207 research outputs found
The adaptive Verlet method
This is the published version, also available here: http://dx.doi.org/10.1137/S1064827595284658.We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the flow, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog--Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scalar nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector field (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem
Accurate and efficient splitting methods for dissipative particle dynamics
We study numerical methods for dissipative particle dynamics (DPD), which is
a system of stochastic differential equations and a popular stochastic
momentum-conserving thermostat for simulating complex hydrodynamic behavior at
mesoscales. We propose a new splitting method that is able to substantially
improve the accuracy and efficiency of DPD simulations in a wide range of the
friction coefficients, particularly in the extremely large friction limit that
corresponds to a fluid-like Schmidt number, a key issue in DPD. Various
numerical experiments on both equilibrium and transport properties are
performed to demonstrate the superiority of the newly proposed method over
popular alternative schemes in the literature
Numerical integrators for motion under a strong constraining force
This paper deals with the numerical integration of Hamiltonian systems in
which a stiff anharmonic potential causes highly oscillatory solution behavior
with solution-dependent frequencies. The impulse method, which uses micro- and
macro-steps for the integration of fast and slow parts, respectively, does not
work satisfactorily on such problems. Here it is shown that variants of the
impulse method with suitable projection preserve the actions as adiabatic
invariants and yield accurate approximations, with macro-stepsizes that are not
restricted by the stiffness parameter
Postprocessed integrators for the high order integration of ergodic SDEs
The concept of effective order is a popular methodology in the deterministic
literature for the construction of efficient and accurate integrators for
differential equations over long times. The idea is to enhance the accuracy of
a numerical method by using an appropriate change of variables called the
processor. We show that this technique can be extended to the stochastic
context for the construction of new high order integrators for the sampling of
the invariant measure of ergodic systems. The approach is illustrated with
modifications of the stochastic -method applied to Brownian dynamics,
where postprocessors achieving order two are introduced. Numerical experiments,
including stiff ergodic systems, illustrate the efficiency and versatility of
the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
Testing and tuning symplectic integrators for Hybrid Monte Carlo algorithm in lattice QCD
We examine a new 2nd order integrator recently found by Omelyan et al. The
integration error of the new integrator measured in the root mean square of the
energy difference, \bra\Delta H^2\ket^{1/2}, is about 10 times smaller than
that of the standard 2nd order leapfrog (2LF) integrator. As a result, the step
size of the new integrator can be made about three times larger. Taking into
account a factor 2 increase in cost, the new integrator is about 50% more
efficient than the 2LF integrator. Integrating over positions first, then
momenta, is slightly more advantageous than the reverse. Further parameter
tuning is possible. We find that the optimal parameter for the new integrator
is slightly different from the value obtained by Omelyan et al., and depends on
the simulation parameters. This integrator could also be advantageous for the
Trotter-Suzuki decomposition in Quantum Monte Carlo.Comment: 14 pages, 6 figure
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