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The coupled index-one constant-step BDF2 equations have a local branch

Statement

Let a compact exact solution segment lie in a common index-one branch chart with $f, g \in C^{1}$ and $g_{z}$ invertible on the graph. For each point $(t, x, ψ (t, x))$ , the coupled constant-step BDF2 root equations

3 X - 4 ξ + ζ - 2 h f (t, X, Z) = 0, g (t, X, Z) = 0

have a unique root $(X, Z)$ in a selected neighborhood of $(x, ψ (t, x))$ whenever $(h, t, ξ, ζ)$ is sufficiently close to $(0, t, x, x)$ . On a compact subsegment in one chart with positive margin, one may choose a common positive step bound after restricting the two input states to neighborhoods of the reference graph.

Facts & Assumptions

Given: The DAE, its common selected branch chart, the compact reference segment, and the constant-step BDF2 residual from Finite-budget coupled constant-step BDF2 for an index-one DAE.

[F1]

A regular index-one DAE has a local reduced ODE branch: $g (t, x, ψ (t, x)) = 0$ and $g_{z} (t, x, ψ (t, x))$ is invertible.

[F2]

The parametrized implicit function theorem with $C^{k}$ regularity: a $C^{1}$ residual whose derivative in its unknown block is invertible at a zero has a unique local root branch for nearby parameters.

Proof

1.1givenF1construct

Define $Φ (h, t, ξ, ζ, X, Z) = (3 X - 4 ξ + ζ - 2 h f (t, X, Z), g (t, X, Z))$ . At $h = 0, ξ = ζ = X = x, Z = ψ (t, x)$ , [F1] gives $Φ = 0$ .

1.2F1algebra

The derivative in $(X, Z)$ at this zero is $(\begin{array}{cc} 3 I_{d} & 0 \\ g_{x} & g_{z} \end{array})$ . If it annihilates $(u, v)$ , then $3 u = 0$ , so $u = 0$ ; the second row then gives $g_{z} v = 0$ , so $v = 0$ by [F1]. Hence the square derivative matrix is invertible.

2.1step 1.1step 1.2F2

Apply [F2] with $(h, t, ξ, ζ)$ as parameters and $(X, Z)$ as unknowns. The implicit function theorem gives a unique nearby root branch for all sufficiently small changes in those parameters.

3.1step 2.1givenconstruct∎

The reference parameter graph ${(0, t, x (t), x (t)) : t \in [a, b]}$ is compact and lies in the open union of the IFT parameter neighborhoods from step 2.1. Compactness gives $δ > 0$ such that the $δ$ -neighborhood of this graph lies in that union. Shrink each associated unknown neighborhood inside the common branch chart. Thus, for all sufficiently small $∣ h ∣$ and pairs of input states sufficiently close to the reference graph, one IFT chart supplies the corresponding unique local root.

Depends on

Used by

Coupled constant-step BDF2 is second order for a regular index-one DAE Theorem

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Sources

Ernst Hairer and Gerhard Wanner, Solving Ordinary Differential Equations II, Chapter VI §VI.1, Solving Index 1 Problems, pp. 372–381 (standard reference, not scraped)
Kendall Atkinson, Weimin Han, and David Stewart, Numerical Solution of Ordinary Differential Equations, Chapter 10 §10.4 (standard reference, not scraped)