Geometric integrators are methods which preserve various phase-flow invariants such as symplectic structure or time-reversal symmetry. The Verlet method is a simple example of symplectic-reversible method, and its continuing popularity in molecular simulation is partly explained by this fact. In this talk, I will present the results of recent research aimed at exploiting geometric structure in more sophisticated timestepping algorithms: (1) Rigid body simulations and (2) adaptive timestep methods.

(1) When small groups are replaced by rigid bodies, or in dipole-dipole simulations, the standard approach is to introduce quaternions and to integrate the resulting equations of motion using a Runge-Kutta or Bulirsch-Stoer method--schemes which preserve none of the available geometric structure. I will describe the efficient implementation of a symplectic-reversible treatment of rigid groups and dipoles based on a variant of SHAKE, and I will give some numerical comparisons with quaternion based techniques.

(2) Variable timesteps are needed whenever a molecular system undergoes rapid time-local events, such as shocks or large deformations. Moreover, in certain types of simulations, the instability arising from a too-large timestep appears as isolated jumps associated to localized rapid motions. I will discuss the use of a new variable stepsize method, the Adaptive Verlet method, which is explicit, second-order accurate, and preserves the time-reversal symmetry, and I will show some preliminary results of numerical simulations using the new approach.

This talk describes joint work with Eric Barth, Ayla Kol, Brian Laird, Sebastian Reich, and Stephen Bond.

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1996-1997 Mathematics in High Performance Computing