Numerical Integration

278. Numerical Integration

Methods for numerically integrating the ODEs of a system-dynamics model. Critical for simulation accuracy.

278.1. Euler method (default for SD)

𝑆_{𝑡 + Δ 𝑡} = 𝑆_{𝑡} + Δ 𝑡 \cdot ({net flow}_{𝑡})

Forward Euler. Simple, fast, error per step $𝑂 (Δ 𝑡^{2})$ , total error $𝑂 (Δ 𝑡)$ .

For SD models, Euler with a small enough $Δ 𝑡$ is usually good enough.

278.2. Time step ( $Δ 𝑡$ ) selection rule

Δ 𝑡 \leq \frac{min (𝜏_{𝑖})}{4}

where $𝜏_{𝑖}$ are the time constants of the system. Rule of thumb: divide the fastest time constant by 4-10.

Why? At $Δ 𝑡 = \frac{𝜏}{4}$ , Euler captures the dominant time scale; faster than that, all time scales are well-resolved.

278.3. Runge-Kutta 2 (RK2 / midpoint)

𝑘_{1} = 𝑓 (𝑆_{𝑡}, 𝑡)

𝑘_{2} = 𝑓 (𝑆_{𝑡} + (Δ \frac{𝑡}{2}) 𝑘_{1}, 𝑡 + Δ \frac{𝑡}{2})

𝑆_{𝑡 + Δ 𝑡} = 𝑆_{𝑡} + Δ 𝑡 \cdot 𝑘_{2}

Error per step $𝑂 (Δ 𝑡^{3})$ , total $𝑂 (Δ 𝑡^{2})$ . Twice the work of Euler per step, better accuracy.

278.4. Runge-Kutta 4 (RK4)

The “classical” RK4:

𝑘_{1} = 𝑓 (𝑆_{𝑡}, 𝑡)

𝑘_{2} = 𝑓 (𝑆_{𝑡} + (Δ \frac{𝑡}{2}) 𝑘_{1}, 𝑡 + Δ \frac{𝑡}{2})

𝑘_{3} = 𝑓 (𝑆_{𝑡} + (Δ \frac{𝑡}{2}) 𝑘_{2}, 𝑡 + Δ \frac{𝑡}{2})

𝑘_{4} = 𝑓 (𝑆_{𝑡} + Δ 𝑡 \cdot 𝑘_{3}, 𝑡 + Δ 𝑡)

𝑆_{𝑡 + Δ 𝑡} = 𝑆_{𝑡} + (Δ \frac{𝑡}{6}) (𝑘_{1} + 2 𝑘_{2} + 2 𝑘_{3} + 𝑘_{4})

Error per step $𝑂 (Δ 𝑡^{5})$ , total $𝑂 (Δ 𝑡^{4})$ . Four times the work per step, much higher accuracy.

278.5. Which to use?

Method	Per-step cost	Per-step error	Best for
Euler	1 eval	$𝑂 (Δ 𝑡^{2})$	Most SD models, small $Δ 𝑡$
RK2	2 evals	$𝑂 (Δ 𝑡^{3})$	Moderate accuracy needs
RK4	4 evals	$𝑂 (Δ 𝑡^{5})$	High-accuracy / oscillatory systems

For smooth SD models with $Δ 𝑡 = \frac{𝜏_{min}}{4}$ : Euler is fine. For oscillatory or stiff systems: RK4 or adaptive methods.

278.6. Stiff systems

When time constants span orders of magnitude (e.g., $𝜏_{1} = 1$ , $𝜏_{2} = 1000$ ), explicit methods (Euler, RK) need tiny $Δ 𝑡$ to stay stable on the fast time scale, wasting effort on the slow.

Implicit methods (backward Euler, BDF) trade per-step cost for stability — large $Δ 𝑡$ allowed.

In SD, stiff systems are rare; explicit methods are usually sufficient.

278.7. Initialization in equilibrium

Common SD practice: set initial stocks so that all flows balance at $𝑡 = 0$ (system at steady state). Then study the response to a perturbation.

This isolates the dynamic response from initial-condition transients.

278.8. See also

Stocks and Flows — the equations being integrated
System Dynamics overview
Differential Equations