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Finite-time global error of the velocity-Verlet method: $∥ z_{n} - Φ_{n h} (z_{0}) ∥ \leq C (e^{L T} - 1) h^{2} / L$

Statement

Use the data, finite field and derivative bounds, and constants $h_{1}, C_{1}$ of The velocity-Verlet step matches the exact flow through order two: local truncation error $O (h^{3})$ on a convex tube on an open bounded convex tube $W$ , with compact $K_{1}$ and positive tube margin. Let $T > 0$ , $0 < ∣ h ∣ \leq h_{1}$ , and $n \in N$ satisfy $n ∣ h ∣ \leq T$ . Suppose $z_{j} = V_{h}^{j} (z_{0})$ and $y_{j} = Φ_{j h} (z_{0})$ both lie in $K_{1}$ for $0 \leq j \leq n$ . With $L = L_{1} = \sup_{W} ∥ D X_{H} ∥$ , the following holds:

$∥ z_{n} - y_{n} ∥ \leq C_{1} h^{2} \frac{e^{L T} - 1}{L} (L > 0), ∥ z_{n} - y_{n} ∥ \leq C_{1} T h^{2} (L = 0) .$

The constants are independent of $n, h, T$ , with the displayed data fixed.

Facts & Assumptions

Given: The stated data, including finite bounds on the convex tube and retention of both sequences in $K_{1}$ .

[F1]

The velocity-Verlet step matches the exact flow through order two: local truncation error $O (h^{3})$ on a convex tube supplies existence of the short exact trajectories from each point of $K_{1}$ in $W$ , and $∥ V_{h} (z) - Φ_{h} (z) ∥ \leq C_{1} ∣ h ∣^{3}$ .

[F2]

On the convex $W$ , the derivative bound and the chain rule give $∥ X_{H} (x) - X_{H} (y) ∥ \leq L ∥ x - y ∥$ ; the exact flow obeys its integral equation (The local flow of an autonomous first-order system, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F3]

Exponential integration and monotonicity follow from The exponential function is smooth and $(\exp)^{'} = \exp$ and The exponential function is strictly increasing.

[F4]

The field $X_{H} = (M^{- 1} p, - \nabla V (q))$ is $C^{1}$ on its phase domain and hence locally Lipschitz. Clause 2 of The local flow is a partial one-parameter group: identity, composition and inverse-time laws applies to its unique maximal autonomous solutions: if $Φ_{s} (z)$ and $Φ_{h} (Φ_{s} (z))$ are defined, then $Φ_{s + h} (z) = Φ_{h} (Φ_{s} (z))$ , for signed real times as well.

Proof

1.1givenF1F2F3algebra

Exact-flow stability. Write $a = ∣ h ∣$ . For short trajectories starting at $x, y \in K_{1}$ , their distance $u (t)$ , with time oriented by the sign of $h$ , satisfies $u (t) \leq u (0) + L \int_{0}^{t} u (s) d s$ on $0 \leq t \leq a$ by [F2]. If $v (t) = u (0) + L \int_{0}^{t} u (s) d s$ , then $v^{'} \leq L v$ ; differentiation of $e^{- L t} v (t)$ proves $v (t) \leq e^{L t} u (0)$ . Thus $∥ Φ_{h} (x) - Φ_{h} (y) ∥ \leq e^{L a} ∥ x - y ∥$ .

2.1givenF1F4step 1.1algebra

Recurrence. Set $e_{j} = ∥ z_{j} - y_{j} ∥$ . For $j < n$ , the datum $y_{j} = Φ_{j h} (z_{0})$ lies in $K_{1}$ , so [F1] supplies $Φ_{h} (y_{j})$ ; [F4] with $s = j h$ gives $Φ_{h} (y_{j}) = Φ_{(j + 1) h} (z_{0}) = y_{j + 1}$ . Thus by [F1], $e_{j + 1} \leq ∥ V_{h} (z_{j}) - Φ_{h} (z_{j}) ∥ + ∥ Φ_{h} (z_{j}) - Φ_{h} (y_{j}) ∥ \leq C_{1} a^{3} + e^{L a} e_{j}$ . Since $e_{0} = 0$ , induction gives $e_{n} \leq C_{1} a^{3} \sum_{k = 0}^{n - 1} e^{L a k}$ .

3.1step 2.1F3algebra∎

Evaluation. For $L > 0$ the sum is $(e^{L n a} - 1) / (e^{L a} - 1)$ , and $e^{L a} - 1 \geq L a$ , obtained by integrating $L e^{L t} \geq L$ on $[0, a]$ . Hence $e_{n} \leq C_{1} a^{2} (e^{L T} - 1) / L$ . For $L = 0$ the sum equals $n$ and $n a \leq T$ gives $e_{n} \leq C_{1} T a^{2}$ .

Remarks

This is a conditional finite-time trajectory estimate. Retention in the compact tube is assumed. The analytic energy theorem does not give exponential-time accuracy of the original trajectory. Negative steps obey the same bound with elapsed time $n ∣ h ∣$ .

Finite-time global error of the velocity-Verlet method: $∥ z_{n} - Φ_{n h} (z_{0}) ∥ \leq C (e^{L T} - 1) h^{2} / L$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Finite-time global error of the velocity-Verlet method: ∥zn−Φnh(z0)∥≤C(eLT−1)h2/L∥zn​−Φnh​(z0​)∥≤C(eLT−1)h2/L

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Finite-time global error of the velocity-Verlet method: $∥ z_{n} - Φ_{n h} (z_{0}) ∥ \leq C (e^{L T} - 1) h^{2} / L$