yoshida4.hpp File Reference

Yoshida's fourth-order symplectic integrator, built from three velocity Verlet steps. More...

#include <string_view>
#include "orrery/core/particle_data.hpp"
#include "orrery/core/types.hpp"
#include "orrery/integrators/acceleration_field.hpp"
#include "orrery/integrators/integrator.hpp"

Go to the source code of this file.

Classes
class	orrery::integrators::Yoshida4
	The fourth-order symplectic composition described above. More...

Detailed Description

Yoshida's fourth-order symplectic integrator, built from three velocity Verlet steps.

Yoshida's construction says that any symmetric second-order method S(h) can be raised to fourth order by composing three of its steps with weights

w1 = 1 / (2 - 2^(1/3)),  w0 = -2^(1/3) / (2 - 2^(1/3))

as S(w1 h) S(w0 h) S(w1 h). Velocity Verlet is such a method, so this class is that composition and nothing else. Writing it out as its own sequence of kicks and drifts would restate arithmetic that already exists one file away, and the two copies would then have to be kept in agreement by attention rather than by construction.

The middle weight is negative: the composition takes a long step forward, a shorter step backwards, and a long step forward again, and 2 w1 + w0 = 1 leaves the net advance equal to h. That is not a quirk of this particular construction. No composition of a symmetric method reaches an order above two with all its substeps positive, which is a theorem rather than a limitation of anyone's ingenuity. The practical consequence is that the intermediate states inside a step are not states the system passes through, so a diagnostic taken mid-step would be measuring nothing, and the largest substep is about 1.35 times the nominal step rather than smaller than it.

Three force evaluations per step against velocity Verlet's one. It earns them where the accuracy per unit of work matters more than the cost per step: at equal work, that is at a third of the timestep, its error falls as the fourth power of a step that is three times smaller rather than as the square, and it remains symplectic while doing so.