Journal of Applied Mathematics

Volume 2012 (2012), Article ID 125926, 17 pages

doi:10.1155/2012/125926

Research Article

Convergence and Stability in Collocation Methods of Equation

Han Yan,¹ Shufang Ma,² Yanbin Liu,¹ and Hongquan Sun¹

¹School of Mathematical Sciences, Heilongjiang University, Harbin 150080, China

²Department of Mathematics, Northeast Forest University, Harbin 150040, China

Received 13 August 2012; Accepted 4 October 2012

Academic Editor: J. C. Butcher

Copyright © 2012 Han Yan 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.

Abstract.

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods for . The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

1. Introduction

This paper deals with the convergence, superconvergence, and stability of the collocation methods of the following differential equation with piecewise continuous argument (EPCA):

where is an integer, , is a given initial value, is an unknown function, and denotes the greatest integer function. The general form of EPCA is

where the argument has intervals of constancy. This kind of equations has been initiated by Wiener [1, 2], Cooke and Wiener [3], and Shah and Wiener [4]. The general theory and basic results for EPCA have by now been thoroughly investigated in the book of Wiener [5].

There are some authors who have considered the stability of numerical solutions for this kind of equations (see [6–8]). Though (1.1) is a delay differential equation (see [9–11]), the delay function is discontinuous. In [12], the convergence and superconvergence of collocation methods for a differential equation with piecewise linear delays is concerned.

Definition 1.1 (see Wiener [5]). A solution of (1.1) on is a function that satisfies the following conditions.

(1) is continuous on .(2)The derivative exists at each point , with the possible exception of the point , where one-sided derivatives exist.(3)(1.1) is satisfied on each interval with integral endpoints.

Theorem 1.2 (see Wiener [5]). Equation (1.1) has on a unique solution

where is the fractional part of and

Equation (1.1) is asymptotically stable (the solution of (1.1) tends to zero as ), for all , if and only if the inequalities

hold.

2. Existence and Uniqueness of Collocation Methods

Let be a given step size with integer and let the mesh on be defined by

Accordingly, the collocation points are chosen as

where denotes a given set of collocation parameters.

We approximate the solution by collocation in the piecewise polynomial spaces

where denotes the set of all real polynomials of degree not exceeding . The collocation solution is the element in this space that satisfies the collocation equation

Let . Then

where

Integrating the above equality, we can get that

where . So

Let , , . We have

where .

Denote , ,  , and for any , if . We have

When the solution of (2.10) has been found, the collocation solution on the interval is determined by

So we can obtain the following theorem.

Theorem 2.1. Assume that the given functions in (1.1) satisfy , where . Then there exists an so that for the mesh with mesh diameter satisfying , and each of the linear algebraic systems (2.10) has a unique solution . Hence the collocation of (2.4) defines a unique collocation solution for the initial-value problem (1.1), and its representation on the subinterval is given by (2.11).

3. Global Convergence Results

In the following, unless otherwise specified, the derivatives of and denote the left derivatives.

Theorem 3.1. Assume the following:

(1) the given functions in (1.1) satisfy , ;(2) is the collocation solution to (1.1) defined by (2.10) and (2.11) with .

Then the estimates

hold for any set of collocation points with . The constants dependent on the collocation parameters and but not on .

Proof. The collocation error satisfies the equation

with . Assumption implies that (at , the derivative of denotes the right derivative and at , which denotes the left derivative) and hence . Thus we have, using Peano's Theorem for on ,

with the Peano remainder term, and Peano kernel are given by

Integration of (3.3) leads to

where

Recalling the local representation (2.5) of the collocation solution on and setting , the collocation error on may be written as

while

Since is continuous in , and hence at the mesh points, we also have the relation

with . The fact that yields

We are now ready to establish the estimates in Theorem 3.1. Let ; since the collocation error satisfies

it follows from (3.7) and (3.8) that

Denote

we can get that

According to Theorem 2.1, this linear system has a unique solution whenever , and hence there exists a constant so that uniformly for . Here, for denotes the matrix (operator) norm induced by the -norm in . Denote , , and . So

Equation (3.14) now leads to the estimate

with obvious meanings of and . By using the discrete Gronwall inequality, its solution is bounded by

and so (3.15) yields

Denote

we have

This concludes the proof of Theorem 3.1.

4. Global Superconvergence Results

Theorem 4.1. Assume that the assumptions of Theorem 3.1 hold, and let be replaced by and , with . If the collocation parameters are subject to the orthogonality condition

then the corresponding collocation solution satisfies, for ,

with depending on the collocation parameters and on but not on . The exponent cannot, in general, be replaced by . For the derivative , we attain only .

Proof. Let

denote the defect (or: residual) associated with the collocation solution to the initial-value problem (1.1). by definition of the collocation solution the defect vanishes on the set as follows:

Moreover, the uniform convergence of and established in Theorem 3.1 implies the uniform boundedness (as ) of on , as well as that of its derivatives of order not exceeding (here the derivatives refer to the left derivatives).

It following from (4.3) that the collocation error satisfies the equation

By Theorem 3.1, there exists a constant , such that

and this holds for any choice of the . On the other hand, the collocation error solves the initial-value problem

For , whose solution is given by

The function denotes the “resolvent” (or: resolvent kernel) of (1.1) as follows:

If , let , , and ; we have

Suppose now that each of the integrals over is approximated by the interpolatory -point quadrature formula with abscissas , then

Here, terms denote the quadrature errors induced by these quadrature approximations. By assumption (4.1) each of these quadrature formulas has degree of precision , and thus the Peano Theorem for quadrature implies that the quadrature errors can be bounded by

because the defect is in on each subinterval . Due to the special choice of the quadrature abscissas, we have , because whenever . Hence

This leads to the estimate

for and , with .

We assume for

Then for , let , and ; we have

Similarly to the case of , we have

This completes the proof.

5. The Local Superconvergence Results on

Theorem 5.1. Assume the following:

(a) and , for some with and value as specified in below, (b)The distinct collocation parameters are chosen so that the general orthogonality condition

holds, with .

Then, for all meshes with , the collocation solution corresponding to the collocation points based on these satisfies

where depends on the collocation parameters and on but not on .

Proof. If , for

with

so

By the induction method similarly to the proof of Theorem 4.1, the assertion of Theorem 5.1 follows.

6. Numerical Stability

In this section, we will discuss the stability of the collocation methods. We introduce the set consisting of all pairs which satisfy the condition

and divide the region into three parts:

By (2.9) and (2.10), we can obtain that

where , , and .

Let and . It is easy to see

where

Let . Then there exists such that

since and is continuous in a neighborhood of zero. In the rest of the paper we define

Definition 6.1 (see [6]). Process (2.11) for (1.1) is called asymptotically stable at if and only if for all and .

(i) is invertible. (ii) for any given relation (6.4) defines that satisfy for .

Definition 6.2 (see [6]). The set of all pairs at which the process (2.11) for (1.1) is asymptotically stable is called asymptotical stability region denoted by .

Theorem 6.3 (see [6]). Suppose that the collocation method is -stable and the stability function is given by the -Padé approximation to the exponential . Then if and only if is even.

Theorem 6.4 (see [6]). Suppose that the stability function of the collocation method is given by the -Padé approximation to the exponential . Then if and only if is even.

Theorem 6.5 (see [6]). For all the collocation methods, we have .

Using the above theorems we can formulate the following result.

Theorem 6.6 (see [6]). Suppose that the collocation method is -stable and the stability function is given by the -Padé approximation to the exponential . Then and if and only if both and are even,

Corollary 6.7. For the -stable higher order collocation methods, it is easy to see from Theorem 6.6.

(i) For the -stage Gauss-Legendre method, if and only if is even.(ii)For the -stage Lobatto IIIA method, if and only if is old. (iii)For the -stage Radau IIA method, if and only if is old and if and only if is even.

7. Numerical Experiments

In order to give a numerical illustration to the conclusions in the paper, we consider the following two problems ([6]):

It can be checked that and .

For illustrating the convergence and superconvergence orders in this paper, we choose and and use the Gauss collocation parameters: , , the Radau IIA collocation parameters: , , the Lobatto IIIA collocation parameters: , , and three sets of random collocation parameters: , ; , ; , , respectively, for ; and we use the Gauss collocation parameters: , , and , the Radau IIA collocation parameters: , , and , the Lobatto IIIA collocation parameters: , , and , and three sets of random collocation parameters: , , ; , , ; , , , respectively, for . In Tables 1, 2, 3, and 4 we list the absolute values of the absolute errors of for the six collocation parameters and for and , respectively, and the ratios of the absolute values of the errors of over that of .

Table 1: The absolute values of absolute errors of for example (7.1) with .

Gauss
Radau IIA
Lobatto IIIA

Ratio

Table 2: The absolute values of absolute errors of for example (7.1) with .

Gauss
Radau IIA
Lobatto IIIA

Ratio

9.4201

Table 3: The absolute values of absolute errors of for example (7.2) with .

Gauss
Radau IIA
Lobatto IIIA

Ratio

Table 4: The absolute values of absolute errors of for example (7.2) with .

Gauss
Radau IIA
Lobatto IIIA

Ratio

From the above tables, we can see that the convergence orders are consistent with our theoretical analysis.

In Figures 1, 2, 3, 4, 5, 6, 7, and 8, we draw the absolute values of the numerical solution of collocation methods. It is easy to see that the numerical solution is asymptotically stable.

Figure 1: The Gauss collocation method with and for (7.1).

Figure 2: The Radau IIA collocation method with and for (7.1).

Figure 3: The Gauss collocation method with and for (7.1).

Figure 4: The Radau IIA collocation method with and for (7.1).

Figure 5: The Gauss collocation method with and for (7.2).

Figure 6: The Radau IIA collocation method with and for (7.2).

Figure 7: The Gauss collocation method with and for (7.2).

Figure 8: The Radau IIA collocation method with and for (7.2).

Acknowledgments

This work is supported by the Research Fund of the Heilongjiang Provincial Education Department (no. 11551363, no. 12511414), the National Nature Science Foundation of China (no. 11101130).

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