Mathematical Sciences: Numerical Methods for Molecular Dynamics

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9600088 Skeel The work involves the analysis, synthesis, and testing of efficient numerical time stepping schemes for the long-time simulation of Hamiltonian systems of differential equations. This is motivated by the desire to improve the performance of computer programs that simulate the dynamics of biomolecules, with particular attention to the MDScope simulation program NAMD/SIgMA. Much of the proposed work is general and applies directly to a variety of applications modeled by Hamiltonian systems, such as astrophysics, some structural dynamics problems, and wave propagation. Some of the work is less general but can be adapted to other applications. Typically, the complicated dynamics of Hamiltonian systems is a combined effect of many simple interactions, often having a range of time scales. A key emphasis of the research is the development of sound "long-timestep" methods, which exploit the slowness present in many components of the motion and the possible regularity present in other components. More specifically, such methods evaluate some interactions at time increments that well exceed half the period of the fast modes of the system. Another emphasis is the design of time stepping schemes that can compute long trajectories having the right qualities. A specific goal is the construction of a multiple time stepping scheme that can be used both with long timesteps and for long-time simulations--existing schemes fail in one respect or the other. Another specific goal is to reduce the cost of certain calculations, such as inverse-square-law forces, by reducing their accuracy without sacrificing qualitative properties. Mathematical formulas are being devised that can predict typical motions of large molecules such as proteins and DNA. The aim is find shortcuts that enable very fast calculations on large computers. These ideas are tested for accuracy and speed, and the more useful ones are put into the software that biologis ts and other scientists use to study the action of molecules. Computer simulations can help the scientist to understand how molecules perform biological functions, which can be very difficult to observe experimentally. Such studies are also useful in the design of molecules for agricultural and medical purposes. Much of the proposed work is general and extends to other applications in which dissipation is negligible, such as astrophysics, some structural dynamics problems, and wave propagation.