Research Article | Open Access
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.
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 ; here , and denotes the th derivative of . To each , there exists a functional , and the corresponding set of all functional is . is the variational derivative of the functional . With the aid of the differential operator , we can define a binary operator on :
If this binary operator satisfies the following conditions: (i) is antisymmetric, (ii) is bilinear, (iii) satisfies the Jacobi identity, for all functionals , then, it is called a Poisson bracket. In this case, is called Hamiltonian operator.
For given a Hamiltonian functional and a Hamiltonian operator , Hamiltonian equation takes the following form: This evolution equation is called an infinite-dimensional Hamiltonian system.
Consider the infinite-dimensional Hamiltonian system of the KdV equation: It has the Hamiltonian structure where is the Hamiltonian operator and is the Hamiltonian functional.
3. Semidiscrete Legendre Polynomials Spectral Approximation
Let be the th degree Legendre polynomial. The Legendre polynomials satisfy the three-term recurrence relation: and the orthogonality relation:
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 be a sequence of orthogonal polynomials. As a general rule, for differential equations with boundary conditions, our task is to look for basis functions in the form where are chosen so that satisfy the homogeneous boundary conditions.
Suppose that , for the fixed homogeneous boundary conditions As , (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 . The -inner product on is defined by
The basis functions can be orthogonalized standard on the - inner product. Thus, we can get the sequence of standard orthogonal basis functions .
After carefully calculation, the orthogonal basis is
Set and set as an orthogonal projection. , where
Denote . The inner product of is usually denoted by Euclidean inner , that is, where . Set ,
Hamiltonian equation has the special Poisson structure; so we can exploit it to design numerical approximations. We can discretize Hamiltonian operator and Hamiltonian functionals; then a numerical bracket can be defined.
The discretization of the Hamiltonian operator is
The discretization of a functional in is
Let be the set of discrete functionals; then we can define a bracket on , which is an approximation of bracket .
Now we define the semidiscrete approximative equation in of the infinite-dimensional Hamiltonian system as
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:
The KdV equation can be written as Hamiltonian form: where the Hamiltonian operator is and the the Hamiltonian functional is .
By above analysis and the chosen orthogonal basis, for , Then
This equations can be written as Hamiltonian form in another way, that is, where the Hamiltonian operator is abd the Hamiltonian functional is .
In the same theory, for , we can get where .
Then The corresponding semidiscrete approximation is
It is easy to verify that is Hamiltonian operator; so the approximating system can be written as adifferent Hamiltonian form, and it can be verified that is not a Hamiltonian operator. As is a constant antisymmetric matrix, is a finite-dimensional Hamiltonian system. So the approximating system can preserve the Poisson structure given by Hamiltonian operator .
Theorem 4.1. The equation is the discretization of the KdV equation ; then has the property of energy conservation law.
are a sequence of standard orthogonal basis,
is a constant antisymmetric matrix.
According to , then .
The function is a conservation law of energy, that is, 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:
It can be rewritten as two forms of the first-order equations:
This equation can be written as Hamiltonian form:
The Hamiltonian operator is , and the corresponding Hamiltonian functional is .
There is another way to write the equation into Hamiltonian form, that is,
The corresponding Hamiltonian operator is , and the Hamiltonian functional is .
In this case, the element in is denoted by . The inner product is denoted by , , where .
Take the orthogonal basis:
Set That is, is -dimensional subspace of .
The orthogonal projection is ,
Denote . The inner product of is usually denoted by Euclidean inner , that is, where and .
The discretization of the Hamiltonian operator is
The discretization of a functionals in is
By the above analysis and the chosen orthogonal basis, for ,
The corresponding semidiscrete approximation is
For the other form,we can also obtain
The corresponding semidiscrete approximation is
Similar to the analysis of the KdV equation, for the situation of , we can verify that and are all Hamiltonian operators; so the approximating system can be written as and , two different Hamiltonian forms. As and both are constant antisymmetric matrix for , are finite-dimensional Hamiltonian systems. The approximating systems can preserve the Poisson structure given by Hamiltonian operators and .
Theorem 5.1. The equations and are the discretizations of the 1-dim wave equation Then and both have the property of energy conservation law.
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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Copyright © 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.