The Takahashi-Imada Method

Recall the Takahashi-Imada method introduced in the last chapter, 

One might recognize the modification as being proportional to one part of the commutator expansion in the Verlet method, in fact 

This is certainly not a coincidence. If we replace the potential C/ by C/ in the Verlet 

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This method can be shown to have effective order four, meaning that there is a change of variables Xh which can be used to transform the Takahashi-Imada method into one of order four using the processing technique of Sect. 2.4.5. The potential energy modification has been specifically chosen to annihilate terms in the local error expansion (after coordinate transformation). 

Since Hh is assumed to be constant along the numerical solution, the coordinate transformations have the result of giving an effective order of four for the energy. It turns out that the Takahashi-Imada method is, more generally, an effective 4th order scheme, i.e. for arbitrary quantities, not just the energy [166],... 

Show that this method is time-reversible, volume-preserving (see Exercise 8), and is a (locally) 0 h ) perturbation of the Takahashi-Imada method. Hint the order can be determined by Taylor-expanding the force evaluations. 

Simplified Takahashi-Imada Method. The following method was first given by Wisdom, Holman and Touma [397] ... 

Hairer, E., McLachlan, R., Skeel, R.D. On energy conservation of the simplified Takahashi-Imada method. Math. ModeU. Numer. Anal. 43, 631-644 (2009). doi 10.1051/m2an/2009019... 

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