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Structure-Preserving Integrators
Why does a worse method give a better orbit?
RK4 is fourth order; velocity Verlet is second. Over a long orbit Verlet wins anyway, and not by a little. This chapter is where accuracy stops being the only axis - some methods approximate the trajectory, and others preserve the geometry the trajectory lives in.
01Why a worse method gives a better orbitadvanced30 min
RK4 is fourth order. Velocity Verlet is second. Over a long orbit Verlet wins comprehensively, and the reason has nothing to do with accuracy.
02Discretising the action instead of the equationsfrontier30 min
Variational integrators derive the update from a discrete least-action principle. Symplecticity and conserved momenta then follow automatically rather than being imposed.