Higher Order Symplectic Methods The Suzuki-Yoshida Method

3 Higher Order Symplectic Methods The Suzuki-Yoshida Method 

Yoshida [398] gives an elegant method for creating a symplectic scheme of arbitrarily high order, using Trotter s results for compositions of linear operators [370], This work is related to methods suggested by Suzuki [354] in the context of Trotter factorization of quantum operators. Consider a scheme with order 2s (for 1) and where the evolution of the system under the method is given by exp(/t /,), where 

We have free reign over constants tq and ti as long as 2ro + ti = 1. Hence we have an opportunity to annihilate the perturbation operator at order by choosing 0 as well. Solving simultaneously, there exists a unique real 

Though the fourth-order version of the scheme was given first by Forest and Ruth [139] (and discovered independently by Yoshida [398] and Candy and Rozmus [66]), we shall simply refer to these higher-order schemes as Yoshida methods, owing to the elegant derivation of schemes of arbitrary order. 

Example 3.1 Consider using velocity Verlet as the base second-order method to build a Yoshida fourth-order method from. As the scheme is second-order, we have 

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