The Complete Characterization of Fourth-Order Symplectic Integrators with Extended-Linear Coefficients

The structure of symplectic integrators up to fourth-order can be completely and analytical understood when the factorization (split) coefficents are related linearly but with a uniform nonlinear proportional factor. The analytic form of these {\it extended-linear} symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and non-forward fourth-order algorithms with arbitrary number of operators. Most fourth-order forward integrators can now be derived analytically from this extended-linear formulation without the use of symbolic algebra.Comment: 12 pages, 2 figures, submitted to Phys. Rev. E, corrected typo

Accurate sampling using Langevin dynamics

We show how to derive a simple integrator for the Langevin equation and illustrate how it is possible to check the accuracy of the obtained distribution on the fly, using the concept of effective energy introduced in a recent paper [J. Chem. Phys. 126, 014101 (2007)]. Our integrator leads to correct sampling also in the difficult high-friction limit. We also show how these ideas can be applied in practical simulations, using a Lennard-Jones crystal as a paradigmatic case

Efficient numerical integrators for stochastic models

The efficient simulation of models defined in terms of stochastic differential equations (SDEs) depends critically on an efficient integration scheme. In this article, we investigate under which conditions the integration schemes for general SDEs can be derived using the Trotter expansion. It follows that, in the stochastic case, some care is required in splitting the stochastic generator. We test the Trotter integrators on an energy-conserving Brownian model and derive a new numerical scheme for dissipative particle dynamics. We find that the stochastic Trotter scheme provides a mathematically correct and easy-to-use method which should find wide applicability.Comment: v

Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation

We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure

A Fundamental Theorem on the Structure of Symplectic Integrators

I show that the basic structure of symplectic integrators is governed by a theorem which states {\it precisely}, how symplectic integrators with positive coefficients cannot be corrected beyond second order. All previous known results can now be derived quantitatively from this theorem. The theorem provided sharp bounds on second-order error coefficients explicitly in terms of factorization coefficients. By saturating these bounds, one can derive fourth-order algorithms analytically with arbitrary numbers of operators.Comment: 4 pages, no figure

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

On the construction of high-order force gradient algorithms for integration of motion in classical and quantum systems

A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders. Contrary to the scheme by Chin and Kidwell [Phys. Rev. E 62, 8746 (2000)], where high-order algorithms are introduced by standard iterations of a force-gradient integrator of order four, the present method allows to reduce the total number of expensive force and its gradient evaluations to a minimum. At the same time, the precision of the integration increases significantly, especially with increasing the order of the generated schemes. The algorithms are tested in molecular dynamics and celestial mechanics simulations. It is shown, in particular, that the efficiency of the new fourth-order-based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12, respectively. The results corresponding to sixth- and eighth-order-based composition schemes are also presented up to the sixteenth order. For orders 14 and 16, such highly precise schemes, at considerably smaller computational costs, allow to reduce unphysical deviations in the total energy up in 100 000 times with respect to those of the standard fourth-order-based iteration approach.Comment: 23 pages, 2 figures; submitted to Phys. Rev.