Enhancements of the Collisional Verlet Method

A further idea introduced in [184] was to use a force decoupling strategy to reduce the cost of computations. The idea is to exploit the observation that impulsive forces ( kicks ) can be supplied without reducing the order of accuracy as long as these have a vanishing component in the direction of the collision vector that is if F = —VUiqc) u = 0. This can be achieved in systems of spheres with pah-potentials (fij only by writing pij = aij + Pij where the derivative of the second term is chosen to vanish at the point of contact between the spheres, i.e. a (a + aj) = 0. If this decomposition is used, then it is possible to build a 2nd order accurate hybrid method that uses only the first part a to define the quadratic Verlet paths for the collision detection scheme, whereas fi is introduced as a standard kick at collision points. 

In Collisional Verlet, the momenta are adjusted at the collision point to preserve the kinetic energy (and hence the total energy). It is possible to project to some other manifold using projection techniques like those mentioned previously, as discussed in [40]. In particular, one may use the backward error analysis to obtain a modified Hamiltonian Hh corresponding to the Verlet method with stepsize h, then to project during collisions not onto the energy surface, but onto the modified energy surface, so that 

Other methods for hard sphere molecular dynamics and polymer modelling based on variational integration strategies, as well as comparative discussion, are provided in [235]. 

The Verlet method applied to the harmonic oscillator H = -[P-/2 results in a linear transformation of the form 

Write out the update schemes Yoshida high-order methods of order 6 and 8. How many force evaluations are required per iteration  

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