# Conservative Difference Scheme for Generalized Rosenau-KdV Equation

Authors: Yan Luo; Youcai Xu; Minfu Feng
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

A conservative Crank-Nicolson finite difference scheme for the initial-boundary
value problem of generalized Rosenau-KdV equation is proposed. The difference scheme
shows a discrete analogue of the main conservation law associated to the equation. On the other
hand the scheme is implicit and stable with second order convergence. Numerical experiments
verify the theoretical results.


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## Body

## 1. Introduction

In this paper, we consider the following initial-boundary value problem of the generalized Rosenau-KdV equation:(1)ut+uxxxxt+ux+(up)x+uxxx=0,
with an initial condition
(2)u(x,0)=u0(x),x∈[xL,xR],
and boundary conditions
(3)u(xL,t)=u(xR,t)=0,ux(xL,t)=ux(xR,t)=0,uxx(xL,t)=uxx(xR,t)=0,dddddddddddddt∈[0,T],
where p≥2 is a integer and u0(x) is a known smooth function. When p=2, (1) is called usual Rosenau-KdV equation:
(4)ut+uxxxxt+ux+uux+uxxx=0.Zuo [1] discussed the solitary wave solutions and periodic solutions for Rosenau-KdV equation. In [2], a conservative nonlinear finite difference scheme for an initial-boundary value problem of Rosenau-Kdv equation is considered.In [3, 4] the solitary solution and invariant for generalized Rosenau-KdV equation are given. In [4] the singular 1-soliton solution is derived by the ansatz method, and the adiabatic parameter dynamics of the water waves is obtained by perturbation theory. In [5, 6], the ansatz method is applied to obtain the topological soliton solution of the generalized Rosenau-KdV equation. The method as well as the exp-function method is also applied to extract a few more solutions to this equation. In [7], Zheng and Zhou give an average linear scheme for the generalized Rosenau-KdV equation. In this paper, we propose a conservative Crank-Nicolson finite difference scheme for an initial-boundary value problem of the generalized Rosenau-Kdv equation.The initial-boundary value problem (1)–(3) possesses the following conservative property [3, 4]:
(5)E(t)=∥u∥L22+∥uxx∥L22=E(0).When-xL≫0, xR≫0, the initial-boundary value problem (1)–(3) and the Cauchy problem (1) are consistent, so that the boundary conditions (3) are reasonable.It is known that the conservative scheme is better than the nonconservative ones. The nonconservative scheme may easily show nonlinear blow-up. A lot of numerical experiments show that the conservative scheme can possess some invariant properties of the original differential equation [7–18]. The conservative scheme is more suitable for long-time calculations. In [18] Pan and Zhang said “… in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation”.The rest of this paper is organized as follows. In Section2, we propose a Crank-Nicolson implicit nonlinear finite difference scheme for the generalized Rosenau-KdV equation and discuss the property of its solution. In Section 3, we prove that the finite difference scheme is of second order convergence. Finally, some numerical tests are given in Section 4 to verify our theoretical analysis.


## 2. Finite Difference Scheme and Its Property

Leth=(xR-xL)/J and let τ be the uniform step size in the spatial and temporal direction, respectively.  Denote  xj=xL+jh  (j=-1,0,1,2,…,J,J+1),  tn=nτ  (n=0,1,2,…,N,N=[T/τ]),  ujn≈u(xj,tn), and Zh0={u=(uj)∣u-1=u0=uJ=uJ+1=0,j=-1,0,1,2,…,J,J+1}. Throughout this paper, we denote C as a generic positive constant independent of h and τ, which may have different values in different occurrences. We introduce the following notations:
(6)(ujn)x=uj+1n-ujnh,(ujn)x¯=ujn-uj-1nh,(ujn)x^=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∥.We propose a conservative Crank-Nicolson finite difference scheme for the solution of (1)–(3):
(7)(ujn)t+(ujn)xxx¯x¯t+(ujn+(1/2))x^+(ujn+(1/2))xx¯x^+pp+1(ujn+(1/2))p-1(ujn+(1/2))x^+[(ujn+(1/2))p]x^,(8)j=1,2,3,…,J-1;n=1,2,3,…,N-1,(9)uj0=u0(xj),j=0,1,2,3,…,J,(10)(u0n)x^=(uJn)x^=0(u0n)xx¯=(uJn)xx¯=0,dddddddddddun∈Zh0,n=1,2,3,…,N.
From the boundary conditions (3), we know that (10) is reasonable.Lemma 1.
It follows from summation by parts that, for any two mesh functionsu,v∈Zh0,
(11)〈ux,v〉=-〈u,vx〉,〈ux^,v〉=-〈u,vx^〉,〈uxx¯,v〉=-〈ux,vx〉.
Then one has
(12)〈uxx¯,u〉=-〈ux,ux〉=-∥ux∥2.
Furthermore, if (u0n)xx¯=(uJn)xx¯=0, then
(13)〈uxxx¯x¯,u〉=∥uxx∥2.To show the existence of the solution for (7)–(10), the following Brouwer fixed point theorem should be introduced. For the proof, see [19].Lemma 2.
LetH be a finite dimensional inner product space, let ∥·∥ be the associated norm, and let g:H→H be continuous. Assume, moreover, that there exists an α>0, for all x∈H and ∥x∥=α, 〈g(x),x〉>0. Then there exists x*∈H such that g(x*)=0 and ∥x*∥≤α.Then one has the following theorem.Theorem 3.
There existsun∈Zh0 which satisfies the difference scheme (7)–(10) (1≤n≤N).Proof.
In order to prove the theorem by the mathematical induction, we assume thatu0,u1,…,un which satisfy (7)–(10) exist for n≤N-1. Next prove that there also exists un+1 which satisfies (7)–(10).
We defineg on Zh0 as follows:
(14)g(v)=2v-2un+2vxxx¯x¯-2uxxx¯x¯n+τvx^+τvxx¯x^+τp1+p[vp-1vx^+(vp)x^].
Taking an inner product of (14) with v and considering
(15)〈vx^,v〉=0,〈vxx¯x^,v〉=0,〈vp-1vx^+(vp)x^,v〉=0,
we have
(16)〈g(v),v〉=2∥v∥2-2〈un,v〉+2∥vxx∥2-2〈uxxn,vxx〉≥2∥v∥2-2∥un∥·∥v∥+2∥vxx∥2-2∥uxxn∥·∥vxx∥≥2∥v∥2-(∥un∥2+∥v∥2)+2∥vxx∥2-(∥uxxn∥2+∥vxx∥2)≥∥v∥2-(∥un∥2+∥uxxn∥2)+∥vxx∥2≥∥v∥2-(∥un∥2+∥uxxn∥2).
Hence, it is obvious that〈g(v),v〉>0 for all  ∀v∈Zh0 with ∥v∥2=∥un∥2+∥uxxn∥2+1. It follows from Lemma 2 that there exists v*∈Zh0 such that g(v*)=0. Let un+1=2v*-un; then un+1 satisfies (7).The difference scheme (7)–(10) simulates the conservation property of the problem (1)–(3) as follows.Theorem 4.
Suppose thatu0∈H02[xL,xR], then the difference scheme (7)–(10) is conservative:
(17)En=∥un∥2+∥uxxn∥2=En-1=⋯=E0.Proof.
Taking an inner product of (7) with 2un+(1/2)(i.e., un+1+un), according to the boundary condition (10) and Lemma 1, we obtain
(18)1τ(∥un+1∥2-∥un∥2)+1τ(∥uxxn+1∥2-∥uxxn∥2)+2〈ux^n+(1/2),un+(1/2)〉+2〈uxx¯x^n+(1/2),un+(1/2)〉+2〈φ(un+(1/2)),un+(1/2)〉=0,
where φ(ujn+(1/2))=(p/(p+1))[(ujn+(1/2))p-1·(ujn+(1/2))x^+((ujn+(1/2))p)x^].
From(19)〈ux^n+(1/2),un+(1/2)〉=0,〈uxx¯x^n+(1/2),un+(1/2)〉=0,(20)〈φ(un+(1/2)),2un+(1/2)〉=2pp+1h∑j=1J-1[ujn+(1/2)+((ujn+(1/2))p)x^2pp+1(ujn+(1/2))p-1·(ujn+(1/2))x^ddddddddd+ujn+(1/2)+((ujn+(1/2))p)x^2pp+1]ujn+(1/2)=pp+1h∑j=1J-1(ujn+(1/2))p-1·(uj+1n+(1/2)-uj-1n+(1/2))+[(uj+1n+(1/2))p-(uj-1n+(1/2))p]ujn+(1/2)=pp+1∑j=1J-1‍[(uj+1n+(1/2))p-1ujn+(1/2)+(ujn+(1/2))p]·uj+1n+(1/2)-pp+1∑j=1J-1‍[(ujn+(1/2))p-1uj-1n+(1/2)+(uj-1n+(1/2))p]·ujn+(1/2)=0,
we have
(21)(∥un+1∥2-∥un∥2)+(∥uxxn+1∥2-∥uxxn∥2)=0.
Then (17) is gotten from (21).In order to prove the bounded quality of the difference solution, we introduce the following lemma.Lemma 5 (discrete Sobolev’s inequality [2]).
There exist two constantsC1 and C2 such that
(22)∥un∥∞≤C1∥un∥+C2∥uxn∥.Theorem 6.
Supposeu0∈H02[xL,xR]; then the solution of (7)–(10) satisfies
(23)∥un∥≤C,∥uxn∥≤C,
which yield
(24)∥un∥∞≤C,∥uxn∥∞≤C(n=1,2,…,N).Proof.
It follows from (17) that
(25)∥un∥≤C,∥uxxn∥≤C.
According to (12) and Schwarz inequality, we get
(26)∥uxn∥2≤∥un∥·∥uxxn∥≤12(∥un∥2+∥uxxn∥2)≤C.
Using Lemma 5, we have
(27)∥un∥∞≤C,∥uxn∥∞≤C.Remark 7.
Theorem6 implies that the solution of difference scheme (7)–(10) is stable in the sense of norm ∥·∥∞.


## 3. Convergence

In order to prove the convergence of the difference scheme, we need to introduce the lemma as follows:Lemma 8 (discrete Gronwall inequality [2]).
Supposew(k) and ρ(k) are nonnegative functions and ρ(k) is nondecreasing. If C>0 and
(28)w(k)≤ρ(k)+Cτ∑l=0k-1‍w(l),∀k,
then
(29)w(k)≤ρ(k)eCτk,∀k.Theorem 9.
Suppose thatu0∈H02[xL,xR]; then the solution un of (1)–(3) satisfies
(30)∥u∥L2≤C,∥ux∥L2≤C,∥u∥L∞≤C,∥ux∥L∞≤C.Proof.
It follows from (5) that
(31)∥u∥L2≤C,∥uxx∥L2≤C.
By Holder inequality and Schwarz inequality, we get
(32)∥ux∥L22=∫xLxR‍uxuxdx=uux|xLxR-∫xLxR‍uuxxdx=-∫xLxR‍uuxxdx≤∥u∥L2·∥uxx∥L2≤12(∥u∥L22+∥uxx∥L22),
which implies
(33)∥ux∥L2≤C.
Using Sobolev inequality, we get
(34)∥u∥L∞≤C,∥ux∥L∞≤C.Letv(x,t) be the solution of problem (1)–(3), vjn=v(xj,tn); then the truncation error of the difference scheme (7)–(10) is
(35)rjn=(vjn)t+(vjn)xxx¯x¯t+(vjn+(1/2))xx^+(vjn+(1/2))xx¯x^+φ(vjn+(1/2)).
Making use of Taylor expansion, we know that rjn=O(τ2+h2) holds if h,τ→0.Theorem 10.
Supposeu0∈H02[xL,xR]; then the solution un of the difference scheme (7)–(10) converges to the solution v(x,t) of the problem (1)–(3) with order O(τ2+h2) in norm ∥·∥∞.Proof.
Subtracting (7) from (35) and letting ejn=vjn-ujn, we have
(36)rjn=(ejn)t+(ejn)xxx¯x¯t+(ejn+(1/2))x^+(ejn+(1/2))xx¯x^+φ(vjn+(1/2))-φ(ujn+(1/2)).
Computing the inner product of (36) with 2en+(1/2), we obtain
(37)〈rn,2en+(1/2)〉=1τ(∥en+1∥2-∥en∥2)+1τ(∥exxn+1∥2-∥exxn∥2)+〈ex^n+1,2en+(1/2)〉+〈exx¯x^n+(1/2),2en+(1/2)〉+〈φ(vn+(1/2))-φ(un+(1/2)),2en+(1/2)〉.
Similar to the proof of (19), we have
(38)〈ex^n+1,2en+(1/2)〉=0,〈exx¯x^n+(1/2),2en+(1/2)〉=0.
This indicates
(39)(∥en+1∥2-∥en∥2)+(∥exxn+1∥2-∥exxn∥2)=τ〈rn,2en+(1/2)〉-τ〈Q1+Q2,2en+1/2〉,
where
(40)Q1=php+1∑j=1J-1[(vjn+(1/2))p-1·(vjn+(1/2))x^dddddddd-(ujn+(1/2))p-1·(ujn+(1/2))x^],Q2=php+1∑j=1J-1[((vjn+(1/2))p)x^-((ujn+(1/2))p)x^].
Noting that
(41)〈Q1,2en+(1/2)〉=-2p1+ph∑j=0J-1[(vjn+(1/2))p-1·(vjn+(1/2))x^dddddddddddd-(ujn+(1/2))p-1·(ujn+(1/2))x^]ejn+(1/2)=-2p1+ph∑j=0J-1[(vjn+(1/2))p-1·(ejn+(1/2))x^ejn+(1/2)]-2p1+ph∑j=0J-1[(vjn+(1/2))p-1-(ujn+(1/2))p-1]×(ujn+(1/2))x^ejn+(1/2)=-2p1+ph∑j=0J-1(vjn+(1/2))p-1·(ejn+(1/2))x^dddddddddddddd×ejn+(1/2)-2p1+ph∑j=0J-1ejn+(1/2)ddddddddddd×∑k=0p-2[(vjn+(1/2))p-2-k(ujn+(1/2))k]ddddddddddd×(ujn+(1/2))x^ejn+(1/2)≤Ch∑j=0J-1|(ejn+(1/2))x^|·|ejn+(1/2)|+Ch∑j=0J-1|(ejn+(1/2))2|≤C(∥exn+(1/2)∥2+∥en+(1/2)∥2)≤C(∥exn+1∥2+∥exn∥2+∥en+1∥2+∥en∥2),〈Q2,2en+(1/2)〉=-2p1+ph∑j=0J-1[((vjn+(1/2))p)x^-((ujn+(1/2))p)x^]ejn+(1/2)=-2p1+ph∑j=0J-1[(vjn+(1/2))p-(ujn+(1/2))p](ejn+(1/2))x^=2p1+ph∑j=0J-1[∑k=0p-1(vjn+(1/2))p-1-k(ujn+(1/2))k]×(ujn+(1/2))x^ejn+(1/2)≤C(∥exn+(1/2)∥2+∥en+(1/2)∥2)≤C(∥exn+1∥2+∥exn∥2+∥en+1∥2+∥en∥2)
and with
(42)〈rn,2en+(1/2)〉=〈rn,en+1+en〉≤∥rn∥2+12(∥en+1∥2+∥en∥2),
we have
(43)(∥en+1∥2-∥en∥2)+(∥exxn+1∥2-∥exxn∥2)≤Cτ(∥en+1∥2+∥en∥2+∥exn+1∥2+∥exn∥2)+τ∥rn∥2.
Similar to the proof of Theorem 6, we have
(44)∥exn+1∥2≤12(∥en+1∥2+∥exxn+1∥2);∥exn∥2≤12(∥en∥2+∥exxn∥2).
This yields
(45)∥en+1∥2+∥exxn+1∥2-(∥en∥2+∥exxn∥2)≤Cτ(∥en+1∥2+∥en∥2+∥exxn+1∥2+∥exxn∥2)+τ∥rn∥2.
Let Bn=∥en∥2+∥exxn∥2. We claim that
(46)Bn+1-Bn≤Cτ(Bn+1+Bn)+τ∥rn∥2,
which yields
(47)(1-Cτ)(Bn+1-Bn)≤2CτBn+τ∥rn∥2.
If τ is sufficiently small which satisfies 1-Cτ>0, then
(48)Bn+1-Bn≤CτBn+Cτ∥rn∥2.
Summing up (48) from 0 to n-1, we have
(49)Bn≤B0+Cτ∑l=0n-1∥rl∥2‍+Cτ∑l=0n-1Bl‍.
Since
(50)τ∑l=0n-1∥rl∥2≤nτmax⁡0≤l≤n-1∥rl∥2≤T·O(τ2+h2)2
and B0=O(τ2+h2)2, we obtain
(51)Bn≤O(τ2+h2)2+Cτ∑l=0n-1Bl.
By Lemma 8, we get
(52)Bn≤O(τ2+h2)2,
which implies
(53)∥en∥≤O(τ2+h2),∥exxn∥≤O(τ2+h2).
From (44), we have
(54)∥exn∥≤O(τ2+h2).
By Lemma 5 we get
(55)∥en∥∞≤O(τ2+h2).Finally, we can similarly prove the results as follows.Theorem 11.
The solutionun of (7)–(10) is unique.


## 4. Numerical Simulations

The difference scheme (7)–(10) is a nonlinear system about ujn+1 that can be easily solved by the Newton iterative algorithm.LetxL=-60,xR=90, and T=40. According to [3, 4], when p=3, the soliton solution is as follows:
(56)u(x,t)=14-15+341sech214×-5+412[x-110(5+41)t],
and the initial condition is
(57)u0(x)=14-15+341sech214-5+412x.
When p=5, the soliton solution is as follows:
(58)u(x,t)=415(-5+34)4sech13×-5+34[x-110(5+34)t],
and the initial condition is
(59)u0(x)=415(-5+34)4sech13-5+34x.
In Table 1 we give the error at various time step. We denote the C-N scheme in this paper as scheme I and the difference scheme in [7] as scheme II. In Table 2 we give the error comparison between scheme I and scheme II. It is easy to see that the calculation results of scheme I are slightly better than scheme II. Using the method in [20, 21], we verified the second convergence of the difference scheme in Table 3. Numerical simulations on the conservation invariant En are given in Table 4.Table 1
The error at various time step.
τ
=
h
=
0.25
τ
=
h
=
0.125
τ
=
h
=
0.0625
p
=
3
p
=
5
p
=
3
p
=
5
p
=
3
p
=
5
t
=
10
2.53343
e
-
3
3.43739
e
-
3
6.35973
e
-
4
8.64177
e
-
4
1.59182
e
-
4
2.16316
e
-
4
t
=
20
4.40914
e
-
3
6.31248
e
-
3
1.10927
e
-
3
1.58976
e
-
3
2.77703
e
-
4
3.98160
e
-
4
t
=
30
6.03109
e
-
3
9.13274
e
-
3
1.51828
e
-
3
2.30402
e
-
3
3.80216
e
-
4
5.77288
e
-
4
t
=
40
7.53941
e
-
3
1.20204
e
-
2
1.89987
e
-
3
3.03743
e
-
3
4.75848
e
-
4
7.61418
e
-
4Table 2
The error comparison whent=40.
τ
=
h
=
0.25
τ
=
h
=
0.125
τ
=
h
=
0.0625
p
=
3
p
=
5
p
=
3
p
=
5
p
=
3
p
=
5
Scheme I
7.53941
e
-
3
1.20204
e
-
2
1.89987
e
-
3
3.03743
e
-
3
4.75848
e
-
4
7.61418
e
-
4
Scheme II
1.34986
e
-
2
1.79985
e
-
2
3.42489
e
-
3
4.56804
e
-
3
8.59570
e
-
4
1.14689
e
-
3Table 3
The verification of the second convergence.
∥
e
n
(
h
,
τ
)
∥
/
∥
e
2
n
(
h
/
2
,
τ
/
2
)
∥
p
=
3
p
=
5
τ
=
h
=
0.1
τ
=
h
=
0.05
τ
=
h
=
0.025
τ
=
h
=
0.1
τ
=
h
=
0.05
τ
=
h
=
0.025
t
=
10
—
3.98355
3.99527
—
3.97764
3.99497
t
=
20
—
3.97483
3.99443
—
3.97073
3.99275
t
=
30
—
3.97233
3.99319
—
3.96382
3.99112
t
=
40
—
3.96837
3.99261
—
3.95742
3.98917Table 4
Numerical simulations on conservation invariantEn.
τ
=
h
=
0.25
τ
=
h
=
0.125
τ
=
h
=
0.0625
p
=
3
p
=
5
p
=
3
p
=
5
p
=
3
p
=
5
t
=
0
1.68252899330
3.11067490241
1.68254308255
3.11070293879
1.68254661109
3.11070996431
t
=
10
1.68252899329
3.11067490241
1.68254308255
3.11070293879
1.68254661108
3.11070996430
t
=
20
1.68252899328
3.11067490240
1.68254308255
3.11070293879
1.68254661095
3.11070996426
t
=
30
1.68252899327
3.11067490240
1.68254308255
3.11070293879
1.68254661102
3.11070996417
t
=
40
1.68252899325
3.11067490240
1.68254308254
3.11070293879
1.68254661095
3.11070996435The wave graph comparison ofu(x,t) at various times is given in Figures 1 and 2 when p=3 and p=5.Figure 1
Whenp=3 and τ=h=0.125, the wave graph of u(x,t) at various times.Figure 2
Whenp=5 and τ=h=0.125, the wave graph of u(x,t) at various times.


## 5. Conclusions

In this paper, we propose a conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of the generalized Rosenau-KdV equation. The two-level finite difference scheme is of second order convergence and unconditionally stable, which can start by itself. From Table2 we conclude that the C-N scheme is more efficient than scheme II in [7]. From Table 3 we conclude that the C-N scheme is of second order convergence obviously. Numerical simulations on the conservation invariant En are given in Table 4. Figures 1 and 2 show that the height of the wave graph at different time is almost identical. Table 4 and Figures 1 and 2 imply that the finite difference scheme is conservative and efficient.



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