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

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

where {t} is the fractional part of t and 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 hold.

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 $$ so that for the mesh I_h with mesh diameter h > 0 satisfying $$, 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 $$ for the initial-value problem (1.1), and its representation on the subinterval [t_n, t_n+1] is given by (2.11).

Theorem 3.1. Assume the following:

• (1)

  the given functions in (1.1) satisfy a, b ∈ ℝ, K ∈ C^m(D);

• (2)

  $$ is the collocation solution to (1.1) defined by (2.10) and (2.11) with $$.

Then the estimates 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

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 Integration of (3.3) leads to where 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 while Since e_h is continuous in [0, T], and hence at the mesh points, we also have the relation with b_j : = β_j(1). The fact that e_h(0) = 0 yields 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 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 $$, and hence there exists a constant D₀ < ∞ so that ∥(I_m×m − hA_n) ⁻¹∥₁ ≤ D₀ uniformly for 0 ≤ n ≤ N − 1. Here, for B ∈ L(ℝ^m), ∥B∥₁ denotes the matrix (operator) norm induced by the l₁-norm in ℝ^m. Denote M_m+1 : = ∥u^(m+1)∥_∞, $$, and $$. So

Equation (3.14) now leads to the estimate

with obvious meanings of γ₀ and γ₁. 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.

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

then the corresponding collocation solution $$ satisfies, for $$, with C₂ depending on the collocation parameters and on $$ but not on h. The exponent m + 1 cannot, in general, be replaced by m + 2. For the derivative $$, we attain only $$.

Proof. Let

denote the defect (or: residual) associated with the collocation solution $$ to the initial-value problem (1.1). by definition of the collocation solution the defect δ_h vanishes on the set X_h as follows: Moreover, the uniform convergence of u_h and $$ 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

By Theorem 3.1, there exists a constant D, such that and this holds for any choice of the {c_i}. On the other hand, the collocation error e_h solves the initial-value problem For t ∈ [k, k + 1], whose solution is given by The function r = r(t, s) denotes the “resolvent” (or: resolvent kernel) of (1.1) as follows: If k = 0, let t = t_l + vh, v ∈ [0,1], and 0 ≤ l ≤ p − 1; we have Suppose now that each of the integrals over [0,1] is approximated by the interpolatory m-point quadrature formula with abscissas {c_i}, then 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 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 $$, because δ_h(t) = 0 whenever t ∈ X_h. Hence This leads to the estimate for 0 ≤ l ≤ p − 1 and v ∈ [0,1], with $$.

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 Similarly to the case of t ∈ [0,1], we have This completes the proof.

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

  $$ ()

holds, with J_κ ≠ 0.

Then, for all meshes I_h with $$, the collocation solution $$ corresponding to the collocation points X_h based on these {c_i} satisfies

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)

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

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.

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,

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.