Convergence and Stability in Collocation Methods of Equation 𝑢^′(𝑡)=𝑎𝑢(𝑡)+𝑏𝑢([𝑡])

Abstract

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.

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): 𝑢^′(𝑡)(𝑡)𝑡[],𝑢(0)=𝑎𝑢+𝑏𝑢([]),𝑡∈0,𝑇=𝑢₀, (1.1) where 𝑇 is an integer, 𝑎,𝑏∈ℝ, 𝑢₀∈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)((𝑡)(𝛼(𝑡𝑢(0)=𝑓𝑡,𝑢,𝑢))),𝑡≥0,=𝑢₀, (1.2) 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.

Definition 1.1 (see Wiener [5]). A solution of (1.1) on [0,∞) is a function 𝑢(𝑡) that satisfies the following conditions.(1)𝑢(𝑡) is continuous on [0,∞).(2)The derivative 𝑢^′(𝑡) exists at each point 𝑡∈[0,∞), with the possible exception of the point [𝑡]∈[0,∞), where one-sided derivatives exist.(3)(1.1) is satisfied on each interval [𝑘,𝑘+1)⊂[0,∞) with integral endpoints.

Theorem 1.2 (see Wiener [5]). Equation (1.1) has on [0,∞) a unique solution 𝑢(𝑡)=𝑚₀({𝑡})𝑏₀^[𝑡]𝑢₀, (1.3) where {𝑡} is the fractional part of 𝑡 and 𝑚₀(𝑡)∶=𝑒^𝑎𝑡+𝑒^𝑎𝑡−1𝑎⁻¹𝑏,𝑏₀∶=𝑚₀(1). (1.4) Equation (1.1) is asymptotically stable (the solution of (1.1) tends to zero as 𝑡→∞), for all 𝑢₀, if and only if the inequalities 𝑒−𝑎^𝑎+1𝑒^𝑎−1<−𝑏<−𝑎 (1.5) hold.

2. 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 𝐼_ℎ∶=𝑡_𝑛∶0=𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑇. (2.1) Accordingly, the collocation points are chosen as 𝑋_ℎ∶=𝑡_𝑛,𝑖=𝑡_𝑛+𝑐_𝑖ℎ∶0<𝑐₁<⋯<𝑐_𝑚≤1(0≤𝑛≤𝑁−1), (2.2) where {𝑐_𝑖} denotes a given set of collocation parameters.

We approximate the solution by collocation in the piecewise polynomial spaces 𝑆_𝑚⁽⁰⁾([0,𝑇])∶=𝑣∈𝐶([0,𝑇])∶𝑣|_{[𝑡𝑛,𝑡𝑛+1]}∈𝜋_𝑚, (2.3) 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 𝑢^′_ℎ(𝑡)=𝑎𝑢_ℎ(𝑡)+𝑏𝑢_ℎ([𝑡]),𝑡∈𝑋_ℎ,𝑢_ℎ(0)=𝑢₀. (2.4) Let 𝑌_𝑛,𝑗∶=𝑢^′_ℎ(𝑡_𝑛+𝑐_𝑗ℎ). Then 𝑢^′_ℎ𝑡_𝑛+𝑣ℎ=^𝑚_𝑗=1𝐿_𝑗(𝑣)𝑌_𝑛,𝑗,𝑣∈(0,1], (2.5) where 𝐿_𝑗(𝑣)∶=^𝑚_{𝑖=1,𝑖≠𝑗}𝑣−𝑐_𝑖𝑐_𝑗−𝑐_𝑖. (2.6) Integrating the above equality, we can get that 𝑢_ℎ𝑡_𝑛+𝑣ℎ=𝑢_ℎ𝑡_𝑛+ℎ^𝑚_𝑗=1𝛽_𝑗(𝑣)𝑌_𝑛,𝑗, (2.7) where 𝛽_𝑗(𝑣)∶=∫^𝑣₀𝐿_𝑗(𝑠)𝑑𝑠. So 𝑌_𝑛,𝑖=𝑎𝑢_ℎ𝑡_𝑛,𝑖+𝑏𝑢_ℎ𝑡_𝑛,𝑖. (2.8) Let 𝑛=𝑘𝑝+𝑙, 𝑘∈ℤ, 𝑙=0,1,2,…,𝑝−1. We have 𝑌_{𝑘𝑝+𝑙,𝑖}=𝑎𝑢_ℎ𝑡_{𝑘𝑝+𝑙,𝑖}+𝑏𝑢_ℎ𝑡_𝑘𝑝=𝑎𝑢_ℎ𝑡_{𝑘𝑝+𝑙}+ℎ^𝑚_𝑗=1𝑎_𝑖𝑗𝑌_{𝑘𝑝+𝑙,𝑗}+𝑏𝑢_ℎ𝑡_𝑘𝑝, (2.9) where 𝑎_𝑖𝑗∶=𝛽_𝑗(𝑐_𝑖).

Denote 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑚, 𝑌_𝑛=(𝑌_𝑛,1,𝑌_𝑛,2,…,𝑌_𝑛,𝑚)^𝑇,  𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑚)^𝑇, 𝑒=(1,1,…,1)^𝑇 and for any 𝑥_𝑗∈ℝ, ∑⁻¹_𝑗=0𝑥_𝑗=0 if 𝑘=0. We have 𝐼_𝑚×𝑚−ℎ𝑎𝐴𝑌_{𝑘𝑝+𝑙}=𝑢_ℎ𝑡_{𝑘𝑝+𝑙}𝑎𝑒+𝑢_ℎ𝑡_𝑘𝑝𝑏𝑒. (2.10) 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_ℎ𝑡_𝑛+𝑣ℎ=𝑢_ℎ𝑡_𝑛+ℎ𝛽^𝑇(𝑣)𝑌_𝑛. (2.11)

So we can obtain the following theorem.

Theorem 2.1. Assume that the given functions in (1.1) satisfy 𝑎,𝑏∈ℝ,𝐾∈𝐶(𝐷), where 𝐷∶={(𝑡,𝑠)∶0≤𝑠≤𝑡≤𝑇}. Then there exists an ℎ>0 so that for the mesh 𝐼_ℎ with mesh diameter ℎ>0 satisfying ℎ<ℎ, and each of the linear algebraic systems (2.10) has a unique solution 𝑌_𝑛∈ℝ^𝑚. Hence the collocation of (2.4) defines a unique collocation solution 𝑢_ℎ∈𝑆_𝑚⁽⁰⁾(𝐼_ℎ) for the initial-value problem (1.1), and its representation on the subinterval [𝑡_𝑛,𝑡_𝑛+1] is given by (2.11).

3. Global Convergence Results

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

Theorem 3.1. Assume the following:(1) the given functions in (1.1) satisfy 𝑎,𝑏∈ℝ, 𝐾∈𝐶^𝑚(𝐷);(2)𝑢_ℎ∈𝑆_𝑚⁽⁰⁾(𝐼_ℎ) is the collocation solution to (1.1) defined by (2.10) and (2.11) with ℎ∈(0,ℎ). Then the estimates 𝑢^(𝜈)−𝑢_ℎ^(𝜈)_∞∶=max_{𝑡∈[0,𝑇]}𝑢^(𝜈)(𝑡)−𝑢_ℎ^(𝜈)(𝑡)≤𝐶_𝜈𝑢^(𝑚+1)_∞ℎ^𝑚(𝜈=0,1) (3.1) hold for any set 𝑋_ℎ(𝑘=1,2,…) of collocation points with 0<𝑐₁<⋯<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 𝑒^′_ℎ(𝑡)=𝑎𝑒_ℎ(𝑡)+𝑏𝑒_ℎ([𝑡]),𝑡∈𝑋_ℎ, (3.2) 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], 𝑢^′𝑡_𝑛+𝑣ℎ=^𝑚_𝑗=1𝐿_𝑗(𝑣)𝑢^′𝑡_𝑛,𝑗+ℎ^𝑚𝑅⁽¹⁾_𝑚+1,𝑛(𝑣),𝑣∈(0,1], (3.3) with the Peano remainder term, and Peano kernel are given by 𝑅⁽¹⁾_𝑚+1,𝑛(𝑣)∶=¹₀𝐾_𝑚(𝑣,𝑧)𝑢^(𝑚+1)𝑡_𝑛𝐾+𝑧ℎ𝑑𝑧,_𝑚()1𝑣,𝑧∶=()!()𝑚−1𝑣−𝑧₊^𝑚−1−^𝑚_𝑗=1𝐿_𝑗(𝑣)𝑐_𝑗−𝑧₊^𝑚−1,𝑣∈(0,1]. (3.4) Integration of (3.3) leads to 𝑢𝑡_𝑛+𝑣ℎ=𝑢𝑡_𝑛+ℎ^𝑚_𝑗=1𝛽_𝑗(𝑣)𝑢^′𝑡_𝑛,𝑗+ℎ^𝑚+1𝑅_𝑚+1,𝑛(𝑣),𝑣∈(0,1], (3.5) where 𝑅_𝑚+1,𝑛(𝑣)∶=^𝑣₀𝑅⁽¹⁾_𝑚+1,𝑛(𝑠)𝑑𝑠. (3.6) 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 𝑒_ℎ𝑡_𝑛+𝑣ℎ=𝑒_ℎ𝑡_𝑛+ℎ^𝑚_𝑗=1𝛽_𝑗(𝑣)𝜀_𝑛,𝑗+ℎ^𝑚+1𝑅_𝑚+1,𝑛(𝑣),𝑣∈(0,1], (3.7) while 𝑒^′_ℎ𝑡_𝑛+𝑣ℎ=^𝑚_𝑗=1𝐿_𝑗(𝑣)𝜀_𝑛,𝑗+ℎ^𝑚𝑅⁽¹⁾_𝑚+1,𝑛(𝑣),𝑣∈(0,1]. (3.8) Since 𝑒_ℎ is continuous in [0,𝑇], and hence at the mesh points, we also have the relation 𝑒_ℎ𝑡_𝑛=𝑒_ℎ𝑡_𝑛−1+ℎ=𝑒_ℎ𝑡_𝑛−1+ℎ^𝑚_𝑗=1𝑏_𝑗𝜀_{𝑛−1,𝑗}+ℎ^𝑚+1𝑅_{𝑚+1,𝑛−1}(1),𝑛=1,…,𝑁−1, (3.9) with 𝑏_𝑗∶=𝛽_𝑗(1). The fact that 𝑒_ℎ(0)=0 yields 𝑒_ℎ𝑡_𝑛=ℎ^𝑚_𝑗=1𝑏_𝑗^𝑛−1_𝑟=0𝜀_𝑟,𝑗+ℎ^{𝑚+1𝑛−1}_𝑟=0𝑅_𝑚+1,𝑟(1),𝑛=1,…,𝑁−1. (3.10) We are now ready to establish the estimates in Theorem 3.1. Let 𝑛=𝑘𝑝+𝑙(𝑙=0,1,…,𝑝−1); since the collocation error satisfies 𝑒^′_ℎ𝑡_{𝑘𝑝+𝑙,𝑖}=𝑎𝑒_ℎ𝑡_{𝑘𝑝+𝑙,𝑖}+𝑏𝑒_ℎ𝑡_𝑘𝑝, (3.11) it follows from (3.7) and (3.8) that 𝜀_{𝑘𝑝+𝑙,𝑖}=𝑒^′_ℎ𝑡_{𝑘𝑝+𝑙,𝑖}=𝑎𝑒_ℎ𝑡_{𝑘𝑝+𝑙,𝑖}+𝑏𝑒_ℎ𝑡_𝑘𝑝=𝑎𝑒_ℎ𝑡_{𝑘𝑝+𝑙}+ℎ^𝑚_𝑗=1𝑎_𝑖𝑗𝜀_{𝑘𝑝+𝑙,𝑗}+ℎ^𝑚+1𝑅_{𝑚+1,𝑘𝑝+𝑙}𝑐_𝑖+𝑏𝑒_ℎ𝑡_𝑘𝑝. (3.12)
Denote 𝜀_𝑛∶=𝜀_𝑛,1,𝜀_𝑛,2,…,𝜀_𝑛,𝑚^𝑇,𝑅_𝑚+1,𝑛∶=𝑅_𝑚+1,𝑛𝑐₁,𝑅_𝑚+1,𝑛𝑐₂,…,𝑅_𝑚+1,𝑛𝑐_𝑚^𝑇, (3.13) we can get that 𝐼_𝑚×𝑚−ℎ𝑎𝐴𝜀_{𝑘𝑝+𝑙}=𝑒_ℎ𝑡_{𝑘𝑝+𝑙}𝑎𝑒+𝑒_ℎ𝑡_𝑘𝑝𝑏𝑒+𝑎ℎ^𝑚+1𝑅_{𝑚+1,𝑘𝑝+𝑙}. (3.14) According to Theorem 2.1, this linear system has a unique solution whenever ℎ∈(0,ℎ), and hence there exists a constant 𝐷₀<∞ so that ‖(𝐼_𝑚×𝑚−ℎ𝐴_𝑛)⁻¹‖₁≤𝐷₀ uniformly for 0≤𝑛≤𝑁−1. Here, for 𝐵∈𝐿(ℝ^𝑚),‖𝐵‖₁ denotes the matrix (operator) norm induced by the 𝑙₁-norm in ℝ^𝑚. Denote 𝑀_𝑚+1∶=‖𝑢^(𝑚+1)‖_∞, 𝐾_𝑚∶=max_𝑣∈[0,1]∫¹₀|𝐾_𝑚(𝑣,𝑧)|𝑑𝑧,𝑏∶=max_{1≤𝑗≤𝑚}|𝑏_𝑗|, and 𝛽∶=max_{1≤𝑖≤𝑚,𝑣∈[0,1]}𝛽_𝑖(𝑣). So 𝑒_ℎ𝑡_𝑛≤ℎ𝑏^𝑛−1_𝑟=0𝜀_𝑟1+ℎ^𝑚𝑇𝐾_𝑚𝑀_𝑚+1. (3.15)
Equation (3.14) now leads to the estimate 𝜀_{𝑘𝑝+𝑙1}≤𝐷₀𝑚|𝑎|ℎ𝑏^{𝑘𝑝+𝑙−1}_𝑟=0𝜀_𝑟1+|𝑎|𝑚ℎ^𝑚𝑇𝐾_𝑚𝑀_𝑚+1+|𝑏|𝑚ℎ𝑏^𝑘𝑝−1_𝑟=0𝜀_𝑟1+|𝑏|𝑚ℎ^𝑚𝑇𝐾_𝑚𝑀_𝑚+1+|𝑎|ℎ^𝑚+1𝑚𝑀_𝑚+1𝐾_𝑚≤𝛾₀ℎ^{𝑘𝑝+𝑙−1}_𝑟=0𝜀_𝑟1+𝛾₁𝑀_𝑚+1ℎ^𝑚, (3.16) with obvious meanings of 𝛾₀ and 𝛾₁. By using the discrete Gronwall inequality, its solution is bounded by 𝜀_𝑛1≤𝛾₁𝑀_𝑚+1ℎ^𝑚exp𝛾₀𝑇=∶𝐵𝑀_𝑚+1ℎ^𝑚, (3.17) and so (3.15) yields 𝑒_ℎ𝑡_𝑛≤𝑏𝐵+𝐾_𝑚𝑇𝑀_𝑚+1ℎ^𝑚. (3.18) Denote Λ_𝑚∶=max_{[]1≤𝑗≤𝑚,𝑣∈0,1}𝐿_𝑗(𝑣), (3.19) we have 𝑒_ℎ𝑡_𝑛+𝑣ℎ≤𝑏𝐵+𝐾_𝑚𝑇𝑀_𝑚+1ℎ^𝑚+ℎ𝛽𝐵𝑀_𝑚+1ℎ^𝑚+ℎ^𝑚+1𝑀_𝑚+1𝐾_𝑚=∶𝐶₀𝑀_𝑚+1ℎ^𝑚,𝑒^′_ℎ𝑡_𝑛+𝑣ℎ≤Λ_𝑚𝜀_𝑛1+ℎ^𝑚𝐾_𝑚𝑀_𝑚+1≤Λ_𝑚𝐵𝑀_𝑚+1ℎ^𝑚+ℎ^𝑚𝐾_𝑚𝑀_𝑚+1=∶𝐶₁𝑀_𝑚+1ℎ^𝑚. (3.20) This concludes the proof of Theorem 3.1.

4. Global Superconvergence Results

Theorem 4.1. Assume that the assumptions (2) of Theorem 3.1 hold, and let (1) be replaced by 𝑎,𝑏∈𝐶^𝑑(𝐼) and 𝐾∈𝐶^𝑑(𝐷), with 𝑑≥𝑚+1. If the 𝑚 collocation parameters {𝑐_𝑖} are subject to the orthogonality condition 𝐽₀∶=¹₀^𝑚_𝑖=1𝑠−𝑐_𝑖𝑑𝑠=0, (4.1) then the corresponding collocation solution 𝑢_ℎ∈𝑆_𝑚⁽⁰⁾(𝐼_ℎ) satisfies, for ℎ∈(0,ℎ), 𝑢−𝑢_ℎ∞≤𝐶₂ℎ^𝑚+1, (4.2) with 𝐶₂ depending on the collocation parameters and on ‖𝑢^(𝑚+2)‖_∞ but not on ℎ. The exponent 𝑚+1 cannot, in general, be replaced by 𝑚+2. For the derivative 𝑢^′_ℎ, we attain only ‖𝑢^′−𝑢^′_ℎ‖_∞=𝑂(ℎ^𝑚).

Proof. Let 𝛿_ℎ(𝑡)∶=−𝑢^′_ℎ(𝑡)+𝑎𝑢_ℎ(𝑡)+𝑏𝑢_ℎ([𝑡]),𝑡∈𝐼, (4.3) denote the defect (or: residual) associated with the collocation solution 𝑢_ℎ∈𝑆_𝑚⁽⁰⁾(𝐼_ℎ) to the initial-value problem (1.1). by definition of the collocation solution the defect 𝛿_ℎ vanishes on the set 𝑋_ℎ as follows: 𝛿_ℎ(𝑡)=0∀𝑡∈𝑋_ℎ. (4.4) Moreover, the uniform convergence of 𝑢_ℎ and 𝑢^′_ℎ established in Theorem 3.1 implies the uniform boundedness (as ℎ→0) of 𝛿_ℎ on 𝐼, as well as that of its derivatives of order not exceeding 𝑑 (here the derivatives refer to the left derivatives).
It following from (4.3) that the collocation error 𝑒_ℎ=𝑢−𝑢_ℎ satisfies the equation 𝛿_ℎ(𝑡)=𝑒^′_ℎ(𝑡)−𝑎𝑒_ℎ(𝑡)−𝑏𝑒_ℎ([𝑡]),𝑡∈𝐼. (4.5) By Theorem 3.1, there exists a constant 𝐷, such that 𝛿_ℎ(𝑡)_∞≤𝐷ℎ^𝑚𝑀_𝑚+1, (4.6) and this holds for any choice of the {𝑐_𝑖}. On the other hand, the collocation error 𝑒_ℎ solves the initial-value problem 𝑒^′_ℎ(𝑡)=𝑎𝑒_ℎ(𝑡)+𝑏𝑒_ℎ([𝑡])+𝛿_ℎ(𝑡),𝑡∈𝐼,𝑒_ℎ(0)=0. (4.7) For 𝑡∈[𝑘,𝑘+1], whose solution is given by 𝑒_ℎ(𝑡)=𝑟(𝑡,𝑘)+^𝑡_𝑘𝑏𝑟(𝑡,𝑠)𝑑𝑠𝑒_ℎ(𝑘)+^𝑡_𝑘𝑟(𝑡,𝑠)𝛿_ℎ(𝑠)𝑑𝑠,𝑡∈𝐼. (4.8) The function 𝑟=𝑟(𝑡,𝑠) denotes the “resolvent” (or: resolvent kernel) of (1.1) as follows: 𝑟(𝑡,𝑠)∶=𝑒^{𝑎(𝑡−𝑠)},with𝑟∈𝐶^𝑚+1(𝐷). (4.9) If 𝑘=0, let 𝑡=𝑡_𝑙+𝑣ℎ, 𝑣∈[0,1], and 0≤𝑙≤𝑝−1; we have 𝑒_ℎ𝑡_𝑙+𝑣ℎ=^𝑡𝑙₀^+𝑣ℎ𝑟𝑡_𝑙+𝑣ℎ,𝑠𝛿_ℎ(𝑠)=𝑑𝑠_{𝑙−1𝑗=0𝑡𝑗+1𝑡𝑗}𝑟𝑡_𝑙+𝑣ℎ,𝑠𝛿_ℎ(𝑠)𝑑𝑠+^𝑡𝑙_𝑡^+ℎ𝑣_𝑙𝑟𝑡_𝑙+𝑣ℎ,𝑠𝛿_ℎ(𝑠)𝑑𝑠=ℎ^𝑙−1_𝑗=0¹₀𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑠𝛿_ℎ𝑡_𝑗+ℎ𝑠𝑑𝑠+ℎ^𝑣₀𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑙+ℎ𝑠𝛿_ℎ𝑡_𝑙+ℎ𝑠𝑑𝑠. (4.10) Suppose now that each of the integrals over [0,1] is approximated by the interpolatory 𝑚-point quadrature formula with abscissas {𝑐_𝑖}, then ¹₀𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑠𝛿_ℎ𝑡_𝑗+ℎ𝑠𝑑𝑠=^𝑚_𝑖=1𝑏_𝑗𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑐_𝑖𝛿_ℎ𝑡_𝑗+ℎ𝑐_𝑖+𝐸_𝑗(𝑣),𝑣∈[0,1]. (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 𝐸_𝑗(𝑣)≤𝑄ℎ^𝑚+1,𝑣∈[0,1], (4.12) 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=1𝑏_𝑗𝑟(𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑐_𝑖)𝛿_ℎ(𝑡_𝑗+ℎ𝑐_𝑖)=0, because 𝛿_ℎ(𝑡)=0 whenever 𝑡∈𝑋_ℎ. Hence 𝑒_ℎ𝑡_𝑙+𝑣ℎ=ℎ^𝑙−1_𝑗=0𝐸_𝑗(𝑣)+ℎ^𝑣₀𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑙+ℎ𝑠𝛿_ℎ𝑡_𝑙+ℎ𝑠𝑑𝑠,𝑣∈[0,1]. (4.13) This leads to the estimate 𝑒_ℎ𝑡_𝑙+𝑣ℎ≤ℎ^𝑙−1_𝑗=0𝑄ℎ^𝑚+1+ℎ𝑟₀𝛿_ℎ∞≤𝑄𝑇ℎ^𝑚+1+𝐷𝑟₀ℎ^𝑚+1𝑀_𝑚+1=∶𝐶₀ℎ^𝑚+1, (4.14) for 0≤𝑙≤𝑝−1 and 𝑣∈[0,1], with 𝑟₀∶=max_𝑡∈𝐼∫^𝑇₀|𝑟(𝑡,𝑠)|𝑑𝑠.
We assume for 𝑡∈[𝑘−1,𝑘]𝑒_ℎ𝑡_{(𝑘−1)𝑝+𝑙}+𝑣ℎ≤𝐶_𝑘−1ℎ^𝑚+1,𝑣∈[0,1],0≤𝑙≤𝑝−1. (4.15) Then for 𝑡∈[𝑘,𝑘+1], let 𝑡=𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑣∈[0,1], and 0≤𝑙≤𝑝−1; we have 𝑒_ℎ𝑡_{𝑘𝑝+𝑙}+𝑣ℎ=𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑘+^{𝑡𝑘𝑝+𝑙}_𝑘^+𝑣ℎ𝑏𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑠𝑑𝑠𝑒_ℎ(𝑘)+^{𝑡𝑘𝑝+𝑙}_𝑘^+𝑣ℎ𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑠𝛿_ℎ(𝑠)=𝑟𝑡𝑑𝑠_{𝑘𝑝+𝑙}+𝑣ℎ,𝑘+^{𝑡𝑘𝑝+𝑙}_𝑘^+𝑣ℎ𝑏𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑠𝑑𝑠𝑒_ℎ(𝑘)+ℎ^{𝑘𝑝+𝑙−1}_{𝑗=𝑘𝑝}¹₀𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑡_𝑗+ℎ𝑠𝛿_ℎ𝑡_𝑗+ℎ𝑠𝑑𝑠+ℎ^𝑣₀𝑟𝑡_{𝑘𝑝+𝑙}+𝑣ℎ,𝑡_{𝑘𝑝+𝑙}+ℎ𝑠𝛿_ℎ𝑡_{𝑘𝑝+𝑙}+ℎ𝑠𝑑𝑠. (4.16) Similarly to the case of 𝑡∈[0,1], we have 𝑒_ℎt_{𝑘𝑝+𝑙}+𝑣ℎ≤𝑟₀+𝑟₀|𝑏|𝐶_𝑘−1ℎ^𝑚+1+𝑝𝑄ℎ^𝑚+2+𝑟₀𝐷𝑀_𝑚+1ℎ^𝑚+1=∶𝐶_𝑘ℎ^𝑚+1. (4.17) This completes the proof.

5. The Local Superconvergence Results on 𝐼_ℎ

Theorem 5.1. Assume the following: (a)𝑎,𝑏∈𝐶^𝑚+𝜅(𝐼) and 𝐾∈𝐶^𝑚+𝜅(𝐷), for some 𝜅 with 1≤𝜅≤𝑚 and value as specified in (𝑏) below, (b)The 𝑚 distinct collocation parameters {𝑐_𝑖} are chosen so that the general orthogonality condition 𝐽_𝜈∶=¹₀𝑠^𝜈𝑚_𝑖=1𝑠−𝑐_𝑖𝑑𝑠=0,𝜈=0,…,𝜅−1 (5.1)holds, with 𝐽_𝜅≠0.
Then, for all meshes 𝐼_ℎ with ℎ∈(0,ℎ), the collocation solution 𝑢_ℎ∈𝑆_𝑚⁽⁰⁾(𝐼_ℎ) corresponding to the collocation points 𝑋_ℎ based on these {𝑐_𝑖} satisfies max𝑢(𝑡)−𝑢_ℎ(𝑡)∶𝑡∈𝐼_ℎ≤𝐶₃ℎ^𝑚+𝜅, (5.2) where 𝐶₃ depends on the collocation parameters and on ‖𝑢^{(𝑚+𝜅+1)}‖_∞ but not on ℎ.

Proof. If 𝑘=0, for 𝑡=𝑡_𝑙(0≤𝑙≤𝑝−1)𝑒_ℎ𝑡_𝑙=^𝑡𝑙₀𝑟𝑡_𝑙,𝑠𝛿_ℎ(𝑠)𝑑𝑠=^𝑙−1_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑟𝑡_𝑙+𝑣ℎ,𝑠𝛿_ℎ(𝑠)𝑑𝑠=ℎ^𝑙−1_𝑗=0¹₀𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑠𝛿_ℎ𝑡_𝑗+ℎ𝑠𝑑𝑠=ℎ^𝑙−1_𝑗=0^𝑚_𝑖=1𝑏_𝑗𝑟𝑡_𝑙+𝑣ℎ,𝑡_𝑗+ℎ𝑐_𝑖𝛿_ℎ𝑡_𝑗+ℎ𝑐_𝑖+𝐸_𝑗(𝑣), (5.3) with 𝐸_𝑗(𝑣)≤𝐶ℎ^𝑚+𝜅, (5.4) so 𝑒_ℎ𝑡_𝑙≤𝐶ℎ^𝑚+𝜅. (5.5) 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 𝐻 consisting of all pairs (𝑎,𝑏)∈ℝ² which satisfy the condition )𝑒𝐻∶=(𝑎,𝑏∶−𝑎^𝑎+1𝑒^𝑎,−1<𝑏<−𝑎 (6.1) and divide the region into three parts: 𝐻₀)∶()},𝐻∶={(𝑎,𝑏𝑎,𝑏∈𝐻,𝑎=0₁)∶()},𝐻∶={(𝑎,𝑏𝑎,𝑏∈𝐻,𝑎<0₂∶={(𝑎,𝑏)∶(𝑎,𝑏)∈𝐻,𝑎>0}. (6.2) By (2.9) and (2.10), we can obtain that 𝑢𝑡_{𝑘𝑝+𝑙+1}=𝑅(𝑥)𝑢𝑡_{𝑘𝑝+𝑙}+𝛼𝑥,𝑦𝑢𝑡_𝑘𝑝,𝑙=0,1,…,𝑝−1, (6.3) where 𝑥∶=ℎ𝑎,𝑦∶=ℎ𝑏, 𝑅(𝑥)∶=1+𝑏^𝑇𝑥(𝐼−𝐴𝑥)⁻¹𝑒, and 𝛼(𝑥,𝑦)∶=𝑦(1+𝑥𝑏^𝑇(𝐼−𝐴𝑥)⁻¹𝑒)=𝑦𝑏^𝑇(𝐼−𝐴𝑥)⁻¹𝑒.

Let 𝑈_𝑘∶=(𝑢_𝑘𝑝,𝑢_𝑘𝑝+1,…,𝑢_{𝑘𝑝+𝑝})^𝑇 and 𝐵∶=∏^𝑝_𝑖=1𝐵_𝑖. It is easy to see 𝑈_𝑘=𝐵𝑈_𝑘+1,𝑘=1,2,…, (6.4) where 𝐵=0⋯0𝑏_1,𝑝+10⋯0𝑏_2,𝑝+1⋯⋯⋯⋯0⋯0𝑏_{𝑝+1,𝑝+1},𝑏_𝑖,𝑝+1=𝑎1+1+𝑏𝑅(𝑥)^𝑖−1()−1,𝑎≠0,1+𝑖−1ℎ𝑏,𝑎=0.𝑖=1,2,…,𝑝+1. (6.5)

Let 𝜑(𝑥)∶=𝑏^𝑇(𝐼−𝑥𝐴)⁻¹𝑒. Then there exists 𝛿>0 such that 𝜑(𝑥)>0∀𝑥with|𝑥|≤𝛿, (6.6) since 𝜑(0)=1 and 𝜑(𝑥) is continuous in a neighborhood of zero. In the rest of the paper we define 𝑀∶=𝑎1,𝑎≤0,𝛿,𝑎>0. (6.7)

Definition 6.1 (see [6]). Process (2.11) for (1.1) is called asymptotically stable at (𝑎,𝑏) if and only if for all 𝑚≥𝑀 and ℎ=1/𝑚. (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.

Theorem 6.3 (see [6]). Suppose that the collocation method is 𝐴₀-stable and the stability function is given by the (𝑟,𝑠)-Padé approximation to the exponential 𝑒^𝑥. Then 𝐻₁⊆𝑆 if and only if 𝑟 is even.

Theorem 6.4 (see [6]). Suppose that the stability function of the collocation method is given by the (𝑟,𝑠)-Padé approximation to the exponential 𝑒^𝑧. Then 𝐻₂⊆𝑆 if and only if 𝑠 is even.

Theorem 6.5 (see [6]). For all the collocation methods, we have 𝐻₀⊆𝑆.

Using the above theorems we can formulate the following result.

Theorem 6.6 (see [6]). Suppose that the collocation method is 𝐴₀-stable and the stability function is given by the (𝑟,𝑠)-Padé approximation to the exponential 𝑒^𝑥. Then 𝐻₀⊆𝑆 and 𝐻⊆𝑆 if and only if both 𝑟 and 𝑠 are even, 𝐻₁⊆𝑆iff𝐻𝑟,₂⊆𝑆iff𝑠iseven. (6.8)

Corollary 6.7. For the 𝐴-stable higher order collocation methods, it is easy to see from Theorem 6.6. (i) For the 𝜈-stage Gauss-Legendre method, 𝐻⊆𝑆 if and only if 𝜈 is even.(ii)For the 𝜈-stage Lobatto IIIA method, 𝐻⊆𝑆 if and only if 𝜈 is old. (iii)For the 𝜈-stage Radau IIA method, 𝐻₁⊆𝑆 if and only if 𝜈 is old and 𝐻₂⊆𝑆 if and only if 𝜈 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]): 𝑢^′₁(𝑡)=−20𝑢₁(𝑡)−10.3𝑢₁([𝑡]),𝑢₁(0)=1,𝑢^′₂(𝑡)=10𝑢₂(𝑡)−10.0001𝑢₂([𝑡]),𝑢₂(0)=1. (7.1)(7.2) It can be checked that (−20,−10.3)∈𝐻₁ and (10,−10.0001)∈𝐻₂.

For illustrating the convergence and superconvergence orders in this paper, we choose 𝑚=2 and 𝑚=3 and use the Gauss collocation parameters: 𝑐₁=(3−√3)/6, 𝑐₂=(3+√3)/6, the Radau IIA collocation parameters: 𝑐₁=1/3, 𝑐₂=1, the Lobatto IIIA collocation parameters: 𝑐₁=0, 𝑐₂=1, and three sets of random collocation parameters: 𝑐₁=1/4, 𝑐₂=1/2; 𝑐₁=1/3, 𝑐₂=2/3; 𝑐₁=2/3, 𝑐₂=1, respectively, for 𝑚=2; and we use the Gauss collocation parameters: 𝑐₁=(5−√15)/10, 𝑐₂=1/2, and 𝑐₃=(5+√15)/10, the Radau IIA collocation parameters: 𝑐₁=(4−√6)/10, 𝑐₂=(4+√6)/10, and 𝑐₃=1, the Lobatto IIIA collocation parameters: 𝑐₁=0, 𝑐₂=1/2, and 𝑐₃=1, and three sets of random collocation parameters: 𝑐₁=1/3, 𝑐₂=1/2, 𝑐₃=2/3; 𝑐₁=1/4, 𝑐₂=1/3, 𝑐₃=1/2; 𝑐₁=1/2, 𝑐₂=2/3, 𝑐₃=1, respectively, for 𝑚=3. In Tables 1, 2, 3, and 4 we list the absolute values of the absolute errors of 𝑢𝑡=10 for the six collocation parameters and for 𝑚=2 and 𝑚=3, respectively, and the ratios of the absolute values of the errors of 𝑁=100 over that of 𝑁=200.

Table 1 

The absolute values of absolute errors of 𝑢_ℎ for example (7.1) with 𝑚=2.

Table 2 

The absolute values of absolute errors of 𝑢_ℎ for example (7.1) with 𝑚=3.

Table 3 

The absolute values of absolute errors of 𝑢_ℎ for example (7.2) with 𝑚=2.

Table 4 

The absolute values of absolute errors of 𝑢_ℎ for example (7.2) with 𝑚=3.

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 𝑚=2 and 𝑝=50 for (7.1).

Figure 2 

The Radau IIA collocation method with 𝑚=2 and 𝑝=100 for (7.1).

Figure 3 

The Gauss collocation method with 𝑚=3 and 𝑝=50 for (7.1).

Figure 4 

The Radau IIA collocation method with 𝑚=3 and 𝑝=1000 for (7.1).

Figure 5 

The Gauss collocation method with 𝑚=2 and 𝑝=50 for (7.2).

Figure 6 

The Radau IIA collocation method with 𝑚=2 and 𝑝=1000 for (7.2).

Figure 7 

The Gauss collocation method with 𝑚=3 and 𝑝=50 for (7.2).

Figure 8 

The Radau IIA collocation method with 𝑚=3 and 𝑝=1000 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).