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An Improved Algorithm for Efficient and Accurate Simulations of Langevin EquationsPDE and Applied Math Seminar
|Speaker: ||Niels Gronbech-Jensen, UC Davis|
|Location: ||1147 MSB|
|Start time: ||Tue, Apr 16 2013, 1:10PM|
The Stormer-Verlet type numerical integrators for simulating equations
of motion are well known and widely used in many contemporary scientific contexts.
We review why the simple "Verlet" algorithm, which is a direct second order finite
difference approximation to a second order differential, is so desirable for initial
value problems with conservation properties.
For Langevin dynamics, where coupling to a heat-bath is included through a dissipation-
fluctuation balance in the equation of motion, the nature of the conservation requirements
change, and the premise for the algorithm is put into question.
We present a simple re-derivation of the Stormer-Verlet algorithm, including linear friction
with associated stochastic noise. We analytically demonstrate that the new algorithm
correctly reproduces diffusive behavior of a particle in a flat potential, and, for a harmonic
oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping,
frequency, and time step for both underdamped and overdamped dynamics within the usual
stability limit of the Verlet algorithm. The method, which is as simple as the conventional
Verlet scheme, is numerically tested on both low-dimensional nonlinear systems as well
as complex systems with many degrees of freedom. Finally, we discuss the opportunities and
benefits of proper thermodynamic properties of the numerical integrator for simultaneous
accuracy and efficiency in, e.g., Molecular Dynamics simulations.