Legendre Polynomials Spectral Approximation for the Infinite-Dimensional Hamiltonian Systems

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.


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 1 discussed the semidiscrete Fourier spectral approximation of infinite-dimensional Hamiltonian systems, Hamiltonian of infinite-dimensional Hamiltonian systems, and Hamiltonian structure.Shen 2 studied the dual-Petrov-Galerbin method for third and higher odd-order equations.Ma and Sun 3 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.

Mathematical Problems in Engineering
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 infinitedimensional 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.

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 is the Hamiltonian functional.

Semidiscrete Legendre Polynomials Spectral Approximation
Let L n x be the nth degree Legendre polynomial.The Legendre polynomials satisfy the three-term recurrence relation: and the orthogonality relation: 1

3.2
As suggested in 4 , 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 {L n } 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 form where a k j j 1, 2, . . ., m are chosen so that φ k x satisfy the m homogeneous boundary conditions.Suppose that U {H x |H x is a smooth function, x ∈ −1, 1 }, for the fixed homogeneous boundary conditions As m 2, 3.3 has the form Using the basic properties of Legendre polynomials and the boundary value conditions, obviously We can verify readily that Easily, we obtain φ 0 x , φ 1 x , φ 2 x , . ... The L 2 -inner product on U is defined by The basis functions φ k x k 1, 2, . . .can be orthogonalized standard on the L 2inner product.Thus, we can get the sequence of standard orthogonal basis functions ψ k x .
After carefully calculation, the orthogonal basis is . . .

3.9
Set and set P as an orthogonal projection. where where q a 0 , a 1 , a 2 , . . ., a N T .Set 3.14

3.17
The discretization of a functional H x in U is

3.18
Let U be the set of discrete functionals; then we can define a bracket on U,

3.19
which is an approximation of bracket {H, G}.

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: The KdV equation can be written as Hamiltonian form: where the Hamiltonian operator is D 1 ∂ x and the the Hamiltonian functional is x − u 3 dx.By above analysis and the chosen orthogonal basis, for N 2,

4.3
Then  Then

4.7
The corresponding semidiscrete approximation is

4.8
It is easy to verify that D 1 is Hamiltonian operator; so the approximating system can be written as Proof.
. ., ψ N are a sequence of standard orthogonal basis, The function H 1 u is a conservation law of energy, that is, d u dt D 1 ∇H 1 u T has the property of energy conservation law.

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:

5.1
It can be rewritten as two forms of the first-order equations: ∂v ∂t ∂u ∂x .

5.2
This equation can be written as Hamiltonian form: The Hamiltonian operator is D 1 0 ∂ x ∂ x 0 , and the corresponding Hamiltonian functional is There is another way to write the equation into Hamiltonian form, that is, The corresponding Hamiltonian operator is D 2 0 1 −1 0 , and the Hamiltonian functional is In this case, the element in U is denoted by u u 1 , u 2 T .The inner product is denoted by u, v Take the orthogonal basis: That is, B is 2N 1-dimensional subspace of U.

Mathematical Problems in Engineering
The orthogonal projection is where p a 0 , a 0 , . . ., a N , a N T and q b 0 , b 0 , . . ., b N , b N T . Set 5.9 Denote P I −1 • P : U → B, u −→ Pu a 0 , a 0 , a 1 , a 1 , a 2 , a 2 , . . ., a N , a N T .

5.10
The discretization of the Hamiltonian operator D is

5.11
The discretization of a functionals H x in U is

5.12
By the above analysis and the chosen orthogonal basis, for N 2,

5.14
The corresponding semidiscrete approximation is

5.15
For the other form, we can also obtain

5.19
Similar to the analysis of the KdV equation, for the situation of N 2, we can verify that D 1 and D 2 are all Hamiltonian operators; so the approximating system can be written as The proof of Theorem 5.1 is similar to that of Theorem 4.1.

2 . 5 . 1 .
d u/dt D 1 ∇H 1 u T and d u/dt D 2 ∇H 2 u T , two different Hamiltonian forms.As D 1 and D 2 both are constant antisymmetric matrix for N > 2, dimensional Hamiltonian systems.The approximating systems can preserve the Poisson structure given by Hamiltonian operators D 1 and D Theorem The equations d u/dt D 1 ∇H 1 u T and d u/dt D 2 ∇H 2 u T are the discretizations of the 1-dim wave equation Then d u/dt D 1 ∇H 1 u T and d u/dt D 2 ∇H 2 u T both have the property of energy conservation law.
A}. δH/δu is the variational derivative of the functional H ∈ F. With the aid of the differential operator D, we can define a binary operator on F: 2 T , . . ., u nT T , and u i denotes the ith derivative of u.To each H u ∈ A, there exists a functional H H u dx, and the corresponding set of all functional is F {H H u dx|H u ∈ Now we define the semidiscrete approximative equation in B of the infinite-dimen-Hamiltonian operator, then 3.20 is exactly a finite-dimensional Hamiltonian system.The function H Pu is a conservation law if and only if 3.20 always preserves conservation law.
The equation d u/dt D 1 ∇H 1 u T is the discretization of the KdV equation u t 6uu x u xxx 0; then d u/dt D 1 ∇H 1 u T has the property of energy conservation law.