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Lie and Strang splitting have global orders one and two, with separate subflow errors

Statement

Let $a, b \in C^{3} (U, R^{d})$ define the additive IVP of Additive autonomous ODE partitions and have exact subflows on a common compact convex tube over $[0, T]$ . Assume $a, b$ are Lipschitz there with constants $L_{a}, L_{b}$ , and the exact solution, all split stages, and all numerical states remain in that tube with positive margin. For all sufficiently small $h = T / N$ and exact initial data, the exact Lie and Strang split maps of Lie and Strang compositions of exact split flows satisfy $\max_{0 \leq n \leq N} ∥ y_{n}^{L i e} - y (t_{n}) ∥_{2} \leq C_{1} h, \max_{0 \leq n \leq N} ∥ y_{n}^{S t r a n g} - y (t_{n}) ∥_{2} \leq C_{2} h^{2} .$ The constants may depend on the tube, the fields, and $T$ . For approximate subflows, suppose $∥ {\hat{Φ}}_{c}^{s} (u) - Φ_{c}^{s} (u) ∥_{2} \leq ϵ_{c} (s)$ for $c \in {a, b}$ and every admitted starting state $u$ . If all approximate compositions stay in the tube, their composed one-step maps are Lipschitz with constant $1 + \hat{L} h$ for some $\hat{L} \geq 0$ , and $ϵ_{b} (h) + \exp (L_{b} h) ϵ_{a} (h) = O (h^{2})$ for Lie and $ϵ_{a} (h / 2) + \exp (L_{a} h / 2) ϵ_{b} (h) + \exp ((L_{a} / 2 + L_{b}) h) ϵ_{a} (h / 2) = O (h^{3})$ for Strang, then the same respective global orders hold. These additional conditions keep subflow-solver errors separate from splitting errors.

Facts & Assumptions

Given: The finite-dimensional fields, interval, exact and numerical paths, and tube hypotheses in the Statement.

[F1]

Lie and Strang splitting errors have commutator expansions: the exact Lie split has a uniform one-step defect $O (h^{2})$ and the exact Strang split has a uniform one-step defect $O (h^{3})$ on the stated tube.

[F2]

Lie and Strang compositions of exact split flows: the Lie and Strang maps are the compositions $Φ_{b}^{h} \circ Φ_{a}^{h}$ and $Φ_{a}^{h / 2} \circ Φ_{b}^{h} \circ Φ_{a}^{h / 2}$ .

[F3]

A first-order initial value problem is equivalent to its Volterra integral equation: a solution of $x^{'} = c (x)$ satisfies $x (t) = x (0) + \int_{0}^{t} c (x (s)) d s$ .

[F4]

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: $∥ \int_{0}^{t} g (s) d s ∥_{2} \leq \int_{0}^{t} ∥ g (s) ∥_{2} d s$ for continuous vector functions and $t \geq 0$ .

[F5]

Gronwall's integral inequality with variable and constant coefficients: if $u (t) \leq A + K \int_{0}^{t} u (s) d s$ with continuous $u \geq 0$ and constants $A, K \geq 0$ , then $u (t) \leq A \exp (K t)$ .

[F6]

The exponential function is smooth and $(\exp)^{'} = \exp$ and The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : the real exponential is positive and has derivative $\exp (x)$ .

[F7]

The exponential function is strictly increasing and The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : on a bounded interval, the exponential mean-value estimate gives $\exp (K h) \leq 1 + K \exp (K h_{0}) h$ for $0 \leq h \leq h_{0}$ and $K \geq 0$ .

[F8]

Global error from uniform one-step defect and stability: a uniform one-step defect $C h^{p + 1}$ and one-step Lipschitz bound $1 + L h$ on a common compact tube imply fixed-time global error $O (h^{p})$ for exact initial data.

[A1]

Exact and approximate subflow stages are defined and remain in the tube; for the approximate case the pointwise subflow errors and the composed-map stability bounds stated in the Statement hold uniformly in the step and starting state.

Proof

1.1A1F2F3F4F5algebra

Let $K = L_{a} + L_{b}$ . For either field $c \in {a, b}$ , let $x (t)$ and $z (t)$ start at $u$ and $v$ . By [F3] and [F4], $∥ x (t) - z (t) ∥_{2} \leq ∥ u - v ∥_{2} + L_{c} \int_{0}^{t} ∥ x (s) - z (s) ∥_{2} d s .$ The constant case of [F5] yields $∥ Φ_{c}^{t} (u) - Φ_{c}^{t} (v) ∥_{2} \leq \exp (L_{c} t) ∥ u - v ∥_{2}$ . Composing these bounds according to [F2] shows both exact split maps are Lipschitz with factor $\exp (K h)$ .

2.1F6F7step 1.1algebra

Fix $h_{0} > 0$ within the admitted step range. For $0 \leq h \leq h_{0}$ , [F6] and the mean value theorem give $\exp (K h) - 1 = K h \exp (ξ) \leq K h \exp (K h_{0})$ for some $ξ \in [0, K h]$ , using monotonicity from [F7]. Thus both exact split maps satisfy a one-step bound $1 + L h$ , with $L = K \exp (K h_{0})$ (and $L = 0$ when $K = 0$ ).

2.2A1F2step 1.1algebra

Fix a starting state $u$ in the tube. For Lie, insert the exact $b$ flow between the two compositions and use its Lipschitz bound from step 1.1: $∥ {\hat{Φ}}_{b}^{h} ({\hat{Φ}}_{a}^{h} (u)) - Φ_{b}^{h} (Φ_{a}^{h} (u)) ∥_{2} \leq ϵ_{b} (h) + \exp (L_{b} h) ϵ_{a} (h) .$ For Strang, inserting the exact outer $a$ and middle $b$ maps in sequence gives $∥ {\hat{Φ}}_{a}^{h / 2} ({\hat{Φ}}_{b}^{h} ({\hat{Φ}}_{a}^{h / 2} (u))) - Φ_{a}^{h / 2} (Φ_{b}^{h} (Φ_{a}^{h / 2} (u))) ∥_{2} \leq ϵ_{a} (h / 2) + \exp (L_{a} h / 2) ϵ_{b} (h) + \exp ((L_{a} / 2 + L_{b}) h) ϵ_{a} (h / 2) .$ The inequalities hold uniformly over all starting states in the tube.

3.1F1F8step 2.1

The local defects are uniformly $O (h^{2})$ for Lie and $O (h^{3})$ for Strang by [F1]. Combine these defects with the stability bounds from step 2.1 and apply [F8] with $p = 1$ and $p = 2$ , respectively. The exact-root Lie and Strang global estimates follow.

4.1A1F1F8step 2.2∎

Under the stated $O (h^{2})$ Lie and $O (h^{3})$ Strang perturbation budgets, step 2.2 and [F1] give the same respective local-defect orders for the approximate compositions. Their one-step Lipschitz constants are assumed to be $1 + \hat{L} h$ in [A1], so [F8] applies with $p = 1$ and $p = 2$ and proves the approximate-subflow global bounds. This does not assert that local subflow accuracy alone implies stability.

Depends on

Used by

Conditional finite-dimensional Lie and Strang splitting residual orders Corollary

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62 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ernst Hairer, Geometric Numerical Integration, Lecture 2: Symplectic Integrators (standard reference, not scraped)
Gilbert Strang, On the Construction and Comparison of Difference Schemes (standard reference, not scraped)