MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation82416710.1155/2011/824167824167Research ArticleLegendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian SystemsLvZhongquan1, 2XueMei1WangYushun1GonçalvesPaulo Batista1Jiangsu Key Laboratory for NSLSCSInstitute of MathematicsSchool of Mathematical SciencesNanjing Normal UniversityNanjing 210046Chinannu.cn2College of ScienceNanjing Forestry UniversityNanjing 210037Chinanjfu.edu.cn20111506201120111110201006042011300420112011Copyright © 2011 Zhongquan Lv et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers a Legendre polynomials spectral approximation for the infinite-dimensional Hamiltonian systems. As a consequence, the Legendre polynomials spectral semidiscrete system is a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

1. Introduction

The numerical method for infinite-dimensional Hamiltonian Systems has been widely developed. One of the great challenges in the numerical analysis of PDEs is the development of robust stable numerical algorithms for Hamiltonian PDEs. For the numerical analysis, we always look for those discretizations which can preserve as much as possible some intrinsic properties of Hamiltonian equations. In fact, for Hamiltonian systems, the most important is its Hamiltonian structure. From this point of view, some semidiscrete numerical methods which are based on spectral methods have been developed. Spectral methods have proved to be particularly useful in infinite-dimensional Hamiltonian. Wang  discussed the semidiscrete Fourier spectral approximation of infinite-dimensional Hamiltonian systems, Hamiltonian of infinite-dimensional Hamiltonian systems, and Hamiltonian structure. Shen  studied the dual-Petrov-Galerbin method for third and higher odd-order equations. Ma and Sun  deliberated the third-order equations by using an interesting Legendre-Petrov-Galerbin method. So we consider that the Legendre polynomials basis is very important to analysis of the discretization of Hamiltonian systems.

In this paper, we consider a Legendre polynomials spectral approximation for the KdV equation and the wave equation. As a consequence, we show that the Legendre polynomials spectral semidiscrete system is also a Hamiltonian system for the Hamiltonian system whose Hamiltonian operator is a constant differential operator.

The paper is organized as follows. In Section 2, we give a brief description of infinite-dimensional Hamiltonian equations. In Section 3, we introduce semidiscrete Legendre polynomials spectral approximation. In the last two sections, we consider the Legendre polynomials spectral approximation for the boundary value problem of the KdV equation and the wave equation. Moreover, we give the conclusion about the Hamiltonian structure.

2. A Brief Description of Infinite-Dimensional Hamiltonian Equations

First, we get familiar with some basic knowledge about the infinite-dimensional Hamilton system.

Let the set A={H[u]H(x,u(n))|Hisainfinitedifferentiablesmoothfunction}; here u(n)=(u1T,u2T,,unT)T, and ui denotes the ith derivative of u. To each H[u]A, there exists a functional =H[u]dx, and the corresponding set of all functional is ={=H[u]dx|H[u]A}. δ/δu is the variational derivative of the functional . With the aid of the differential operator 𝒟, we can define a binary operator on : {H,G}=δHδuDδGδu,H,GF.

If this binary operator satisfies the following conditions:

{  ,  } is antisymmetric, {H,G}=-{G,H},

{  ,  } is bilinear, {αH+βG,K}=α{H,K}+β{G,K},  α,  βR,

{  ,  } satisfies the Jacobi identity, {{H,G},K}+{{G,K},H}+{{K,H},G}=0, for all functionals ,𝒢,𝒦, then, it is called a Poisson bracket. In this case, 𝒟 is called Hamiltonian operator.

For given a Hamiltonian functional and a Hamiltonian operator 𝒟, Hamiltonian equation takes the following form: ut=DδHδu. This evolution equation is called an infinite-dimensional Hamiltonian system.

Consider the infinite-dimensional Hamiltonian system of the KdV equation: ut+6uux+uxxx=0. It has the Hamiltonian structure ut=DδH[u], where 𝒟=x is the Hamiltonian operator and H[u]=-11(12ux2-u3)dx is the Hamiltonian functional.

3. Semidiscrete Legendre Polynomials Spectral Approximation

Let Ln(x) be the nth degree Legendre polynomial. The Legendre polynomials satisfy the three-term recurrence relation: L0(x)=1,  L1(x)=x,(n+1)Ln+1(x)=(2n+1)xLn(x)-nLn-1(x),n1 and the orthogonality relation: -11Lk(x)Lj(x)dx=1k+(1/2)δkj,Ln(±1)=(±1)n.

As suggested in , the choice of compact combinations of orthogonal polynomials as basis functions to minimize the bandwidth and the conditions number of the coefficient matrix is very important. Let {Ln} be a sequence of orthogonal polynomials. As a general rule, for differential equations with m boundary conditions, our task is to look for basis functions in the formϕk(x)=Lk(x)+j=1maj(k)Lk+j(x), where aj(k)(j=1,2,,m) are chosen so that ϕk(x) satisfy the m homogeneous boundary conditions.

Suppose that U={H(x)|H(x)  isasmoothfunction,  x[-1,1]}, for the fixed homogeneous boundary conditions H(-1)=H(1)=0. As m=2, (3.3) has the form ϕk(x)=Lk(x)+a1Lk+1(x)+a2Lk+2(x). Using the basic properties of Legendre polynomials and the boundary value conditions, obviously ϕk(-1)=0,ϕk(1)=0. We can verify readily that ϕk(x)=Lk(x)-Lk+2(x). Easily, we obtain ϕ0(x),ϕ1(x),ϕ2(x),. The L2-inner product on U is defined by (p,q)=-11pqdx,p,  qU.

The basis functions ϕk(x)(k=1,2,) can be orthogonalized standard on the L2- inner product. Thus, we can get the sequence of standard orthogonal basis functions ψk(x).

After carefully calculation, the orthogonal basis isψ0=14(15-3x2),ψ1=14(105x-105x3),ψ2=18(-35+245x2-215x4),

Set B=span{ψ0,ψ1,ψ2,,ψN}U, and set P as an orthogonal projection. P:UB, uu¯=Pu=a0ψ0+a1ψ1++aNψN, where an=-11u(x)ψn(x)dx,n=1,2,3,.

Denote B̂={û=(a0,a1,,aN)TRN+1}. The inner product of B̂ is usually denoted by Euclidean inner ·,  ·, that is, p̂,q̂=a0ã0+a1ã1+a2ã2++aNãN,p̂,q̂B,̂ where q̂=(ã0,ã1,ã2,,ãN)T. Set I:B̂B, ûu¯=Iû=a0ψ0+a1ψ1++aN.

Denote P̂=I-1P:UB̂, uP̂u=(a0,a1,a2,,aN)T.

Hamiltonian equation ut=DδHδu has the special Poisson structure; so we can exploit it to design numerical approximations. We can discretize Hamiltonian operator 𝒟 and Hamiltonian functionals; then a numerical bracket can be defined.

The discretization of the Hamiltonian operator 𝒟 is D̂=P̂DI:B̂B̂,D̂(p̂)q̂=p̂D̂(Ip̂)Iq̂,p̂,q̂B̂.

The discretization of a functional H(x) in U is Ĥ(û)=-11H(Iû)dx,ûB̂.

Let U¯ be the set of discrete functionals; then we can define a bracket on U¯, {Ĥ,Ĝ}=ĤD̂(Ĝ)T,Ĥ=(Ĥû1,Ĥû2,,ĤûN), which is an approximation of bracket {H,G}.

Now we define the semidiscrete approximative equation in B̂ of the infinite-dimensional Hamiltonian system ut=𝒟δδu asdûdt=D̂(H(û))T.

If 𝒟̂ is still a Hamiltonian operator, then (3.20) is exactly a finite-dimensional Hamiltonian system. The function Ĥ(Pu) is a conservation law if and only if (3.20) always preserves conservation law.

4. Legendre Polynomials Spectral Approximation for the Boundary Value Problem of the KdV Equation

We consider the KdV equation with the fixed boundary conditions discussed above: ut+6uux+uxxx=0,x(-1,  1),t[0,T],u(-1,t)=u(1,t)=0,t[0,T].

The KdV equation can be written as Hamiltonian form: ut=xδH1δu, where the Hamiltonian operator is 𝒟1=x and the the Hamiltonian functional is 1=-11((1/2)ux2-u3)dx.

By above analysis and the chosen orthogonal basis, for N=2,D̂1=(07220-7202120-2120),Ĥ1(û)=-11[12(a0ψ0+a1ψ1+a2ψ2)x2-(a0ψ0+a1ψ1+a2ψ2)3]  dx=54a02+214a12+32a0a2+514a22+31514a03-31514a02a2+15522a12a2+42352002a23+152a0a12+695154a0a22. Then Ĥ1T=(52a0+32a2+91514a02-357a0a2+152a12+5154a22212a1+15511a1a2+157a0a132a0+512a2-3514a02+15522a12+126952002a22+69577a0a2).

This equations can be written as Hamiltonian form in another way, that is, ut=(xxx+4ux+2uxI)δH2δu, where the Hamiltonian operator is 𝒟2=xxx+4ux+2uxI abd the Hamiltonian functional is 2=-11(-(1/2)u2)dx.

In the same theory, for N=2, we can get D̂2=(-2157a1-1372+4157a0-657a21057a1-7-2157a0-457a20R-457a1-5212-657a0-1611157a2811157a1)Ĥ2(û)=-11[-12(a0ψ0+a1ψ1+a2ψ2)2]dx=-12(a02+a12+a22) where   denotes-2021+1057a0+810157a2.

Then Ĥ2T=(-a0,  -a1,  -a2). The corresponding semidiscrete approximation is dûdt=D̂(H(û))T,û=(a0,a1,a2)T.

It is easy to verify that 𝒟̂1 is Hamiltonian operator; so the approximating system can be written as dûdt=D̂1(H1(û))T, adifferent Hamiltonian form, and it can be verified that 𝒟̂2 is not a Hamiltonian operator. As 𝒟̂1 is a constant antisymmetric matrix, dûdt=P̂xIδH1δu(Iû) is a finite-dimensional Hamiltonian system. So the approximating system can preserve the Poisson structure given by Hamiltonian operator 𝒟̂1.

Theorem 4.1.

The equation dû/dt=𝒟̂1(H1(û))T is the discretization of the KdV equation ut+6uux+uxxx=0; then dû/dt=𝒟̂1(H1(û))T has the property of energy conservation law.

Proof.

D̂=P̂DI:B̂B̂,I:B̂B,ûu¯=Iû=a0ψ0+a1ψ1++aNψN,P:UB,uu¯=Pu=a0ψ0+a1ψ1++aNψN,P̂=I-1P:UB̂,uP̂u=(a0,a1,a2,,aN)T,D̂=((Dψ1,ψ1)(ψ1,ψ1)(Dψ2,ψ1)(ψ1,ψ1)(DψN,ψ1)(ψ1,ψ1)(Dψ1,ψ2)(ψ2,ψ2)(Dψ2,ψ2)(ψ2,ψ2)(DψN,ψ2)(ψ2,ψ2)(Dψ1,ψN)(ψN,ψN)(Dψ2,ψN)(ψN,ψN)(DψN,ψN)(ψN,ψN)).ψ0,ψ1,ψ2,,ψN are a sequence of standard orthogonal basis, (ψi,ψi)=1,i=1,2,,N,(Dψi,ψj)=-(Dψj,ψi),(ij),(Dψi,ψi)=(xψi,ψi)=0,i=1,2,,N.𝒟̂1 is a constant antisymmetric matrix.

According to {Ĥ,Ĝ}=Ĥ𝒟̂(Ĝ)T, then {Ĥ1,Ĥ1}=Ĥ1𝒟̂(Ĥ1)T=0.

The function Ĥ1(û) is a conservation law of energy, that is, dûdt=𝒟̂1(H1(û))T has the property of energy conservation law.

5. Legendre Polynomials Spectral Approximation for the Boundary Value Problem of the Wave Equation

Now we consider the wave equation with the fixed boundary conditions discussed above: 2ut2=2ux2,x(-1,1),t[0,T],u(-1,t)=u(1,t)=0,t[0,T].

It can be rewritten as two forms of the first-order equations: ut=vx,vt=ux.

This equation can be written as Hamiltonian form: u¯t=D1δH1δu,u¯=(uv).

The Hamiltonian operator is 𝒟1=(0xx0), and the corresponding Hamiltonian functional is H1=1/2-11(u2+v2)dx.

There is another way to write the equation into Hamiltonian form, that is, u¯t=D2δH1δu,u¯=(uv).

The corresponding Hamiltonian operator is 𝒟2=(01-10), and the Hamiltonian functional is H2=1/2-11(ux2+v2)dx.

In this case, the element in U is denoted by u=(u1,u2)T. The inner product is denoted by (u,v)=i=12(ui,vi), u,vU, where (ui,vi)=-11uividx.

Take the orthogonal basis: (ψ00),(0ψ0),(ψ10),(0ψ1),(ψ20),(0ψ2),.

Set B=span{(ψ00),(0ψ0),(ψ10),(0ψ1),,(ψN0),(0ψN)}U. That is, B is 2N+1-dimensional subspace of U.

The orthogonal projection is P:UB, uu¯=Pu=a0(ψ00)+ã0(0ψ0)++aN(ψN0)+ãN(0ψN).

Denote B̂={û=(a0,ã0,a1,ã1,,aN,ãN)TR2N+2}. The inner product of B̂ is usually denoted by Euclidean inner ·,·, that is,p̂,q̂=a0b0+ã0b̃0++aNbN+ãNb̃N, where p̂=(a0,ã0,,aN,ãN)T and q̂=(b0,b̃0,,bN,b̃N)T.

Set I:B̂B, ûu¯=Iû=a0(ψ00)+ã0(0ψ0)++aN(ψN0)+ãN(0ψN).

Denote P̂=I-1P:UB̂, uP̂u=(a0,ã0,a1,ã1,a2,ã2,,aN,ãN)T.

The discretization of the Hamiltonian operator 𝒟 is D̂=P̂DI:B̂B̂,D̂(p̂)q̂=p̂D̂(Ip̂)Iq̂,p̂,q̂B̂.

The discretization of a functionals H(x) in U is Ĥ(û)=-11H(Iû)dx,ûB̂.

By the above analysis and the chosen orthogonal basis, for N=2, D̂1=(00720000007200-7200021200-7200021200-212000000-21200),Ĥ1(û)=12-11  [(a0ψ0+a1ψ1+a2ψ2)2+(ã0ψ0+ã1ψ1+ã2ψ2)2]dx=12(a02+ã02+a12+ã12+a22+ã22).

Then Ĥ1T=(a0,ã0,a1,ã1,a2,ã2).

The corresponding semidiscrete approximation is da0dt=72a1,dã0dt=72ã1,da1dt=-72a0+212a2,dã1dt=-72ã0+212ã2,da2dt=-212a1,dã2dt=-212ã1.

For the other form,  we can also obtain D̂2=(010000-10000000010000-10000000010000-10),Ĥ2(û)=12-11[(a0ψ0+a1ψ1+a2ψ2)x2+(ã0ψ0+ã1ψ1+ã2ψ2)2]dx=12(52a02+ã02+212a12+ã12+3a0a2+ã22).

Then Ĥ2=(52a0+32a2,ã0,212a1,ã1,512a2+32a0,ã2).

The corresponding semidiscrete approximation is da0dt=ã0,dã0dt=-52a0-32a2,da1dt=ã1,dã1dt=-212a1,da2dt=ã2,dã2dt=-32a0-512a2.

Similar to the analysis of the KdV equation, for the situation of N=2, we can verify that 𝒟̂1 and 𝒟̂2 are all Hamiltonian operators; so the approximating system can be written as dû/dt=𝒟̂1(H1(û))T and dû/dt=𝒟̂2(H2(û))T, two different Hamiltonian forms. As D1 and 𝒟̂2 both are constant antisymmetric matrix for N>2, dûdt=P̂(0xx0)IδH1δu(Iû),dûdt=P̂(01-10)IδH2δu(Iû). are finite-dimensional Hamiltonian systems. The approximating systems can preserve the Poisson structure given by Hamiltonian operators 𝒟̂1 and 𝒟̂2.

Theorem 5.1.

The equations dû/dt=𝒟̂1(H1(û))T and dû/dt=𝒟̂2(H2(û))T are the discretizations of the 1-dim wave equation ut=vx,vt=ux. Then dû/dt=𝒟̂1(H1(û))T and dû/dt=𝒟̂2(H2(û))T both have the property of energy conservation law.

The proof of Theorem 5.1 is similar to that of Theorem 4.1.

Acknowledgments

The topic is proposed by the project team of Professor Yongzhong Song and Professor Yushun Wang. The aythors gratefully acknowlege their considerable help by means of suggestion, comments and criticism. The work is supported by the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China under Grant no. 200720, the National Natural Science Foundation of China under Grant no. 10971102, and “333 Project” Foundation of Jiangsu Province.

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