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Distributed-delay midpoint error separates history and quadrature errors

Statement

Let $I_{K} [y] (t)$ be the distributed-delay integral from Distributed delays through a finite-window history integral, let $h = τ / m$ , and let $r$ be a positive multiple of $m$ with $Δ = τ / r$ . At a grid query time $t_{n}$ , suppose the exact integrand $g_{n} (θ) = K (θ) y (t_{n} - θ)$ is $C^{2}$ on every quadrature cell with $∥ g_{n}^{''} (θ) ∥_{2} \leq M_{g}$ , including one-sided endpoint derivatives. If $\sup_{s \in [- τ, T]} ∥ {\hat{y}}_{h} (s) - y (s) ∥_{2} \leq E_{h}$ , then the composite midpoint rule of Composite midpoint quadrature for a distributed delay history satisfies $∥ I_{K} [y] (t_{n}) - {\hat{I}}_{K, h, r} (t_{n}) ∥_{2} \leq E_{h} Δ \sum_{j = 0}^{r - 1} ∣ K (θ_{j}) ∣ + \frac{M_{g} τ Δ^{2}}{24} .$ Thus the first term is the error from the stored history at the quadrature nodes, while the second is the quadrature remainder for the exact history. In particular, Error bound for piecewise-linear stored delay histories supplies $E_{h} \leq E + M h^{2} / 8$ under its piecewise- $C^{2}$ hypotheses.

Facts & Assumptions

Given: A continuous kernel, the exact and stored histories, the grid-aligned composite midpoint rule, and the cellwise bound on $g_{n}^{''}$ .

[F1]

The distributed-delay integral is $I_{K} [y] (t) = \int_{0}^{τ} K (θ) y (t - θ) d θ$ (Distributed delays through a finite-window history integral).

[F2]

The algorithm evaluates ${\hat{I}}_{K, h, r} (t_{n}) = Δ \sum_{j < r} K (θ_{j}) {\hat{y}}_{h} (t_{n} - θ_{j})$ at midpoint nodes $θ_{j} = (j + 1 / 2) Δ$ (Composite midpoint quadrature for a distributed delay history).

[F3]

The stored history has the cellwise error $\sup_{s} ∥ {\hat{y}}_{h} (s) - y (s) ∥_{2} \leq E_{h}$ ; one sufficient bound is $E_{h} = E + M h^{2} / 8$ (Error bound for piecewise-linear stored delay histories).

[F4]

The norm of a vector integral is at most the integral of the norm, and a bounded derivative makes a vector function Lipschitz (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[F5]

The Euclidean norm satisfies the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

Proof

1.1F1F4F5givenalgebra

For each quadrature cell with midpoint $c_{j}$ , the bound on $g_{n}^{''}$ and [F4] give $g_{n} (θ) = g_{n} (c_{j}) + (θ - c_{j}) g_{n}^{'} (c_{j}) + R_{j} (θ)$ with $∥ R_{j} (θ) ∥_{2} \leq M_{g} ∣ θ - c_{j} ∣^{2} / 2$ . The linear term integrates to zero on the symmetric cell, so the error of this cell's midpoint rule is at most $\int_{- Δ / 2}^{Δ / 2} M_{g} s^{2} / 2 d s = M_{g} Δ^{3} / 24$ . Summing the $r$ cell errors by [F5] gives $∥ I_{K} [y] (t_{n}) - Δ \sum_{j < r} g_{n} (θ_{j}) ∥_{2} \leq M_{g} r Δ^{3} / 24 = M_{g} τ Δ^{2} / 24$ .

1.2F2F3F5algebra

By [F2], [F3], and the triangle inequality [F5], the difference between the exact-history midpoint sum and the stored-history midpoint sum is bounded by $Δ \sum_{j < r} ∣ K (θ_{j}) ∣ ∥ y (t_{n} - θ_{j}) - {\hat{y}}_{h} (t_{n} - θ_{j}) ∥_{2} \leq E_{h} Δ \sum_{j < r} ∣ K (θ_{j}) ∣$ .

2.1F5step 1.1step 1.2algebra∎

The triangle inequality [F5] applied to steps 1.1 and 1.2 gives the stated sum of the history interpolation error and the exact-history quadrature remainder.

Depends on

Used by

Distributed-delay midpoint quadrature separates two history errors Example

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Sources

Nicola Guglielmi and Ernst Hairer, Solving Delay Differential Equations, Chapter 1 (standard reference, not scraped)
Roland Pulch, Equivalent Systems for Differential Equations with Polynomially Distributed Delay, version 3 (standard reference, not scraped)