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A transverse backward boundary hit has a certified time and trace error bound

Statement

Let $J = [a, b]$ be a compact bracket, let $y$ be an exact incoming characteristic on $J$ , and let $g (t, x)$ be a boundary guard that is $L_{g}$ -Lipschitz in its state argument on a tube containing the exact path and a dense reconstruction $U$ . Suppose $h (t) : = g (t, y (t))$ has a root $τ \in J$ and a declared orientation $σ \in {- 1, 1}$ with $σ h^{'} (t) \geq ν > 0$ throughout $J$ . If $\sup_{t \in J} ∥ y (t) - U (t) ∥_{2} \leq η, ∣ g (\hat{τ}, U (\hat{τ})) ∣ \leq ε, \hat{τ} \in J,$ then the matched first-hit time satisfies $∣ \hat{τ} - τ ∣ \leq Δ : = \frac{L_{g} η + ε}{ν} .$ If the boundary value $β (t, x)$ is Lipschitz in time and state with constants $L_{t}^{β}, L_{x}^{β}$ on the same tube and $∥ y^{'} (t) ∥_{2} \leq M$ , then $∣ β (\hat{τ}, U (\hat{τ})) - β (τ, y (τ)) ∣ \leq L_{t}^{β} Δ + L_{x}^{β} (η + M Δ) .$ This is a local one-hit bound. It gives no conclusion at grazing, unresolved, or simultaneous events.

Facts & Assumptions

Given: A smooth mode characteristic, a boundary guard and orientation, a matched transverse first-hit bracket, a dense-output error bound, a reported guard residual, and the stated boundary-data Lipschitz bounds.

[F1]

Smooth hybrid ODE modes, guards, resets, and deterministic event priority: a smooth mode follows its ODE until a selected guard event; the event orientation is part of the guard data.

[F2]

The derivative of a guard along an ODE mode is its Lie derivative: along $y^{'} = b (t, y)$ , the guard derivative is $h^{'} (t) = \partial_{t} g (t, y (t)) + D_{x} g (t, y (t)) b (t, y (t))$ .

[F3]

A transverse hybrid event time is stable under dense-output and root error: dense state error $η$ , guard residual $ε$ , guard state-Lipschitz constant $L_{g}$ , and transverse margin $ν$ imply $∣ \hat{τ} - τ ∣ \leq (L_{g} η + ε) / ν$ on the matched bracket.

[F4]

A first-order initial value problem is equivalent to its Volterra integral equation: a mode trajectory increment equals the oriented integral of its vector field.

[F5]

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: for ordered endpoints $c \leq d$ , $∥ \int_{c}^{d} f ∥_{2} \leq \int_{c}^{d} ∥ f ∥_{2}$ ; reversed oriented integrals must first be rewritten with ordered endpoints.

Proof

1.1givenalgebra

If $a = b$ , then $\hat{τ} = τ = a$ , so the time error is zero and hence at most $Δ$ . The stated bounds give $∥ U (a) - y (a) ∥_{2} \leq η$ ; the state-Lipschitz bound for $β$ gives $∣ β (a, U (a)) - β (a, y (a)) ∣ \leq L_{x}^{β} η \leq L_{t}^{β} Δ + L_{x}^{β} (η + M Δ)$ . Here $Δ \geq 0$ from the error and residual bounds, $ν > 0$ , and the nonnegative Lipschitz constant $L_{g}$ , while $M, L_{t}^{β}, L_{x}^{β} \geq 0$ . Assume $a < b$ below.

1.2givenF1F2F3

By [F2], the transversality hypothesis is the mode derivative condition required by [F3]. Apply [F3] to this matched bracket with the stated $η, ε, L_{g}, ν$ ; it yields the unique transverse root and the displayed bound $∣ \hat{τ} - τ ∣ \leq Δ$ .

2.1givenF4F5step 1.2algebra∎

By [F4], the increment is the oriented integral from $τ$ to $\hat{τ}$ . Set $c = \min (τ, \hat{τ})$ and $d = \max (τ, \hat{τ})$ ; changing orientation only negates the vector integral, so [F5] gives $∥ y (\hat{τ}) - y (τ) ∥_{2} \leq \int_{c}^{d} ∥ y^{'} (r) ∥_{2} d r \leq M (d - c) = M ∣ \hat{τ} - τ ∣ \leq M Δ$ . The triangle inequality with the dense-output bound gives $∥ U (\hat{τ}) - y (τ) ∥_{2} \leq η + M Δ$ . Apply the time and state Lipschitz inequalities for $β$ to obtain the second displayed bound.

Remarks

An inflow/exit implementation must use a certified first-hit bracket and return unresolved when the bracket or orientation is not certified. This local result does not propagate errors through resets or an event list; those require the full finite-separated-event hypotheses of Finite separated transverse events give an event-aligned hybrid trajectory error bound.

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Sources

European Centre for Medium-Range Weather Forecasts, Numerical Methods, Meteorological Training Course Lecture Series, 2002 (standard reference, not scraped)
Lukas Einkemmer and Alexander Ostermann, On the error propagation of semi-Lagrange and Fourier methods for advection problems, Computers & Mathematics with Applications 69 (2015), 170–179 (standard reference, not scraped)