Research Article | Open Access
Cuicui Liao, Xiaohua Ding, "Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs", Journal of Applied Mathematics, vol. 2012, Article ID 705179, 22 pages, 2012. https://doi.org/10.1155/2012/705179
Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs
We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein-Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.
It is a fundamental approach to develop the discrete multisymplectic numerical methods based on the discrete Hamilton’s principle, because it leads in a natural way to multisymplectic integrators . The discrete Euler-Lagrange equation is produced in the discrete variational principle [2–4]; meanwhile, the discrete multisymplectic structure is also generated [5, 6]. In the other words, the discrete variational integrators are multisymplectic automatically.
1.1. Multisymplectic Structure of Discrete Variational Integrators
By the Hamilton’s principle [6–9], the discrete multisymplectic structure which is preserved by the discrete variational integrator, is described by Poincaré-Cartan forms, in a differential geometric language. In paper , Marsden et al. showed how to obtain this structure directly from the variational principle, on the Lagrangian side. They defined it as the multisymplectic form formula, and they showed that it was conserved by the discrete variational integrator.
Lemma 1.1. If is a solution of discrete Euler-Lagrange equation and , are first variations of , then the following discrete multsisymplectic form formula holds:
The details of this conclusion could be referred to papers [5, 6]. This conclusion states that the discrete variational principles produce discrete variational integrators, and the multisymplecticity of these variational integrators is presented by the discrete multisymplectic form formula (1.1).
Vankerschaver et al.  revisited the multisymplectic form formula , showing that it could be obtained from the boundary Lagrangian that they defined in their paper. They presented an easy way to derive discrete multisymplectic form formula from discrete variational principle, using the notations of Poincaré-Cartan forms. In this paper, we follow the same way to derive the discrete multisymplectic form formulas of our discrete variational integrators.
When we use the discrete variational principle, we need to make a approximation of the Lagrangian. Here, in our paper, we would use nonstandard finite difference methods, instead of standard finite difference, to approximate the Lagrangian function, and derive the corresponding discrete variational integrators.
1.2. Nonstandard Finite Difference Methods
The nonstandard finite difference schemes are well developed by Mickens [11–15] in the past decades. These schemes are developed for compensating the weaknesses that may be caused by standard finite difference methods, for example, the numerical instabilities. Regarding the positivity of solutions, boundedness, and monotonicity of solutions, nonstandard finite difference schemes have a better performance than standard finite difference schemes, due to its flexibility to construct a nonstandard finite difference scheme that can preserve certain properties and structures, which are obeyed by the original equations. Also, the dynamic consistency could be presented well by nonstandard finite difference scheme. These advantages of nonstandard finite difference methods have been shown in many numerical applications. GonzLez-Parra et al. [16–18] developed nonstandard finite difference methods to solve population or biological models. The positivity condition and the conservation law of population dynamics are preserved by nonstandard finite difference schemes. Jordan  and Malek  constructed nonstandard finite difference schemes for heat transfer problems. For the symplectic systems, Mickens  derived the nonstandard finite difference variational integrator for symplectic ODEs. Ma et al.  developed the nonstandard finite difference variational integrator in stochastic ordinary differential equations.
The initial foundation of nonstandard finite difference methods is formed by the exact finite difference schemes . After generalizing these results, Mickens summarizes the following three basic rules to construct nonstandard finite difference schemes. (1) The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives appearing in the differential equations.Note. If the orders of the discrete derivatives are larger than those occurring in the differential equations, then numerical instabilities will in general occur.(2) Discrete representations for derivatives must, in general, have nontrivial denominator functions.Note. For example, the discrete first-derivative is generally represented by where the numerator functions and the denominator functions satisfy (3) Both linear and nonlinear terms should be represented by nonlocal discrete representations on the discrete computational lattice.Note. For example,
In our paper, we combine the advantages of nonstandard finite difference methods and discrete variational principles to construct nonstandard finite difference variational integrators, for two multisymplectic PDEs. These integrators are multysimplectic and their multysimplecticity are presented by their discrete multisymplectic form formulas, respectively.
In Section 2, we consider a simple linear wave equation. With the triangle discretization, we define the discrete Lagrangian using the idea of nonstandard finite difference and derive discrete variational integrator and the corresponding multisymplectic form formula, by discrete variational principle. The convergence of this method is analyzed. In Section 3, for the nonlinear Klein-Gordon equation, triangle discretization and square discretization are considered to obtain the nonstandard finite difference variational integrators. The discrete multisymplectic structures are presented, respectively. The convergence orders of the two methods are also discussed, and the convergence orders are shown in error tables in the numerical experiment section. Section 4 is devoted to showing the numerical behaviors of the developed nonstandard finite difference variational integrators.
2. Nonstandard Finite Difference Variational Integrator for Linear Wave Equation
We first consider a simple linear wave equation, where is a scalar field function with two independent variables, and .
This linear wave equation is actually a multisymplectic PDE. As a classical and simple multisymplectic example, wave equation and its multisymplectic structure have been studied from both Hamiltonian [23–28] and Lagrangian viewpoints [3, 5, 6, 10]. Based on the Lagrangian viewpoint, we could also obtain (2.1) from the Euler-Lagrange equation with the Lagrangian function ,
Assume that we have a uniform quadrangular mesh in the base space, with mesh lengths and . The nodes in this mesh are denoted by , corresponding to the points in . We denote the value of the field at the node by . We label the triangle at with three ordered triple as , and we define to be the set of all such triangles. Then the discrete jet bundle [6, 10] is defined as follows: which is equal to .
Now we use nonstandard finite difference to define the discrete Lagrangian on , which is the discrete version of Lagrangian density [10, 29], where denominator functions and are defined according the exact solution of wave equation [5, 6, 12]
We have followed the rules of constructing nonstandard finite difference schemes in Mickens’ papers [11–15]. (1)The discrete first-derivative is represented by where denominator functions , are defined in (2.6). Using Taylor series expansion, Then the denominator functions satisfy (2)Nonlocal representation on the discrete computational lattice are used here by
After some simple calculations, the discrete Euler-Lagrange equation (2.12) becomes
We could find that this scheme is symmetric in and , and . This is the nonstandard finite difference variational integrator, for the linear wave equation.
As we mentioned in Section 1 and Lemma 1.1, the advantages of deriving the multisymplectic numerical schemes from discrete variational principle are that they are naturally multisymplectic and the discrete multisymplectic structures are also generated in the variational principle. Now it is meaningful to show the multisymplectic structure of this discrete variational integrator (2.14) based on nonstandard finite difference method.
Since we consider triangulation discretization here, we focus on three adjacent triangles around , denote this area by . Following the idea in , the discrete boundary Lagrangian is given by where
Taking twice exterior derivative of both sides, we have, by the fact that , the discrete multisymplectic form formula with following form : where (for ). The discrete Poincaré-Cartan forms , , and are defined by and similarly for and . Thus, for the linear wave equation (2.1), the multisymplectic form formula of this scheme (2.14) based on nonstandard finite difference method can be obtained as follows:
Now we have the first conclusion.
We now discuss the convergence of this variational integrator (2.14) based on the nonstandard finite difference method. From the Lax equivalence theorem we know that, for a well-posed linear initial value problem, the consistent finite difference method is convergent if and only if it is stable.
By Taylor series expansion, we have
The above two equations show that the scheme is consistent and the truncation error for the integrator (2.14) is .
To explore the stability of the nonstandard finite difference variational integrator (2.14), we introduce the following notations:
Then the three-level explicit integrator (2.14) is equivalent to the following two-level scheme:
By using the Von Neumann method , we could get the amplification matrix of the above scheme, where . Note that, in the above matrix, . Let . We have the characteristic equation and the eigenvalues
When ,, the scheme (2.14) satisfies the Von Neumann conditions, which is a necessary condition of the stability of the scheme (2.14). If , , where is an integer, then has two different eigenvalues. If , , then is an identity matrix, but has two different eigenvalues . So is the sufficient condition of the stability for integrator (2.14). Note that, if , there is an unbounded solution , . So the scheme (2.14) is not stable when . Now, we find the necessary and sufficient condition of the stability for integrator (2.14), which is
With the consistence and stability conditions, we have following conclusion.
Theorem 2.2. The nonstandard finite difference variational integrator (2.14) is convergent, when the step sizes and satisfy .
We have shown the idea of using the nonstandard finite difference method to get the discrete variational integrator and the corresponding discrete multisymplectic form formula. In the next section, we will consider the discrete variational integrators for a more complicated example, the nonlinear Klein-Gordon equation.
3. Nonstandard Finite Difference Variational Integrators for Nonlinear Klein-Gordon Equation
In this section, we consider the nonlinear Klein-Gordon equation ,
As known, this equation can be obtained by Euler-Lagrange equation (2.2) with the Lagrangian function
Now we consider the triangle discretization and square discretization, respectively, to get the nonstandard finite difference variational integrators.
3.1. Triangle Discretization
Following the steps in last section and using the idea of nonstandard finite difference, we define the discrete Lagrangian as based on the following constructing rules, (1)The discrete first derivative is represented by where the denominator functions are defined by (2.6), and (2)Nonlocal representations for and are given by where , , , and are positive parameters. Such discretizations for and guarantee the symmetric property of the discrete Lagrangian function .
Similarly, we define discrete Lagrangians on other two adjoint triangles,
Now, the discrete variational integrator with nonstandard finite difference methods could be obtained by discrete Euler-Lagrange equation (2.12):
Substituting , , and into above equation, we arrive at
Using the definition of discrete Lagrangian functions, one can find that this scheme is symmetric with respect to and , and ; that it is multisymplectic, and that it preserves the multisymplectic structure of the original equation.
Its corresponding discrete multisymplectic form formula can be obtained from (2.17), that is, where
It shows the multisymplectic structure of scheme (3.9), and the relations between the field values on the three adjoint triangles are around .
The above results are summarized in the following theorem.
Theorem 3.1. The nonstandard finite difference variational integrator (3.9) for the nonlinear Klein-Gordon equation (3.1) is multisymplectic, and its truncation error is . The discrete multisymplectic structure of this scheme is presented by (3.10).
3.2. Square Discretization
Following the philosophy of the nonstandard finite difference method, we define the discrete Lagrangian on as
In this case,(1)the discrete first-derivative is represented by where the denominator functions are defined by (2.6), and (2)nonlocal representations for and are
Similarly, we have the definitions of on the other three squares adjoint to :
Substituting the discrete Lagrangian , , , and into the previous equation, we arrive at
After simple calculations, it becomes
It is multisymplectic and symmetric in and , and . Similarly, we have the discrete multisymplectic form formula:
To study the truncation error of the integrator, we do the Taylor expansion which leads to
Combing these equations, we can readily observe that the nonstandard finite difference variational integrator (3.21) has truncation error . To verify this conclusion, we investigate the numerical convergence order in our numerical experiments. See Section 4.
We summarize our conclusion in the following theorem.
Theorem 3.2. The nonstandard finite difference variational integrator (3.21) for the nonlinear Klein-Gordon equation (3.1) is multisymplectic, and its truncation error is . The discrete multisymplectic structure of this scheme is presented by (3.22).
4. Numerical Simulations
In this section, we report the performance of the nonstandard finite difference variational integrator (2.14) for solving linear wave equation (2.1) and the nonstandard finite difference variational integrators (3.21) and (3.9) for the nonlinear Klein-Gordon equation (3.1).
4.1. Linear Wave Equation
For linear wave equation (2.1), we consider the initial conditions and the periodic boundary conditions
The nonstandard finite difference variational integrator (2.14) is an explicit five points scheme. We choose the denominator functions and in as and .
From Figure 1, we can see that the nonstandard finite difference variational integrator (2.14) for the linear wave equation performs very well and the periodicity of the linear wave equation is preserved accurately.
4.2. Nonlinear Klein-Gordon Equation
We now consider the nonlinear Klein-Gordon equation (3.1) with the initial condition and periodic boundary conditions.
We use nonstandard finite difference variational integrator (3.21) to simulate this problem with amplitude . The nonstandard finite difference variational integrator (2.14) is an implicit nine-points nonstandard finite difference scheme. The denominator functions and are defined the same as before.
As depicted in Figure 2, the nonstandard finite difference variational integrator (3.21) simulates the wave propagation perfectly at the beginning. After a long time simulation, the integrator still performs very accurate and stable, without showing any blowup. With periodic boundary condition, the wave going out the computational domain shows up in the other direction periodically.
(a) Waveforms at the beginning from to
(b) Waveforms from to
To further investigate the numerical convergence of the proposed schemes, we conduct a series of numerical tests of our nonlinear integrators. In this example, we consider the nonlinear Klein-Gordon equation (3.1) with the initial boundary conditions as follows: where . The exact solution of the problem is
The nonstandard finite difference integrators (3.9) and (3.21) are applied to simulate the Klein-Gordon equation. In the integrator (3.9), we choose here. The -norm errors at , , and are listed in Tables 1 and 2. The orders in the tables are calculated with the formula [35, 36]