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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.

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 [

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

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

Let the set

If this binary operator satisfies the following conditions:

For given a Hamiltonian functional

Consider the infinite-dimensional Hamiltonian system of the KdV equation:

Let

As suggested in [

Suppose that

The basis functions

After carefully calculation, the orthogonal basis is

Set

Denote

Denote

Hamiltonian equation

The discretization of the Hamiltonian operator

The discretization of a functional

Let

Now we define the semidiscrete approximative equation in

If

We consider the KdV equation with the fixed boundary conditions discussed above:

The KdV equation can be written as Hamiltonian form:

By above analysis and the chosen orthogonal basis, for

This equations can be written as Hamiltonian form in another way, that is,

In the same theory, for

Then

It is easy to verify that

The equation

According to

The function

Now we consider the wave equation with the fixed boundary conditions discussed above:

It can be rewritten as two forms of the first-order equations:

This equation can be written as Hamiltonian form:

The Hamiltonian operator is

There is another way to write the equation into Hamiltonian form, that is,

The corresponding Hamiltonian operator is

In this case, the element in

Take the orthogonal basis:

Set

The orthogonal projection is

Denote

Set

Denote

The discretization of the Hamiltonian operator

The discretization of a functionals

By the above analysis and the chosen orthogonal basis, for

Then

The corresponding semidiscrete approximation is

For the other form,

Then

The corresponding semidiscrete approximation is

Similar to the analysis of the KdV equation, for the situation of

The equations

The proof of Theorem

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.