Convergence and Stability in Collocation Methods of Equation u ′

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods for u′ t au t bu t . 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.


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 T is an integer, a, b ∈ R, u 0 ∈ C d is a given initial value, u t ∈ C d is an unknown function, and · denotes the greatest integer function. The general form of EPCA is u t f t, u t , u α t , t ≥ 0, Journal of Applied Mathematics where the argument α t 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 t − t is discontinuous. In 12 , the convergence and superconvergence of collocation methods for a differential equation with piecewise linear delays is concerned.

Existence and Uniqueness of Collocation Methods
Let h : 1/p be a given step size with integer p ≥ 1 and let the mesh on I be defined by Accordingly, the collocation points are chosen as where {c i } denotes a given set of collocation parameters. We approximate the solution by collocation in the piecewise polynomial spaces S 0 m 0, T : v ∈ C 0, T : v| t n ,t n 1 ∈ π m , 2.3 Mathematics   3 where π m denotes the set of all real polynomials of degree not exceeding m. The collocation solution u h is the element in this space that satisfies the collocation equation

Journal of Applied
Let Y n,j : u h t n c j h . Then Integrating the above equality, we can get that When the solution Y n of 2.10 has been found, the collocation solution on the interval t n , t n 1 is determined by u h t n vh u h t n hβ T v Y n .

2.11
So we can obtain the following theorem.

Global Convergence Results
In the following, unless otherwise specified, the derivatives of u and u h denote the left derivatives.
hold for any set X h k 1, 2, . . . of collocation points with 0 < c 1 < · · · < c m ≤ 1. The constants C ν dependent on the collocation parameters {c i } and but not on h.
Proof. The collocation error e h : u − u h satisfies the equation with e h 0 0. Assumption 1 implies that u ∈ C m 1 t n , t n 1 at t n , the derivative of u denotes the right derivative and at t n 1 , which denotes the left derivative and hence u ∈ C m t n , t n 1 . Thus we have, using Peano's Theorem for u on t n , t n 1 , with the Peano remainder term, and Peano kernel are given by Recalling the local representation 2.5 of the collocation solution u h on t n , t n 1 and setting ε n,j : u t n,j − Y n,j , the collocation error e h : u − u h on t n , t n 1 may be written as Since e h is continuous in 0, T , and hence at the mesh points, we also have the relation 3.14 According to Theorem 2.1, this linear system has a unique solution whenever h ∈ 0, h , and hence there exists a constant D 0 < ∞ so that I m×m − hA n −1 .14 now leads to the estimate with obvious meanings of γ 0 and γ 1 . By using the discrete Gronwall inequality, its solution is bounded by

3.20
This concludes the proof of Theorem 3.1.

Global Superconvergence Results
with C 2 depending on the collocation parameters and on u m 2 ∞ but not on h. The function r r t, s denotes the "resolvent" or: resolvent kernel of 1.1 as follows: r t, s : e a t−s , with r ∈ C m 1 D .

4.11
Here, terms E j v denote the quadrature errors induced by these quadrature approximations. By assumption 4.1 each of these quadrature formulas has degree of precision m, and thus the Peano Theorem for quadrature implies that the quadrature errors can be bounded by Journal of Applied Mathematics 9 because the defect δ h is in C m 1 on each subinterval t n , t n 1 . Due to the special choice of the quadrature abscissas, we have m i 1 b j r t l vh, t j hc i δ h t j hc i 0, because δ h t 0 whenever t ∈ X h . Hence r t l vh, t l hs δ h t l hs ds, v ∈ 0, 1 .

4.13
This leads to the estimate .14 for 0 ≤ l ≤ p − 1 and v ∈ 0, 1 , with r 0 : max t∈I T 0 |r t, s |ds. We assume for t ∈ k − 1, k Then for t ∈ k, k 1 , let t t kp l vh, v ∈ 0, 1 , and 0 ≤ l ≤ p − 1; we have

4.16
Similarly to the case of t ∈ 0, 1 , we have

4.17
This completes the proof.

The Local Superconvergence Results on I h
where C 3 depends on the collocation parameters and on u m κ 1 ∞ but not on h. By the induction method similarly to the proof of Theorem 4.1, the assertion of Theorem 5.1 follows.

Numerical Stability
In this section, we will discuss the stability of the collocation methods. We introduce the set H consisting of all pairs a, b ∈ R 2 which satisfy the condition H : a, b : −a e a 1 e a − 1 < b < −a , 6.1 and divide the region into three parts:   0 · · · 0 b 1,p 1 0 · · · 0 b 2,p 1 · · · · · · · · · · · · 0 · · · 0 b p 1,p 1 i I − xA is invertible.
ii for any given u i 1 ≤ i ≤ m relation 6.4 defines U k k 1, 2, . . . that satisfy U k → 0 for k → ∞. Definition 6.2 see 6 . The set of all pairs a, b at which the process 2.11 for 1.1 is asymptotically stable is called asymptotical stability region denoted by S. Using the above theorems we can formulate the following result.

Numerical Experiments
In order to give a numerical illustration to the conclusions in the paper, we consider the following two problems 6 : For illustrating the convergence and superconvergence orders in this paper, we choose 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.