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Exact RATTLE has second order global accuracy on compact tubes

Statement

Let the constrained vector field and exact RATTLE map be those of Regular holonomic phase space and constrained dynamics and Scaled classical RATTLE step map. Suppose the exact solution on $[0, T]$ stays in a compact subset of a convex coordinate chart whose closure lies inside a regular chart, with $A (q)^{- 1}$ uniformly bounded. Let $h = T / N > 0$ , let $z_{0}^{h} \in M$ satisfy $∥ z_{0}^{h} - z (0) ∥ \leq C_{0} h^{2}$ in these coordinates, and let $z_{n + 1}^{h} = Ψ_{h} (z_{n}^{h})$ use the selected exact branches. For all sufficiently large $N$ , the numerical path remains in the regular chart and $\max_{0 \leq n \leq N} ∥ z_{n}^{h} - z (t_{n}) ∥ \leq C_{T} h^{2},$ where $C_{T}$ is independent of $N$ .

Facts & Assumptions

Given: A smooth constrained vector field, its exact local flow, an exact RATTLE branch, the compact-tube hypothesis, and the initial error bound in the Statement.

[F1]

The constrained Hamiltonian vector field has a unique local flow: the vector field is $X (q, p) = (M^{- 1} p, - \nabla V (q) - J (q)^{T} Λ (q, p))$ and has a unique local flow.

[F2]

Scaled RATTLE equations have unique local multiplier branches: the selected multipliers are smooth in local phase coordinates, have common step ranges and bounded derivatives on compact phase tubes, and at $h = 0$ both equal $Λ (q, p) / 2$ .

[F3]

Exact RATTLE preserves both constraints and is symmetric: $Ψ_{- h} = Ψ_{h}^{- 1}$ on the local branches.

[F4]

Multivariable Taylor formula with a Lagrange remainder along a line segment: a $C^{3}$ map has a quadratic Taylor polynomial with a cubic remainder on a compact convex chart.

[F5]

On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ : a uniform derivative bound on a convex coordinate neighborhood implies the corresponding Lipschitz bound.

Proof

1.1F1F2algebra

By [F1] and [F2], differentiating the RATTLE equations at $h = 0$ gives $D_{h} Ψ_{0} (z) = X (z)$ : the position derivative is $M^{- 1} p$ , and the momentum derivative is $- \nabla V - J^{T} (Λ / 2 + Λ / 2)$ . The selected map is $C^{3}$ in $(h, z)$ by [F2].

1.2F2F3algebra

In chart coordinates choose a slightly larger compact phase neighborhood around the exact path. The revised compact-subset conclusion of [F2] gives one step range and uniform $C^{3}$ bounds there. Write $Ψ_{h} (z) = z + h X (z) + h^{2} a (z) + O (h^{3})$ uniformly on this neighborhood. Substitution into the inverse equation gives $Ψ_{h}^{- 1} (z) = z - h X (z) + h^{2} (D X (z) X (z) - a (z)) + O (h^{3})$ . By [F3], this equals $Ψ_{- h} (z) = z - h X (z) + h^{2} a (z) + O (h^{3})$ . Comparing quadratic coefficients yields $a (z) = \frac{1}{2} D X (z) X (z)$ .

1.3F2F5

Since $Ψ_{0}$ is the identity and $Ψ_{h}$ is $C^{3}$ , its state derivative is $D_{z} Ψ_{h} = I + O (h)$ uniformly on the larger compact chart. Thus $∥ D_{z} Ψ_{h} ∥ \leq 1 + L h$ for some $L \geq 0$ and small $h > 0$ . Applying [F5] in the convex chart gives $∥ Ψ_{h} (x) - Ψ_{h} (y) ∥ \leq (1 + L h) ∥ x - y ∥$ there.

2.1F1F2F4step 1.2

The exact flow satisfies $\dot{z} = X (z)$ and hence $\ddot{z} = D X (z) X (z)$ . Its third time derivative is bounded on the compact tube. Taylor's theorem [F4] therefore gives $z (t + h) = z (t) + h X (z (t)) + \frac{1}{2} h^{2} D X (z (t)) X (z (t)) + O (h^{3})$ . Comparing with step 1.2 proves the local defect $∥ Ψ_{h} (z (t)) - z (t + h) ∥ \leq C h^{3}$ uniformly in $t$ .

3.1givenstep 2.1step 1.3algebra∎

Until a numerical state leaves the larger chart, steps 2.1 and 1.3 give $e_{n + 1} \leq (1 + L h) e_{n} + C h^{3}$ , where $e_{n} = ∥ z_{n}^{h} - z (t_{n}) ∥$ . Induction yields $e_{n} \leq (1 + L h)^{n} e_{0} + C h^{3} \sum_{j = 0}^{n - 1} (1 + L h)^{j} \leq e^{L T} (C_{0} + C T) h^{2}$ . For sufficiently small $h$ , this bound is less than the positive distance from the exact compact path to the boundary of the larger chart, so a first-exit index cannot exist. The estimate therefore holds for every $n \leq N$ .

Depends on

Used by

RATTLE gives an exact step and a cubic phase defect for a free rigid bond Example

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Sources

Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, 2nd ed., Chapter VII.1.4, pp. 245–247 (standard reference, not scraped)
Robert I. McLachlan, Klas Modin, Olivier Verdier, and Matt Wilkins, Geometric Generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339–370, §§4.3 and 4.5 (standard reference, not scraped)