# C-N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

Authors: Jinsong Hu; Bing Hu; Youcai Xu
Journal: Mathematical Problems in Engineering
 (2011)

---

## Abstract

We study the initial-boundary problem of dissipative symmetric regularized
long wave equations with damping term. Crank-Nicolson nonlinear-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 the theoretical analysis.


---

## Body

## 1. Introduction

A symmetric version of regularized long wave equation (SRLWE),(1.1)uxxt-ut=ρx+uux,ρt+ux=0,
has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [1]. The sech2 solitary wave solutions are (1.2)u(x,t)=3(v2-1)vsech212v2-1v2(x-vt),ρ(x,t)=3(v2-1)v2sech212v2-1v2(x-vt).
The four invariants and some numerical results have been obtained in [1], where v is the velocity, v2>1. Obviously, eliminating ρ from (1.1), we get a class of SRLWE(1.3)utt-uxx+12(u2)xt-uxxtt=0.
Equation (1.3) is explicitly symmetric in the x and t 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 well-posedness 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 [10–15]).Because of gravity and resistance of propagation medium and air, the principle of dissipation must be considered when studying the move of nonlinear wave. 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(1.4)uxxt-ut+υuxx=ρx+uux,ρt+ux+γρ=0,
where υ,γ are positive, υ>0 is the dissipative coefficient, γ>0 is the damping coefficient. Equation (1.4) are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered (see [16–20]). Existence, uniqueness and well-posedness of global solutions to (1.4) are presented (see [16–20]). But it is difficult to find the analytical solution to (1.4), which makes numerical solution important.In this paper, we study (1.4) with(1.5)u(x,0)=u0(x),ρ(x,0)=ρ0(x),x∈[xL,xR]
and the boundary conditions(1.6)u(xL,t)=u(xR,t)=0,ρ(xL,t)=ρ(xR,t)=0,t∈[0,T].
In [21] we proposed a three-level implicit finite difference scheme to (1.4)–(1.6) with second-order convergence. But the three-level 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 u1,ρ1. Then, reusing initial vale u0,ρ0, we can work out u2,ρ2,u3,ρ3,…. Since the form of SRLW equations is similar ro the Rosenau equation and Rosenau-Burgers equation, the established difference schemes in [22, 23] for solving Rosenau equation and Rosenau-Burgers equation are helpful to investigate the SRLWEs. We propose the Crank-Nicolson finite difference scheme for (1.4)–(1.6) which can start by itself. We will show that this difference scheme is uniquely solvable, convergent and stable in both theoretical and numerical senses.


## 2. Finite Difference Scheme and Its Error Estimation

Leth and τ be the uniform step size in the spatial and temporal direction, respectively. Denote xj=xL+jh(j=0,1,2,…,J), tn=nτ(n=0,1,2,…,N), N=[T/τ], ujn≈u(xj,tn),  ρjn≈ρ(xj,tn) and Zh0={u=(uj)∣u0=uJ=0, j=0,1,2,…,J}. Throughout this paper, we will denote C as a generic constant independent of h and τ that varies in the context. We define the difference operators, inner product, and norms that will be used in this paper as follows: (2.1)(ujn)x=uj+1n-ujnh,(ujn)x¯=ujn-uj-1nh,(ujn)x̂=uj+1n-uj-1n2h,(ujn)t=ujn+1-ujnτ,ujn+1/2=ujn+1+ujn2,〈un,vn〉=h∑j=1J-1ujnvjn,‖un‖2=〈un,un〉,‖un‖∞=max⁡1≤j≤J-1|ujn|.Then, the Crank-Nicolson finite difference scheme for the solution of (1.4)–(1.6) is as follows:(2.2)(ujn)t-(ujn)xx¯t+(ρjn+1/2)x̂-υ(ujn+1/2)xx¯+13(uj+1n+1/2+ujn+1/2+uj-1n+1/2)(ujn+1/2)x̂=0,(2.3)(ρjn)t+(ujn+1/2)x̂+γρjn+1/2=0,(2.4)uj0=u0(xj),ρj0=ρ0(xj),1≤j≤J-1,(2.5)u0n=uJn=0,ρ0n=ρJn=0,1≤n≤N.Lemma 2.1.
It follows summation by parts [12, 23] that for any two discrete functions u,v∈Zh0,
(2.6)〈ux,v〉=-〈u,vx¯〉,〈uxx¯,v〉=-〈ux,vx〉.Lemma 2.2 (Discrete Sobolev's inequality [12, 23]).
There exist two constantsC1 and C2 such that
(2.7)‖un‖∞≤C1‖un‖+C2‖uxn‖.Lemma 2.3 (Discrete Gronwall’s inequality [12, 23]).
Supposew(k), ρ(k) are nonnegative function and ρ(k) is nondecreasing. If C>0 and w(k)≤ρ(k)+Cτ∑l=0k-1w(l), then
(2.8)w(k)≤ρ(k)eCτk.Theorem 2.4.
Ifu0∈H1, ρ0∈L2, then the solution of (2.2)–(2.5) satisfies
(2.9)‖un‖≤C,‖uxn‖≤C,‖ρn‖≤C,‖un‖∞≤C,(n=1,2,…,N).Proof.
Taking an inner product of (2.2) with 2un+1/2 (i.e., un+1+un) and considering the boundary condition (2.5) and Lemma 2.1, we obtain
(2.10)1τ(‖un+1‖2-‖un‖2)+1τ(‖uxn+1‖2-‖uxn‖2)+〈ρx̂n+1/2,2un+1/2〉-υ〈uxx¯n+1/2,2un+1/2〉+〈Pn+1/2,2un+1/2〉=0,
where Pjn+1/2=(1/3)(uj+1n+1/2+ujn+1/2+uj-1n+1/2)·(ujn+1/2)x̂.
Since(2.11)〈ρx̂n+1/2,2un+1/2〉=-2〈ux̂n+1/2,ρn+1/2〉,〈uxx¯n+1/2,2un+1/2〉=-2‖uxn+1/2‖2,(2.12)〈Pn+1/2,2un+1/2〉=2h∑j=1J-1Pjn+1/2ujn+1/2=23h∑j=1J-1(uj+1n+1/2+ujn+1/2+uj-1n+1/2)⋅(ujn+1/2)x̂ujn+1/2=13∑j=1J-1(uj+1n+1/2+ujn+1/2+uj-1n+1/2)⋅(uj+1n+1/2-uj-1n+1/2)ujn+1/2=13∑j=1J-1(uj+1n+1/2+ujn+1/2)uj+1n+1/2ujn+1/2-13∑j=1J-1(ujn+1/2+uj-1n+1/2)uj-1n+1/2ujn+1/2=0,
we obtain
(2.13)1τ(‖un+1‖2-‖un‖2)+1τ(‖uxn+1‖2-‖uxn‖2)-2〈ux̂n+1/2,ρn+1/2〉+2υ‖uxn+1/2‖2=0.
Taking an inner product of (2.3) with 2ρn+1/2 (i.e., ρn+1+ρn) and considering the boundary condition (2.5) and Lemma 2.1, we obtain
(2.14)1τ(‖ρn+1‖2-‖ρn‖2)+〈ux̂n+1/2,2ρn+1/2〉+2γ‖ρn+1/2‖2=0.
Adding (2.13) to (2.14), we have
(2.15)(‖un+1‖2-‖un‖2)+(‖uxn+1‖2-‖uxn‖2)+(‖ρn+1‖2-‖ρn‖2)=-2τ(υ‖uxn+1/2‖2+γ‖ρn+1/2‖2)≤0,
which implies
(2.16)‖un‖2+‖uxn‖2+‖ρn‖2≤‖un-1‖2+‖uxn-1‖2+‖ρn-1‖2≤⋯≤‖u0‖2+‖ux0‖2+‖ρ0‖2=C.
Then, it holds
(2.17)‖un‖≤C,‖uxn‖≤C,‖ρn‖≤C.
By Lemma 2.2, we obtain ∥un∥∞≤C.Theorem 2.5.
Assume thatu0∈H2, ρ0∈H1, the solution of difference scheme (2.2)–(2.5) satisfies:
(2.18)‖ρxn‖≤C,‖uxxn‖≤C,‖uxn‖≤C,‖ρn‖∞≤C,(n=1,2,…,N).Proof.
Differentiating backward (2.2)–(2.5) with respect to x, we obtain
(2.19)(ujn)xt-(ujn)xxx¯t+(ρjn+1/2)xx̂-υ(ujn+1/2)xxx¯+(Pjn+1/2)x=0,(2.20)(ρjn)xt+(ujn+1/2)xx̂+γ(ρjn+1/2)x=0,(2.21)(uj0)x=u0,x(xj),(ρj0)x=ρ0,x(xj),1≤j≤J-1,(2.22)(u0n)x=(uJn)x=0,(ρ0n)x=(ρJn)x=0,1≤n≤N,
where (Pjn+1/2)x=(1/3)[(uj+1n+1/2+ujn+1/2+uj-1n+1/2)(ujn+1/2)x̂]x.
Computing the inner product of (2.19) with 2uxn+1/2 (i.e., uxn+1+uxn) and considering (2.22) and Lemma 2.1, we obtain(2.23)1τ(‖uxn+1‖2-‖uxn‖2)+1τ(‖uxxn+1‖2-‖uxxn‖2)+〈ρxx̂n+1/2,2uxn+1/2〉-υ〈uxxx¯n+1/2,2uxn+1/2〉+〈Pxn+1/2,2uxn+1/2〉=0.
It follows from Theorem 2.4 that
(2.24)|uj+1n+1/2+ujn+1/2+uj-1n+1/2|≤C,(j=0,1,2,…,J).
By the Schwarz inequality and Lemma 2.1, we get
(2.25)〈Pxn+1/2,2uxn+1/2〉=-2〈Pn+1/2,uxx¯n+1/2〉=-23h∑j=1J-1(uj+1n+1/2+ujn+1/2+uj-1n+1/2)⋅(ujn+1/2)x̂(ujn+1/2)xx¯≤Ch∑j=1J-1|(ujn+1/2)x̂|⋅|(ujn+1/2)xx¯|≤C(‖uxn+1/2‖2+‖uxxn+1/2‖2)≤C(‖uxn+1‖2+‖uxn‖2+‖uxxn+1‖2+‖uxxn‖2).
By
(2.26)〈ρxx̂n+1/2,2uxn+1/2〉=-2〈uxx̂n+1/2,ρxn+1/2〉,〈uxxx¯n+1/2,2uxn+1/2〉=-2‖uxxn+1/2‖2,
it follows from (2.23) that
(2.27)(‖uxn+1‖2-‖uxn‖2)+(‖uxxn+1‖2-‖uxxn‖2)-2τ〈uxx̂n+1/2,ρxn+1/2〉≤-2υτ‖uxxn+1/2‖2+Cτ(‖uxn+1‖2+‖uxn‖2+‖uxxn+1‖2+‖uxxn‖2).
Computing the inner product of (2.20) with 2ρxn+1/2 (i.e., ρxn+1+ρxn) and considering (2.22) and Lemma 2.1, we obtain
(2.28)(‖ρxn+1‖2-‖ρxn‖2)+2τ〈uxx̂n+1/2,ρxn+1/2〉+2γτ‖ρxn+1/2‖2=0.
Adding (2.28) to (2.27),we have
(2.29)(‖uxn+1‖2-‖uxn‖2)+(‖uxxn+1‖2-‖uxxn‖2)+(‖ρxn+1‖2-‖ρxn‖2)≤-2υτ‖uxxn+1/2‖2-2γτ‖ρxn+1/2‖2+Cτ(‖uxn+1‖2+‖uxn‖2+‖uxxn+1‖2+‖uxxn‖2)≤Cτ(‖uxn+1‖2+‖uxn‖2+‖uxxn+1‖2+‖uxxn‖2+‖ρxn+1‖2+‖ρxn‖2).
Let An=∥uxn∥2+∥uxxn∥2+∥ρxn∥2, we obtain An+1-An≤Cτ(An+1+An).
Choosing suitableτ which is small enough to satisfy 1-Cτ>0, we get(2.30)An+1-An≤CτAn.
Summing up (2.30) from 0 to n-1, we have
(2.31)An≤A0+Cτ∑l=0n-1Al.
By Lemma 2.3, we get An≤C, which implies ∥ρxn∥≤C, ∥uxxn∥≤C. It follows from Theorem 2.4 and Lemma 2.2 that ∥uxn∥∞≤C, ∥ρn∥∞≤C.


## 3. Solvability, Convergence, and Stability

The following Brouwer fixed point theorem will be needed in order to show the existence of solution for (2.2)–(2.5). For the proof, see [24].Lemma 3.1 (Brouwer fixed point theorem).
LetH be a finite dimensional inner product space, suppose that g:H→H is continuous and there exists an α>0 such that <g(x),x>0 for all x∈H with ∥x∥=α. Then there exists x*∈H such that g(x*)=0 and ∥x*∥≤α.LetZΔ={v=(v1,v2)=(v1,j,v2,j)∣v1,0=v1,J=v2,0=v2,J=0, j=0,1,2,…,J}, equipped with the inner product 〈v,v′〉=〈(v1,v2),(v1′,v2′)〉=〈v1,v1′〉+〈v2,v2′〉 and the norm ∥v∥2=∥v1∥2+∥v2∥2.Theorem 3.2.
There exists(un,ρn)∈ZΔ which satisfies the difference scheme (2.2)–(2.5).Proof.
In order to prove the theorem by the mathematical induction, we assume that(u0,ρ0),(u1,ρ1),…,(un,ρn)∈ZΔ satisfying (2.2)–(2.5). Next prove there exists (un+1,ρn+1) which satisfies (2.2)–(2.5).
Letg=(g1,g2) be a operator on ZΔ defined by(3.1)g1(v)=2v1-2un-2v1xx¯+2uxx¯n+τv2x̂-υτv1xx¯+τW,g2(v)=2v2-2ρn+τv1x̂+γτv2,∀v=(v1,v2)∈ZΔ,
where Wj=(1/3)(v1,j+1+v1,j+v1,j-1)(v1,j)x̂. Computing the inner product of (3.1) with v=(v1,v2), similarly to (2.11) and (2.12), we obtain
(3.2)〈v2x̂,v1〉=-〈v1x̂,v2〉,〈W,v1〉=0.
By (2.5) and the Schwarz inequality, we obtain
(3.3)〈g(v),v〉=〈g1(v),v1〉+〈g2(v),v2〉=2‖v1‖2-2〈un,v1〉+2‖v1x‖2-2〈uxn,v1x〉+υτ‖v1x‖2+2‖v2‖2-2〈ρn,v2〉+γτ‖v2‖2≥2‖v1‖2-(‖un‖2+‖v1‖2)+2‖v1x‖2-(‖uxn‖2+‖v1x‖2)+υτ‖v1x‖2+2‖v2‖2-(‖ρn‖2+‖v2‖2)+γτ‖v2‖2=‖v1‖2+‖v2‖2+‖v1x‖2-‖un‖2-‖uxn‖2-‖ρn‖2+υτ‖v1x‖2+γτ‖v2‖2≥‖v‖2-(‖un‖2+‖uxn‖2+‖ρn‖2).
Hence it is obvious that <g(v),v>0 for all v∈ZΔ with ∥v∥2=(∥un∥2+∥uxn∥2+∥ρn∥2)+1. It follows from Lemma 3.1 that there exists v*∈ZΔ such that g(v*)=0. If we take un+1=2v1-un, ρn+1=2v2*-ρn, then (un+1,ρn+1) satisfies (2.2)–(2.5). This completes the proof.Next we show that the difference scheme (2.2)–(2.5) is convergent and stable.Letv(x,t) and ø(x,t) be the solution of problem (1.4)–(1.6), that is, vjn=u(xj,tn),  øjn=ρ(xj,tn), then the truncation of the difference scheme (2.2)–(2.5) is(3.4)rjn=(vjn)t-(vjn)xx¯t+(øjn+1/2)x̂-υ(vjn+1/2)xx¯+13(vj+1n+1/2+vjn+1/2+vj-1n+1/2)(vjn+1/2)x̂,(3.5)sjn=(øjn)t+(vjn+1/2)x̂+γøjn+1/2.
Making use of Taylor expansion, we know that |rjn|+|sjn|=O(τ2+h2) hold if h, τ→0.Lemma 3.3.
Suppose thatu0∈H1, ρ0∈L2, the solution of (1.4)–(1.6) satisfies ∥u∥L2≤C, ∥ux∥L2≤C, ∥ρ∥L2≤C and ∥u∥L∞≤C.Proof.
See Lemma 1.1 in [21].Theorem 3.4.
Supposeu0∈H1, ρ0∈L2, then the solution un and ρn to the difference scheme (2.2)–(2.5) converges to the solution of problem (1.4)–(1.6), and the rate of convergence is O(τ2+h2).Proof.
Subtracting (2.2) from (3.4) and subtracting (2.3) from (3.5), letting ejn=vjn-ujn, ηjn=ϕjn-ρjn, we have
(3.6)rjn=(ejn)t-(ejn)xx¯t+(ηjn+1/2)x̂-υ(ejn+1/2)xx¯+Qjn+1/2,(3.7)sjn=(ηjn)t+(ejn+1/2)x̂+γηjn+1/2,
where
(3.8)Qjn+1/2=13(vj+1n+1/2+vjn+1/2+vj-1n+1/2)⋅(vjn+1/2)x̂-13(uj+1n+1/2+ujn+1/2+uj-1n+1/2)⋅(ujn+1/2)x̂.
Computing the inner product of (3.6) with 2en+1/2 we get
(3.9)〈rn,2en+1/2〉=1τ(‖en+1‖2-‖en‖2)+1τ(‖exn+1‖2-‖exn‖2)+2υ‖exn+1/2‖2+〈ηx̂n+1/2,2en+1/2〉+〈Qn+1/2,2en+1/2〉.
Similarly to (2.11), we have
(3.10)〈ηx̂n+1/2,2en+1/2〉=-2〈ex̂n+1/2,ηn+1/2〉.
Then (3.9) can be changed to
(3.11)(‖en+1‖2-‖en‖2)+(‖exn+1‖2-‖exn‖2)-2τ〈ex̂n+1/2,ηn+1/2〉=-2υτ‖exn+1/2‖2+τ〈rn,2en+1/2〉+τ〈-Qn+1/2,2en+1/2〉.
It follow from Lemma 3.3, Theorems 2.4 and 2.5 that
(3.12)|vj+1n+1/2+vjn+1/2+vj-1n+1/2|≤C,|(ujn+1/2)|≤C,(j=0,1,2,…,J).
Since(3.13)〈-Qn+1/2,2en+1/2〉=-23h∑j=1J-1(vj+1n+1/2+vjn+1/2+vj-1n+1/2)(vjn+1/2)x̂ejn+1/2+23h∑j=1J-1(uj+1n+1/2+ujn+1/2+uj-1n+1/2)(ujn+1/2)x̂ejn+1/2=-23h∑j=1J-1(vj+1n+1/2+vjn+1/2+vj-1n+1/2)(ejn+1/2)x̂ejn+1/2-23h∑j=1J-1(ej+1n+1/2+ejn+1/2+ej-1n+1/2)(ujn+1/2)x̂ejn+1/2,
and the Schwarz inequality, we obtain
(3.14)〈-Qn+1/2,2en+1/2〉≤23Ch∑j=1J-1|(ejn+1/2)x̂|⋅|ejn+1/2|+23Ch∑j=1J-1(|ej+1n+1/2|+|ejn+1/2|+|ej-1n+1/2|)|ejn+1/2|≤C(‖en+1/2‖2+‖exn+1/2‖2)≤C(‖en+1‖2+‖en‖2+‖exn+1‖2+‖exn‖2).
According to
(3.15)〈rn,2en+1/2〉=〈rn,en+1+en〉≤‖rn‖2+12[‖en+1‖2+‖en‖2].
It follows from (3.14), (3.15), and (3.11) that
(3.16)(‖en+1‖2-‖en‖2)+(‖exn+1‖2-‖exn‖2)-2τ〈ex̂n+1/2,ηn+1/2〉≤Cτ[‖en+1‖2+‖en‖2+‖exn+1‖2+‖exn‖2]+τ‖rn‖2.
Computing the inner product of (3.7) with 2ηn+1/2, we obtain
(3.17)(∥ηn+1∥2-∥ηn∥2)+2τ〈ex̂n+1/2,ηn+1/2〉=τ〈sn,2ηn+1/2〉-2γτ‖ηn+1/2‖2≤Cτ[‖ηn+1‖2+‖ηn‖2]+τ‖sn‖2.
Adding (3.17) to (3.16) we have
(3.18)(‖en+1‖2-‖en‖2)+(‖exn+1‖2-‖exn‖2)+‖ηn+1‖2-‖ηn‖2≤τ‖rn‖2+τ‖sn‖2+Cτ[‖en+1‖2+‖en‖2+‖exn+1‖2+‖exn‖2+‖ηn+1‖2+‖ηn‖2].
Let Bn=∥en∥2+∥exn∥2+∥ηn∥2, we get
(3.19)Bn+1-Bn≤τ‖rn‖2+τ‖sn‖2+Cτ(Bn+1+Bn).
If τ is sufficiently small which satisfies 1-Cτ>0, then
(3.20)Bn+1-Bn≤CτBn+Cτ‖rn‖2+Cτ‖sn‖2.
Summing up (3.20) from 0 to n-1, we have
(3.21)Bn≤B0+Cτ∑l=0n-1‖rl‖2+Cτ∑l=0n-1‖sl‖2+Cτ∑l=0n-1Bl.
Since
(3.22)τ∑l=0n-1‖rl‖2≤nτmax⁡0≤l≤n-1‖rl‖2≤T⋅O(τ2+h2)2,τ∑l=0n-1‖sl‖2≤nτmax⁡0≤l≤n-1‖sl‖2≤T⋅O(τ2+h2)2,
and B0=O(τ2+h2)2, we obtain
(3.23)Bn≤O(τ2+h2)2+Cτ∑l=0n-1Bl.
From Lemma 2.3, we get
(3.24)Bn≤O(τ2+h2)2,
which implies
(3.25)‖en‖≤O(τ2+h2),‖exn‖≤O(τ2+h2),‖ηn‖≤O(τ2+h2).
Using Lemma 2.2, we get
(3.26)‖en‖∞≤O(τ2+h2).Similarly to Theorem3.4, we can prove the results as follows.Theorem 3.5.
Under the conditions of Theorem3.4, the solution un and ρn of (2.2)–(2.5) is stable in the senses of norm ∥·∥∞ and ∥·∥L2, respectively.Theorem 3.6.
The solutionun of (2.2)–(2.5) is unique.


## 4. Numerical Simulations

The difference scheme (2.2)–(2.5) is a nonlinear system about ujn+1 that can be easily solved by Newton iterative algorithm. When t=0, the damping does not effect and the dissipative term will not appear. So the initial conditions (1.4)–(1.6) are same as those of (1.1): (4.1)u0(x)=52sech256x,ρ0(x)=53sech256x,(v=1.5).
Let xL=-20, xR=20, T=5.0. Since we do not know the exact solution of (1.4), an error estimates method in [23] 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  τ=h=1/160 as the reference solution. We denote the C-N scheme in this paper as Scheme I and the difference scheme in [21] as Scheme II. In Tables 1 and 2 we give the ratios in the sense of l∞ at various time step when υ=γ=0.2 and υ=γ=0.5, respectively.Table 1
The  error comparison in the sense ofl∞ at various time step when υ=γ=0.2.
τ=h=0.1τ=h=0.05τ=h=0.025Scheme  IScheme  IIScheme  IScheme  IIScheme  IScheme  IIut=12.051828e-33.031228e-35.086997e-47.715042e-41.212062e-41.963616e-4t=23.658861e-34.444161e-39.062598e-41.130545e-32.159826e-42.891308e-4t=34.659523e-35.248700e-31.154171e-31.332236e-32.749807e-43.363167e-4t=45.230463e-35.817171e-31.295681e-31.476326e-33.087002e-43.736609e-4t=55.509947e-36.570922e-31.365011e-31.660745e-33.252241e-44.213116e-4ρt=11.672623e-31.981718e-34.146233e-45.009105e-49.882705e-51.274281e-4t=22.775247e-33.231087e-36.880969e-48.110353e-41.640128e-42.074091e-4t=33.619022e-34.495888e-38.971716e-41.144555e-32.138326e-42.912403e-4t=44.150387e-35.169257e-31.028962e-31.314334e-32.452495e-43.329702e-4t=54.434692e-35.792717e-31.100089e-31.466326e-32.621296e-43.716609e-4Table 2
The  error comparison in the sense ofl∞ at various time step when υ=γ=0.5.
τ=h=0.1τ=h=0.05τ=h=0.025Scheme  IScheme  IIScheme  IScheme  IIScheme  IScheme  IIut=11.795244e-32.268843e-34.445000e-45.685554e-41.059029e-41.454071e-4t=22.544783e-33.102494e-36.298484e-47.925500e-41.500033e-42.032788e-4t=33.422326e-33.705244e-38.474794e-49.482211e-42.018808e-42.430733e-4t=43.779389e-34.207447e-39.355479e-41.079082e-32.223954e-42.799107e-4t=54.151289e-34.729322e-31.027876e-31.216704e-32.453155e-43.162708e-4ρt=11.303459e-31.963695e-33.234571e-45.083883e-47.706741e-51.294953e-4t=21.868933e-32.928023e-34.633380e-47.506240e-41.104061e-41.907795e-4t=32.462890e-33.453731e-36.110918e-48.823517e-41.453867e-42.266268e-4t=43.016856e-34.169390e-37.518839e-41.066308e-31.813333e-42.721379e-4t=53.446674e-34.513283e-38.743460e-41.141807e-32.126268e-42.918805e-4In Tables3 and 4 we verify the second convergence of the scheme I using the method in [25] when υ=γ=0.2 and υ=γ=0.5, respectively.Table 3
The verification of the second convergence whenυ=γ=0.2.
∥en(h,τ)∥∞/∥e2n(h/2,τ/2)∥∞∥ηn(h,τ)∥∞/∥η2n(h/2,τ/2)∥∞τ=h=0.1τ=h=0.05τ=h=0.025τ=h=0.1τ=h=0.05τ=h=0.025t=1—4.033474.19698—4.034084.19544t=2—4.037324.19596—4.033224.19539t=3—4.037114.19728—4.033814.19567t=4—4.036844.19721—4.033574.19557t=5—4.036564.19714—4.031214.19674Table 4
The verification of the second convergence whenυ=γ=0.5.
∥en(h,τ)∥∞/∥e2n(h/2,τ/2)∥∞∥ηn(h,τ)∥∞/∥η2n(h/2,τ/2)∥∞τ=h=0.1τ=h=0.05τ=h=0.025τ=h=0.1τ=h=0.05τ=h=0.025t=1—4.038794.19724—4.029774.19707t=2—4.040314.19807—4.033264.19667t=3—4.038244.19792—4.030314.20322t=4—4.039764.20669—4.012404.14642t=5—4.038714.19002—3.942004.11211Whenυ=γ=0.2 and υ=γ=0.5, a wave figure comparison of u and ρ at various time step with τ=h=0.05 is as follow: (see Figures 1, 2, 3, and 4).Figure 1
Whenυ=γ=0.2, the wave graph of u at various time.Figure 2
Whenυ=γ=0.2, the wave graph of ρ at various time.Figure 3
Whenυ=γ=0.5, the wave graph of u at various time.Figure 4
Whenυ=γ=0.5, the wave graph of ρ at various time.


## 5. Conclusion

In this paper, we propose Crank-Nicolson nonlinear-implicit finite difference scheme of the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. The two-levels finite difference scheme is of second-order convergence and unconditionally stable, which can start by itself. From the Tables1 and 2 we conclude that the C-N scheme is more efficient than the Scheme 2 in [21]. From the Tables 3 and 4 we conclude that the C-N scheme is of second-order convergence obviously. Figures 1–4 show that the height of wave crest is more and more low with time elapsing due to the effect of damping term and dissipative term and when υ, γ become bigger the droop of the height of wave crest is faster.



---

*Source: 651642-2011-04-19.xml*