Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stability. This timestep problem is challenging if strictly faster methods with the same all-atom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multiple-timescale contexts --- where the fast motions are rapidly decaying or largely decoupled from others --- have not been as successful for biomolecules, where vibrational coupling is strong.

This talk will review general issues that limit the timestep and describe available methods (constrained, reduced-variable, implicit, symplectic,multiple-timestep, and normal-mode-based schemes). Our dual timestep method LN (for its origin in a Langevin/Normal Modes algorithm) will also be presented and recent results presented (joint work with E. Barth and M. Mandziuk). LN relies on an approximate linearization of the equations of motion every t interval (5fs or less), the solution of which is obtained by explicit integration at the inner timestep (e.g., 0.5fs). Since this subintegration process does not require new force evaluations, LN is computationally competitive, providing 4-5 speedup factors, and results in good agreement, in comparison to 0.5fs trajectories. In combination with force splitting techniques, even further computational gains can be achieved for large systems.

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1996-1997 Mathematics in High Performance Computing