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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Explicit midpoint loses its uniform second-order defect bound at a time kink

Example

Define the continuous time-dependent field on $R \times R$ by $f (t, y) = {\begin{cases} 0, & t \leq 0, \\ t, & t > 0. \end{cases}$ With initial value $y (0) = 0$ , its exact solution is $y (t) = 0$ for $t \leq 0$ and $y (t) = t^{2} / 2$ for $t \geq 0$ . For every $h > 0$ , start explicit midpoint at $t_{0} = - h / 2$ with the exact value $y (t_{0}) = 0$ .

Facts & Assumptions

Given: The continuous field and exact solution in the Example, a step $h > 0$ , and the explicit midpoint tableau.

[F1]

The explicit Euler, midpoint, and classical fourth-order Runge–Kutta tableaus and Explicit Runge–Kutta tableaus, stage maps, and failure status: explicit midpoint has $c = (0, 1 / 2)$ , weights $b = (0, 1)$ , stage state $Y_{2} = y + (h / 2) f (t, y)$ , and output $y + h f (t + h / 2, Y_{2})$ .

[F2]

First-order systems, initial value problems, and solutions on intervals: an exact solution satisfies $y^{'} (t) = f (t, y (t))$ on its time interval.

[F3]

One-step ODE methods, local defect, uniform order, and step stability: the local defect is the method output from the exact starting value minus the exact value at the next time; a uniform local order-two claim requires a bound $C h^{3}$ with $C$ independent of the admitted start time.

Verification

1.1F1givenalgebra

On the first midpoint stage, $k_{1} = f (- h / 2, 0) = 0$ , so the second-stage state remains $0$ . Its time is $- h / 2 + h / 2 = 0$ , where the definition of $f$ gives $k_{2} = f (0, 0) = 0$ . Thus the method output is $0$ .

1.2F2givenalgebra

The displayed piecewise function satisfies $y (0) = 0$ and $y^{'} (t) = f (t, y (t))$ for $t < 0$ and $t > 0$ . At $t = 0$ its two one-sided derivatives are both $0 = f (0, y (0))$ , so it is an exact solution there as well. Its value at $h / 2$ is $h^{2} / 8$ , hence the local defect is $0 - h^{2} / 8 = - h^{2} / 8$ .

2.1F3step 1.2algebra∎

If one constant $C$ gave the uniform order-two estimate in [F3], these admitted exact-start steps would satisfy $h^{2} / 8 \leq C h^{3}$ , hence $1 / (8 h) \leq C$ for every $h > 0$ . The left side is unbounded as $h ↓ 0$ , so no such uniform constant exists. The field is continuous but not differentiable in time at $0$ , identifying the missing smoothness condition.

Depends on

Used by

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Dependency tree · two levels

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Sources

Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)
MIT OpenCourseWare, 18.330 Introduction to Numerical Analysis, Chapter 5 (standard reference, not scraped)