Comparison of different numerical integration methods for acceleration/velocity of 2D bodies in Godot.
| method | symplectic1 | error order | variable Δt2 | samples per step3 |
|---|---|---|---|---|
| Basic Euler | ❌ | O(Δt) | ❌ | 1 |
| Symplectic Euler / Rigidbody2D | ✔️ | O(Δt) | ❌ | 1 |
| Velocity Verlet | ✔️ | O(Δt^2) | ✔️ | 14 |
| Leapfrog | ✔️ | O(Δt^2) | ❌ | 14 |
| Midpoint | ❌ | O(Δt^2) | ❌ | 2 |
| RK4 | ❌ | O(Δt^4) | ❔ | 4 |
| Forest Ruth | ✔️ | O(Δt^4) | ✔️ | 3 |
| PEFRL | ✔️ | O(Δt^4)5 | ✔️ | 4 |
Testing scenarios:
- Ballistic trajectory - constant acceleration
- Orbital mechanics - acceleration depends on position
- Harmonic oscillator - acceleration depends on position
Analysis (in progress) - A more detailed comparison based on this project research
References with more accurate mathematical descriptions of the techniques
- Error analysis compared to analytical solution for orbital scene
- Yoshida integrator
- Adaptive timestep techniques
Footnotes
-
The system conserves energy if the underlying system does (ie. planets orbit does not change total energy on average) ↩
-
Is the error constant with a variable timestep? While a workaround is to have a fixed timestep, this is still a preferred property that enables changing the speed of the simulation at runtime and integrating at higher timesteps to predict the future. This might just be a consequence of the first two properties, so take it with a grain of salt. ↩
-
Actual number is doubled when integrating both acceleration and velocity. Note that this is not an accurate indicator of performance, but a ballpark estimate ↩
-
Theoretically it's 2, but one sample can be cached from one frame to the next as it is the same value ↩ ↩2
-
order is the same, but in practice PEFRL is 26 times (!!!) more accurate than FR (Forest Ruth) ↩