# Two Energy Conserving Numerical Schemes for the Klein-Gordon-Zakharov Equations

Authors: Juan Chen; Luming Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Two new difference schemes are proposed for an initial-boundary-value problem of the Klein-Gordon-Zakharov (KGZ) equations. They have the advantage that there is a discrete energy which is conserved. Their stability and convergence of difference solutions are proved in orderO(h2+τ2) on the basis of the prior estimates. Results of numerical experiments demonstrate the efficiency of the new schemes.


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## Body

## 1. Introduction

In this paper, we consider the following initial-boundary-value problem of the KGZ equations (see [1]):
(1)Utt-Uxx+U+NU+|U|2U=0,hhhh-xL<x<xR,0≤t≤T,(2)Ntt-Nxx=(|U|2)xx,-xL<x<xR,hhhhhhhhhhhhhhhhhhhhhh≤t≤T,(3)U|t=0=U0(x),Ut|t=0=U1(x),N|t=0=N0(x),Nt|t=0=N1(x),(4)U|x=xL=U|x=xR=0,N|x=xL=N|x=xR=0,
where a complex unknown functionU(x,t)denotes the fast time scale component of electric field raised by electrons and a real unknown functionN(x,t)denotes the deviation of ion density from its equilibrium;U0(x),U1(x),N0(x), andN1(x)are known smooth functions.The solutionsU(x,t)andN(x,t)of the initial-boundary-value problem (1)–(4) formally satisfy the following energy identity:
(5)E=∫xLxR[12|Ut|2+|Ux|2+|U|2+N|U|2+12|V|2+12|N|2+12|U|4]=const,
where the potential functionVis defined as
(6)V=-fx,fxx=Nt.In [2] Ozawa et al. proved the well-posedness of the equations in three-dimensional space. Adomian discussed the existence of its nonperturbative solutions (see [3]). In [4] Guo and Yuan studied the global smooth solutions for the Cauchy problem of these equations. Furthermore, in [5, 6] the authors proposed three difference schemes for the KGZ equations. It is well known that a conservative scheme performs better than a nonconservative one; for example, Zhang et al. in [7] pointed out that the nonconservative schemes may easily show nonlinear blowup and Li and Vu-Quoc also 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" (see [8]). Up to now, many conservative finite difference schemes have been studied for the Klein-Gordon equation, Klein-Gordon-Schödinger equations, Sine-Gordon equation, Zakharov equations, and so on (see [9–25]). Numerical results of all the schemes are very good. Therefore, in this paper we will generalize the technique of these methods to propose two new conservative difference schemes which are unconditionally stable and more accurate for the KGZ equations.The paper is organized as follows. In Section2, a new difference scheme (i.e., Scheme A) is proposed, and its discrete conservative law is discussed. In Section 3, some prior estimates for difference solutions are made. In Section 4, convergence and stability for the new scheme are proved using discrete energy method. In Section 5, another conservative scheme (i.e., Scheme B) is constructed, and its discrete conservative law is discussed. In Section 6, some prior estimates of Scheme B are obtained by induction, then convergence of the scheme is analyzed. Finally, in Section 7, some numerical results are provided to demonstrate the theoretical results.


## 2. Finite Difference Scheme and Its Conservative Law

Before we propose the new difference scheme for the KGZ (1)–(4), we give some notations as follows:
(7)xj=xL+jh,0≤j≤J=[xR-xLh],tn=nτ,n=0,1,2,…,[Tτ],(wjn)x=wj+1n-wjnh,(wjn)x¯=wjn-wj-1nh,(wjn)t=wjn+1-wjnτ,(wjn)t¯=wjn-wjn-1τ,
wherehandτare step size of space and time, respectively.Also we define the following inner product and norms:(8)(wn,un)=h∑j=0J‍wjnu¯jn,∥wn∥pp=h∑j=0J‍|wjn|p,∥wn∥∞=sup0≤j≤J|wjn|.In this paper,Cstands for a general positive constant which may take different values on different occasions. For briefness, we omit the subscript2of∥wn∥2.Lemma 1.
For any two mesh functions{wj}and{vj},j=0,1,…,J, there is the identity
(9)h∑j=1J-1‍wj(vj)xx¯=-h∑j=0J-1‍(wj)x(vj)x-w0(v0)x+wJ(vJ)x¯.It is easy to prove this lemma directly.Now, we consider the following difference scheme for the KGZ equations (1)–(4).Scheme A.We consider the following:
(10)(Ujn)tt¯+h212(Ujn)xx¯tt¯-12(Ujn+1+Ujn-1)xx¯+12(Ujn+1+Ujn-1)+12Njn(Ujn+1+Ujn-1)+14(|Ujn+1|2+|Ujn-1|2)(Ujn+1+Ujn-1)=0,(11)(Njn)tt¯+h212(Njn)xx¯tt¯-12(Njn+1+Njn-1)xx¯=(|Ujn|2)xx¯,(12)Uj0=U0(xj),Nj0=N0(xj),U0n=UJn=0,N0n=NJn=0,(13)Uj1-Uj-1=2τU1(xj),Nj1-Nj-1=2τN1(xj).
By (10), (12), (11), and (13), we obtain the following:
(14)2τ2(Uj1-Uj0-τU1(xj))+h26τ2(Uj1-Uj0-τU1(xj))xx¯-(Uj1-τU1(xj))xx¯+(Uj1-τU1(xj))+Nj0(Uj1-τU1(xj))+12(|Uj1|2+|Uj1-2τU1(xj)|2)×(Uj1-τU1(xj))=0,2τ2(Nj1-Nj0-τN1(xj))+h26τ2(Nj1-Nj0-τN1(xj))xx¯-(Nj1-τN1(xj))xx¯=(|Uj0|2)xx¯.
Here, we also define the potential function{fjn}to be such that
(15)(fjn)xx¯=(Njn)t,j=1,2,…,J-1,f0n=fJn=0.Theorem 2.
The difference scheme (10)–(15) possesses the following invariant:
(16)En=En-1=⋯=E0=const,
where
(17)En=∥Utn∥2-h212∥Uxtn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+12h∑j=1J-1‍(Njn|Ujn+1|2+Njn+1|Ujn|2)+14(∥Un+1∥44+∥Un∥44)+12∥fxn∥2-h224∥Ntn∥2+14(∥Nn+1∥2+∥Nn∥2).Proof.
Computing the inner product of (10) withUn+1-Un-1and taking the real part, we have
(18)∥Utn∥2-∥Utn-1∥2-h212(∥Uxtn∥2-∥Uxtn-1∥2)+12(∥Uxn+1∥2-∥Uxn-1∥2)+12(∥Un+1∥2-∥Un-1∥2)+12h∑j=1J-1‍Njn(|Ujn+1|2-|Ujn-1|2)+14(∥Un+1∥44-∥Un-1∥44)=0.
Next, computing the inner product of (11) with(1/2)(fn+fn-1)and using (15), we obtain
(19)12(∥fxn∥2-∥fxn-1∥2)-h224(∥Ntn∥2-∥Ntn-1∥2)+14(∥Nn+1∥2-∥Nn-1∥2)+12h∑j=1J-1‍|Ujn|2(Njn+1-Njn-1)=0.
In the computation of (18) and (19), we have used the boundary conditions and Lemma 1. Then, result (16) follows from (18) and (19).


## 3. Some Prior Estimates for Difference Solutions

In this section, we will estimate the difference solutions of Scheme A after introducing two important lemmas proved in [26].Lemma 3 (discrete Sobolev's inequality).
For any discrete functionuh={uj∣j=0,1,…,J}on the finite interval[0,l]and for any givenε>0, there exists a constant C, depending only onε, such that
(20)∥δkuh∥p≤ε∥δnuh∥2+C∥uh∥2,
where2≤p≤∞,0≤k<n.Lemma 4 (Gronwall's inequality).
Suppose that the nonnegative mesh functions{w(n),ρ(n),n=1,2,…,N,Nτ=T}satisfy the inequality
(21)w(n)≤ρ(n)+τ∑l=1n‍Blw(l),
whereBl(l=1,2,…,N)are nonnegative constant. Then, for any0≤n≤N, there is
(22)w(n)≤ρ(n)exp(nτ∑l=1n‍Bl).Theorem 5.
Assume thatU0(x)∈H1,U1(x)∈L2,N0(x)∈H1, and N1(x)∈L2;then the following estimates hold:
(23)∥Utn∥≤C,∥Uxn∥≤C,∥Un∥≤C,∥Un∥∞≤C,∥fxn∥≤C,∥Nn∥≤C,∥Un∥4≤C.Proof.
Applying Young's inequality, it is easy to see that(24)-h212∥Uxtn∥2=-h312∑j=1J-1‍|(Ujn)xt|2=-h312∑j=1J-1‍(Ujn)xt(U¯jn)xt=-h12∑j=1J-1‍(Uj+1n-Ujn)t(U¯j+1n-U¯jn)t≥-h6∑j=1J-1‍(|(Uj+1n)t|2+|(Ujn)t|2)=-13∥Utn∥2,
and by (15), we have
(25)-h224∥Ntn∥2=-h324∑j=1J-1‍[(Njn)t]2=-h324∑j=1J-1‍[(fjn)xx-]2=-h24∑j=1J-1‍[(fjn)x-(fj-1n)x]2≥-h12∑j=1J-1‍{[(fjn)x]2+[(fj-1n)x]2}=-16∥fxn∥2;
then from (16) we get
(26)23∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+12h∑j=1J-1‍(Njn|Ujn+1|2+Njn+1|Ujn|2)+14(∥Un+1∥44+∥Un∥44)+13∥fxn∥2+14(∥Nn+1∥2+∥Nn∥2)≤En=C.
Since
(27)|12h∑j=1J-1‍Njn|Ujn+1|2|≤14h∑j=1J-1‍((Njn)2+|Ujn+1|4)≤14(∥Nn∥2+∥Un+1∥44),|12h∑j=1J-1‍Njn+1|Ujn|2|≤14(∥Nn+1∥2+∥Un∥44),
it follows from (26) that
(28)23∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+13∥fxn∥2≤C.
Therefore
(29)∥Utn∥≤C,∥Uxn∥≤C,∥Un∥≤C,∥fxn∥≤C.
Besides, we can obtain the following estimates by Lemma 3:
(30)∥Un∥∞≤C,∥Un∥4≤C.
On the other hand, by inequalityab≤(1/4)a2+b2, we have
(31)|h∑j=1J-1‍Njn|Ujn+1|2|≤h∑j=1J-1‍[14(Njn)2+|Ujn+1|4]=14∥Nn∥2+∥Un+1∥44,(32)|h∑j=1J-1‍Njn+1|Ujn|2|≤h∑j=1J-1‍[14(Njn+1)2+|Ujn|4]=14∥Nn+1∥2+∥Un∥44.
Thus, it follows from (26) that
(33)∥Nn∥≤C.
This completes the proof.


## 4. Convergence and Stability of the Difference Scheme

In this section, we will discuss the convergence and stability of the difference scheme (10)–(15). First, we define the truncation errors by
(34)rjn=(U(xj,tn))tt¯+h212(U(xj,tn))xx¯tt¯-12(U(xj,tn+1)+U(xj,tn-1))xx¯+12(U(xj,tn+1)+U(xj,tn-1))+12N(xj,tn)(U(xj,tn+1)+U(xj,tn-1))+14(|U(xj,tn+1)|2+|U(xj,tn-1)|2)×(U(xj,tn+1)+U(xj,tn-1)),(35)σjn=(N(xj,tn))tt¯+h212(N(xj,tn))xx¯tt¯-12(N(xj,tn+1)+N(xj,tn-1))xx¯-(|U(xj,tn)|2)xx¯.Lemma 6.
Assume that the conditions of Theorem5 are satisfied andU(x.t)∈C4,4,N(x,t)∈C4,4;then the truncation errors of the difference scheme (10)–(15) satisfy|rjn|+|σjn|=O(τ2+h2)asτ→0,h→0.By Taylor's expansion, Lemma6 can be proved directly. Besides, we note that the approximations of the initial conditions (13) have truncation errors of orderO(τ2), which are consistent with the scheme.Now, we are going to analyze the convergence of the difference scheme (10)–(15).Set the following:(36)ejn=U(xj,tn)-Ujn,ηjn=N(xj,tn)-Njn,1h2(Fj+1n-2Fjn+Fj-1n)=1τ(ηjn+1-ηjn),hhhhhhhhhhhhhhhhhj=1,2,…,J-1,F0n=FJn=0.Theorem 7.
Assume that the conditions of Lemma6 are satisfied; then the solutions of the difference scheme (10)–(15) converge to the solutions of the problem stated in (1)–(4) with orderO(τ2+h2)in theL∞norm forUnand in theL2norm forNn.Proof.
Subtracting (10) from (34), we obtain
(37)rjn=(ejn)tt¯+h212(ejn)xx¯tt¯-12(ejn+1+ejn-1)xx¯+12(ejn+1+ejn-1)+12ηjn(U(xj,tn+1)+U(xj,tn-1))+12Njn(ejn+1+ejn+1)+L1;
that is,
(38)(ejn)tt¯+h212(ejn)xx¯tt¯-12(ejn+1+ejn-1)xx¯+12(ejn+1+ejn-1)=rjn-12ηjn(U(xj,tn+1)+U(xj,tn-1))-12Njn(ejn+1+ejn+1)-L1,
where
(39)L1=14(|U(xj,tn+1)|2+|U(xj,tn-1)|2)×(U(xj,tn+1)+U(xj,tn-1))-14(|Ujn+1|2+|Ujn-1|2)(Ujn+1+Ujn-1)=14(|U(xj,tn+1)|2+|U(xj,tn-1)|2)(ejn+1+ejn-1)+14[|U(xj,tn+1)|2+|U(xj,tn-1)|2-|Ujn+1|2-|Ujn-1|2]·(Ujn+1+Ujn-1)=14(|U(xj,tn+1)|2+|U(xj,tn-1)|2)(ejn+1+ejn-1)+14[U(xj,tn+1)e¯jn+1+ejn+1U¯jn+1+ejn-1U¯jn-1+U(xj,tn-1)e¯jn-1]·(Ujn+1+Ujn-1).
Then computing the inner product of (38) withen+1-en-1and taking the real part, we have
(40)∥etn∥2-∥etn-1∥2-h212(∥extn∥2-∥extn-1∥2)+12(∥exn+1∥2-∥exn-1∥2)+12(∥en+1∥2-∥en-1∥2)≤Cτ(∥rn∥2+∥ηn∥2+∥Nn(en+1+en-1)∥2+∥etn∥2+∥etn-1∥2+∥en+1∥2+∥en-1∥2).
From Lemma 3 and Theorem 5 it follows that
(41)∥Nn(en+1+en-1)∥≤C∥Nn∥·∥en+1+en-1∥∞≤C∥exn+1+exn-1∥+C∥en+1+en-1∥≤C(∥exn+1∥+∥exn-1∥+∥en+1∥+∥en-1∥).
So, substituting (41) into (40), we have
(42)∥etn∥2-∥etn-1∥2-h212(∥extn∥2-∥extn-1∥2)+12(∥exn+1∥2-∥exn-1∥2)+12(∥en+1∥2-∥en-1∥2)≤Cτ(∥rn∥2+∥ηn∥2+∥exn+1∥2≤Cτ(+∥exn-1∥2+∥en+1∥2+∥en-1∥2+∥etn∥2+∥etn-1∥2).
Next, subtracting (11) from (35), we obtain
(43)(ηjn)tt¯+h212(ηjn)xx¯tt¯-12(ηjn+1+ηjn-1)xx¯=σjn+(|U(xj,tn)|2-|Ujn|2)xx¯.
Computing the inner product of (43) with(1/2)(Fn+Fn-1), we get
(44)-12τ(∥Fxn∥2-∥Fxn-1∥2)+h224τ(∥ηtn∥2-∥ηtn-1∥2)-14τ(∥ηn+1∥2-∥ηn-1∥2)=(σjn,12(Fn+Fn-1))+(12(|U(xj,tn)|2-|Ujn|2)xx¯,12(Fn+Fn-1)).
Note that
(45)((|U(xj,tn)|2-|Ujn|2)xx¯,12(Fn+Fn-1))≤C(∥Fxn∥2+∥Fxn-1∥2+∥exn∥2+∥en∥2),(σn,12(Fn+Fn-1))=12h∑j=1J-1σjn(Fjn+Fjn-1)≤C(∥σn∥2+∥Fxn∥2+∥Fxn-1∥2).
Then substituting (45) into (44) we have
(46)∥Fxn∥2-∥Fxn-1∥2-h212(∥ηtn∥2-∥ηtn-1∥2)+12(∥ηn+1∥2-∥ηn-1∥2)≤Cτ(+∥Fxn-1∥2+∥exn∥2+∥en∥2∥σn∥2+∥Fxn∥2+∥Fxn-1∥2+∥exn∥2+∥en∥2).
Now, adding (42) to(46), we get
(47)∥etn∥2-∥etn-1∥2-h212(∥extn∥2-∥extn-1∥2)+12(∥exn+1∥2-∥exn-1∥2)+12(∥en+1∥2-∥en-1∥2)+∥Fxn∥2-∥Fxn-1∥2-h212(∥ηtn∥2-∥ηtn-1∥2)+12(∥ηn+1∥2-∥ηn-1∥2)≤Cτ(∥Fxn-1∥2∥rn∥2+∥ηn∥2+∥σn∥2+∥exn+1∥2+∥exn∥2+∥exn-1∥2+∥en+1∥2+∥en∥2+∥en-1∥2+∥etn∥2+∥etn-1∥2+∥Fxn∥2+∥Fxn-1∥2).
Let
(48)Bn=∥etn∥2-h212∥extn∥2+12(∥exn+1∥2+∥exn∥2)+12(∥en+1∥2+∥en∥2)+∥Fxn∥2-h212∥ηtn∥2+12(∥ηn+1∥2+∥ηn∥2).
It is easy to see that
(49)Bn≥23∥etn∥2+12(∥exn+1∥2+∥exn∥2)+12(∥en+1∥2+∥en∥2)+23∥Fxn∥2+12(∥ηn+1∥2+∥ηn∥2).
Then by (47) and Lemma 6 we have
(50)Bn-Bn-1≤Cτ(∥rn∥2+∥σn∥2)+Cτ(Bn+Bn-1)≤Cτ(h2+τ2)2+Cτ(Bn+Bn-1).
Summing (50) up fornand applying Lemma 4, we get
(51)BN≤(B0+C(h2+τ2)2)exp(CNτ)≤C(B0+(h2+τ2)2).
Therefore, it follows from (49) that
(52)23∥etN∥2+12(∥exN+1∥2+∥exN∥2)+12(∥eN+1∥2+∥eN∥2)+23∥Fxn∥2+12(∥ηN+1∥2+∥ηN∥2)≤C(B0+(h2+τ2)2).
Note thate0,e1,η0, and η1are two-order precision and∥Fx0∥=O(h2+τ2)(see [22]). ThusB0=O(h2+τ2)2. Hence, the following inequalities can be obtained by (52):
(53)∥exN∥≤O(h2+τ2),∥eN∥≤O(h2+τ2),(54)∥etN∥≤O(h2+τ2),∥ηN∥≤O(h2+τ2).
Then, applying Lemma 3, we get
(55)∥exN∥∞≤O(h2+τ2).
So the proof of Theorem7 is complete.In the same way, we can prove that the solutions of the difference schemes (10)–(15) are unconditionally stable for initial data.


## 5. Another Conservative Difference Scheme

In this section, we will propose another conservative difference scheme for the problem given in (1)–(4) and discuss the discrete conservative law of this scheme.Now, we consider the finite difference simulations for (1) and (2) as follows.Scheme B. We consider the following:
(56)(Ujn)tt¯-12(Ujn+1+Ujn-1)xx¯+12(Ujn+1+Ujn-1)+12Njn(Ujn+1+Ujn-1)+18(|Ujn+1|2+2|Ujn|2+|Ujn-1|2)×(Ujn+1+Ujn-1)=0,(57)12(Njn)tt¯+14(Nj+1n+Nj-1n)tt¯-12(Njn+1+Njn-1)xx¯=(|Ujn|2)xx¯.In addition, the initial and boundary conditions (3) and (4) are also, respectively, approximated as
(58)Uj0=U0(xj),Nj0=N0(xj),U0n=UJn=0,N0n=NJn=0,(59)Uj1-Uj-1=2τU1(xj),Nj1-Nj-1=2τN1(xj).We also define the function{fjn}by
(60)(fjn)xx¯=(Njn)t,j=1,2,…,J-1,f0n=fJn=0.
In (56) and (57), letn=0. Then eliminatingU-1andN-1from (58) and (59), we get
(61)2τ2(Uj1-Uj0-τU1(xj))-(Uj1-τU1(xj))xx¯+(Uj1-τU1(xj))+Nj0(Uj1-τU1(xj))+14(|Uj1|2+2|Uj0|2+|Uj1-2τU1(xj)|2)×(Uj1-τU1(xj))=0,1τ2(Nj1-Nj0-τN1(xj))+12τ2(Nj+11-Nj+10-τN1(xj+1)+Nj-11-Nj-10-τN1(xj-1))-(Nj1-τN1(xj))xx¯=(|Uj0|2)xx¯.Theorem 8.
Scheme B admits the following invariant:(62)E~n=E~n-1=⋯=E~0=const,
where
(63)E~n=∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+14∥fxn∥2+12h∑j=1J-1‍(Njn+1|Ujn|2+Njn|Ujn+1|2)+14h∑j=1J-1‍(fjn)x(fj-1n)x+14(∥Nn+1∥2+∥Nn∥2)+18(∑j=1J-1‍∥Un+1∥44+∥Un∥44+2h∑j=1J-1‍|Ujn+1|2|Ujn|2).Proof.
Computing the inner product of (56) withUn+1-Un-1and taking the real part, we have
(64)∥Utn∥2-∥Utn-1∥2+12(∥Uxn+1∥2-∥Uxn-1∥2)+12(∥Un+1∥2-∥Un-1∥2)+12h∑j=1J-1‍Njn(|Ujn+1|2-|Ujn-1|2)+L1=0,
where
(65)L1=18Re{h∑j=1J-1‍(|Ujn+1|2+2|Ujn|2+|Ujn-1|2)18ReL1{×(Ujn+1+Ujn-1)(Ujn+1-Ujn-1)h∑j=1J-1‍}=18h∑j=1J-1‍(|Ujn+1|2+2|Ujn|2+|Ujn-1|2)hhhhh×(|Ujn+1|2-|Ujn-1|2)=18h∑j=1J-1‍(|Ujn+1|4+2|Ujn+1|2|Ujn|2=18h∑j=1J-1‍-2|Ujn|2|Ujn-1|2-|Ujn-1|4)=18(∥Un+1∥44+2h∑j=1J-1‍|Ujn+1|2|Ujn|2=18-2h∑j=1J-1‍|Ujn|2|Ujn-1|2-∥Un-1∥44).
Next, computing the inner product of (57) with(1/2)(fn+fn-1)and by (60), we obtain
(66)-14τ(∥fxn∥2-∥fxn-1∥2)+L2-14τ(∥Nn+1∥2-∥Nn-1∥2)=h2τ∑j=1J-1‍(Njn+1|Ujn|2-Njn-1|Ujn|2),
where
(67)L2=14h∑j=1J-1‍(Nj+1n+Nj-1n)tt¯·12(fjn+fjn-1)=14τh∑j=1J-1‍[(fj+1n+fj-1n)xx¯-(fj+1n-1+fj-1n-1)xx¯]∑[·12(fjn+fjn-1)=-14τh∑j=1J-1‍[(fj+1n+fj-1n)x-(fj+1n-1+fj-1n-1)x]∑[·12(fjn+fjn-1)x=-14τh∑j=1J-1‍[(fjn)x(fj-1n)x-(fjn-1)x(fj-1n-1)x].
Then
(68)14(∥fxn∥2-∥fxn-1∥2)+14h∑j=1J-1‍[(fjn)x(fj-1n)x-(fjn-1)x(fj-1n-1)x]+14(∥Nn+1∥2-∥Nn-1∥2)+h2τ∑j=1J-1‍(Njn+1|Ujn|2-Njn-1|Ujn|2)=0.
Hence, result (62) is obtained by adding (68) to (64). This completes the proof.


## 6. Convergence and Stability of the Scheme

Before we prove the convergence of Scheme B, we estimate the difference solutions of this scheme.Theorem 9.
Assume that the conditions of Theorem5 are satisfied; then the following estimates hold:
(69)∥Utn∥≤C,∥Uxn∥≤C,∥Un∥≤C,∥Un∥∞≤C,∥Nn∥≤C,∥Un∥4≤C.Proof (by induction).
First, because of the inequality(70)a·b≤1p(εa)p+1q(1εb)q,1p+1q=1,p,q,ε>0,
we have
(71)12|h∑j=1J-1‍Njn|Ujn+1|2|≤ε12∥Nn∥2+18ε1∥Un+1∥44,12|h∑j=1J-1‍Njn+1|Ujn|2|≤18∥Nn+1∥2+12∥Un∥44.
Then, substituting (71) into (62) and choosingε1=2ε>1, we get the following inequality:
(72)∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+14∥fxn∥2+14h∑j=1J-1‍(fjn)x(fj-1n)x+14(∥Nn+1∥2+∥Nn∥2)+18(∥Un+1∥44+∥Un∥44+2h∑j=1J-1‍|Ujn+1|2|Ujn|2)≤C+18∥Nn+1∥2+12∥Un∥44+ε12∥Nn∥2+18ε1∥Un+1∥44.
That is,
(73)∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+14∥fxn∥2+14h∑j=1J-1‍(fjn)x(fj-1n)x+18∥Nn+1∥2+14∥Nn∥2+(18-18ε1)∥Un+1∥44+18∥Un∥44+2h8∑j=1J-1|Ujn+1|2|Ujn|2≤C+12∥Un∥44+ε12∥Nn∥2.
Note that
(74)|14h∑j=1J-1‍(fjn)x(fj-1n)x|≤14h∑j=1J-1‍12[(fjn)x2+(fj-1n)x2]=14∥fxn∥2,
so we have
(75)∥Utn∥2+12(∥Uxn+1∥2+∥Uxn∥2)+12(∥Un+1∥2+∥Un∥2)+18∥Nn+1∥2+14∥Nn∥2+(18-18ε1)∥Un+1∥44+18∥Un∥44+2h8∑j=1J-1|Ujn+1|2|Ujn|2≤C+12∥Un∥44+ε12∥Nn∥2.
Obviously, by (58), (59), and the conditions of Theorem 5, the following inequalities hold:
(76)∥Ut-1∥≤C,∥Ut0∥≤C,∥Ux0∥≤C,∥U0∥≤C,∥U0∥∞≤C,∥N0∥≤C,∥U0∥4≤C.
Assume that Theorem9 holds whenn=k; that is,
(77)∥Utk-1∥≤C,∥Uxk∥≤C,∥Uk∥≤C,∥Uk∥∞≤C,∥Nk∥≤C,∥Uk∥4≤C.
By (75) and (77), we get
(78)∥Utk∥2+12(∥Uxk+1∥2+∥Uxk∥2)+12(∥Uk+1∥2+∥Uk∥2)+18∥Nk+1∥2+14∥Nk∥2+(18-18ε1)∥Uk+1∥44+18∥Uk∥44+2h8∑j=1J-1|Ujk+1|2|Ujk|2≤C+12∥Uk∥44+ε12∥Nk∥2≤C,
from which the following inequalities are obtained,
(79)∥Utk∥≤C,∥Uxk+1∥≤C,∥Uk+1∥≤C,∥Nk+1∥≤C,∥Uk+1∥4≤C,
and applying Lemma 3, we have
(80)∥Uk+1∥∞≤C.
Then, for anyn∈{0,1,2,…,N}, the following estimates are obtained:
(81)∥Utn∥≤C,∥Uxn∥≤C,∥Un∥≤C,∥Un∥∞≤C,∥Nn∥≤C,∥Un∥4≤C.
This completes the proof.Theorem 10.
Assume that the conditions of Lemma6 are satisfied; then the solutions of the difference scheme (56)–(61) converge to the solutions of the problem given in (1)–(4) with orderO(τ2+h2)in theL∞norm forUnand in theL2norm forNn.Here, we omit details of the proof of this theorem because it can be proved in the same way as that used to prove Theorem7.


## 7. Numerical Experiments

In this section, we compute the following numerical example to demonstrate the effectiveness of our two difference schemes:(82)Utt-Uxx+U+NU+|U|2U=0,hhhI-20<x<20,0≤t≤T,Ntt-Nxx=(|U|2)xx,-20<x<20,0≤t≤T,U(-20,t)=U(20,t)=0,N(-20,t)=N(20,t)=0,hhhhhhhhhhhh0≤t≤T,U(x,0)=10-22sech(1+52x)·exp[i(21+5x)],-20≤x≤20,N(x,0)=-2sech2(1+52x),-20≤x≤20.The analytic solution of KGZ equations, which is derived in [27], will be used in our computation for comparison. The solution can be written as
(83)U(x,t)=10-22sech(1+52x-t)·exp[i(21+5x-t)],(84)N(x,t)=-2sech2(1+52x-t).In order to quantify the numerical results, we define the “error” functions and “rate of convergence” as(85)e(h,τ)=∥en∥∞=sup0≤j≤J|U(xj,tn)-Ujn|,rate1=log2(e(h,τ)e(h/2,τ/2)),(86)η(h,τ)=∥ηn∥=h∑j=0J|U(xj,tn)-Njn|2,rate2=log2(η(h,τ)η(h/2,τ/2)).
For the two iterative schemes, we use an error restrictorε≤0.000001to control the iterative procedures.Firstly, in Figures1 and 2, the solitary waves computed by Scheme A and Scheme B are compared with the waves of analytic solution, respectively. From (12) and (58), we will see that the the boundary conditions discretization produces no error in computation, so it is harmless to discrete energyEn. The curves of discrete energyEnobtained by the two schemes are plotted in Figure 3. Secondly, Tables 1 and 2 give the errors and the rates of convergence for Scheme A and Scheme B with varioushandτ. Finally, errors produced by our two schemes and the schemes in [5, 6] are compared in Tables 3 and 4.Table 1
Errors and rates of convergence for Scheme A with differenth and τ at time T=1.
(
h, τ)
e
(
h
,
τ
)
rat
e
1
η
(
h, τ)
rat
e
2
(0.2, 0.1)
1.1043
e
-
002
—
2.8775
e
-
002
—
(0.1, 0.05)
2.8652
e
-
003
1.95
7.4419
e
-
003
1.95
(0.05, 0.025)
7.2560
e
-
004
1.98
1.8886
e
-
003
1.98
(0.025, 0.0125)
1.8673
e
-
004
1.96
4.7767
e
-
004
1.98Table 2
Errors and rates of convergence for Scheme B with differenth and τ at time T=5.
(
h, τ)
e
(
h, τ)
rat
e
1
η
(
h, τ)
rat
e
2
(0.1, 0.1)
5.4233
e
-
002
—
0.1237
—
(0.05, 0.05)
1.4050
e
-
002
1.95
3.1279
e
-
002
1.98
(0.025, 0.025)
3.4925
e
-
003
2.00
7.7549
e
-
003
2.01
(0.0125, 0.0125)
8.0729
e
-
004
2.11
1.8386
e
-
003
2.08Table 3
Error comparison for Scheme A and scheme in [6] with θ=0.5 at time T=5.
Spatial step size
Time step size
Scheme A
Scheme in [6]
e
(
h
,
τ
)
×
1
0
2
η
(
h, τ)×102
e
(
h, τ)×102
η
(
h, τ)×102
h
=
0.1
τ
=
0.05
1.3527
3.8829
1.6527
5.5376
τ
=
0.025
0.4430
1.2845
1.1721
3.6144
h
=
0.05
τ
=
0.025
0.3390
0.9655
0.4148
1.3785
τ
=
0.0125
0.1096
0.3189
0.2960
0.9170
h
=
0.025
τ
=
0.0125
0.0849
0.2428
0.1081
0.3728Table 4
Error comparison for Scheme B and Scheme I in [5] at time T=10.
Spatial step size
Time step size
Scheme A
Scheme I in [5]
e
(
h, τ)×102
η
(
h
,
τ
)
×
1
0
2
e
(
h, τ)×102
η
(
h, τ)×102
h
=
0.1
τ
=
0.01
0.9742
2.4407
2.0623
6.3507
h
=
0.05
τ
=
0.025
0.5894
1.8270
4.2819
8.0020
τ
=
0.0125
0.3483
0.5278
2.1097
4.1693
h
=
0.025
τ
=
0.025
0.4753
1.4006
4.3735
7.7397
τ
=
0.0125
0.1839
0.3526
2.1330
3.8573|
U
|andNcomputed by Scheme A withh=τ=0.1.Comparison between analytic solution and numerical solution at timeT=1.
(a)
(b)|
U
|andNcomputed by Scheme B withh=0.1,τ=0.05.Comparison between analytic solution and numerical solution at timeT=5.
(a)
(b)Discrete energyEncomputed by Scheme A and Scheme B withh=0.05,τ=0.05.
(a)
(b)In Figures1 and 2, it is obvious that the solitary waves computed by Scheme A and Scheme B agree with the ones computed by exact solutions quite well. Figure 3 shows that both Schemes A and B possess satisfactory conservative property. Tables 1 and 2 verify the second-order convergence and good stability for the two schemes. Furthermore, Tables 3 and 4 show that our two schemes are more accurate than schemes in [5, 6]. Therefore, it is clear that our two new difference schemes are efficient and accurate for the studied problem.


## 8. Conclusion

In this paper, we study the finite difference method for the KGZ equations. We propose two difference schemes, both of them are conservative on discrete energy law. The two schemes are shown to possess second-order accuracy forUinL∞norm and forNinL2norm. Numerical results demonstrate that the two schemes are accurate and efficient. It is worth mentioning that our methods can be directly extended to two dimensions and/or three dimensions, but some new techniques are required to be used to deal with the prior estimates.



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*Source: 462018-2013-11-30.xml*