A semi-analytical approach to molecular dynamics

Abstract

Despite numerous computational advances over the last few decades, molecular dynamics still favors explicit (and thus easily-parallelizable) time integrators for large scale numerical simulation. As a consequence, computational efficiency in solving its typically stiff oscillatory equations of motion is hampered by stringent stability requirements on the time step size. In this paper, we present a semianalytical integration scheme that offers a total speedup of a factor 30 compared to the Verlet method on typical MD simulation by allowing over three orders of magnitude larger step sizes. By efficiently approximating the exact integration of the strong (harmonic) forces of covalent bonds through matrix functions, far improved stability with respect to time step size is achieved without sacrificing the explicit, symplectic, time-reversible, or fine-grained parallelizable nature of the integration scheme. We demonstrate the efficiency and scalability of our integrator on simulations ranging from DNA strand unbinding and protein folding to nanotube resonators.

Symbolic-Numeric Integration of the Dynamical Cosserat Equations

“…The vector Λ(t) describes external forces acting on the system at time t. We are searching for functions x(t), υ(t) = ẋ(t), and a(t) = ẍ(t) satisfying ( 12) for all t with initial conditions x(t 0 ) = x 0 and υ(t 0 ) = υ 0 . For the employment of the generalized α-method, we can write the integration scheme with respect to (12) as follows:…”

“…Our approach combines the ideas of numerical exponential integration (see e.g. [8,12] and references therein) and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of the arbitrary vector function in the general solution to the kinematic part in terms of the module of the twist vector function. The symbolic part of the integration is performed by means of Maple.…”

Full discretization error analysis of exponential integrators for semilinear wave equations

“…For quasilinear Maxwell's equations, the numerical experiments in [31] confirm this computational potential. Further, their efficiency was demonstrated in [26,28] in the application to molecular dynamics and nonlinear coupled oscillators. Besides the Krylov methods to compute a matrix function applied to a vector, a different approach is considered in [22].…”

Explicit Exponential Rosenbrock Methods and their Application in Visual Computing

“…Typical examples are models in molecular dynamics (see e.g. [36]), chemical kinetics, combustion, mechanical vibrations (mass-spring-damper models), visual computing (specially in computer animation), computational fluid dynamics, meteorology, etc., just to name Vu Thai Luan Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75275-0156, USA, e-mail: vluan@smu.edu Dominik L. Michels Computational Sciences Group, Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, 23955, KSA, e-mail: dominik.michels@kaust.edu.sa a few. They are usually formulated as systems of stiff differential equations which can be cast in the general form…”