Advanced Integrators Summary

Summary of what we've learned about time integrators

Overview

Here’s a handy table of what we’ve learned over the last few articles.

Method	Position Error	Velocity Error	Symplectic	A-stable
Explicit Euler	$\mathcal{O}(h)$	$\mathcal{O}(h)$	No	No
Symplectic Euler	$\mathcal{O}(h)$	$\mathcal{O}(h)$	Yes	No
Velocity Verlet	$\mathcal{O}(h^{2})$	$\mathcal{O}(h^{2})$	Yes	No
Backward Euler	$\mathcal{O}(h)$	$\mathcal{O}(h)$	No	Yes
Implicit Midpoint Method	$\mathcal{O}(h)$	$\mathcal{O}(h)$	Yes	Yes

So if your simulation is not very heavy on physics Symplectic Euler is probably okay. Otherwise you’ll want to go with Verlet. Implicit methods like the Gauss–Legendre methods (of which implicit midpoint is the simplest) should be reserved for particularly stubborn differential equations as they are expensive.