and
, where
is a generic positive constant that varies in the context.

Proof.

Multiplying (1.4) by

and integrating over

, we have

Then, multiplying (1.5) by

and integrating over

, we have

Adding (1.14) to (1.11), we obtain

So
is decreasing with respect to
, which implies that
,
. Then, it indicates that
,
, and
. It is followed from Sobolev inequality that
.