Multiple Time Step Method Research Articles

A stabilized multiple time step method for coupled Stokes-Darcy flows and transport model

<abstract><p>A stabilized finite element algorithm with different time steps on different physical variables for the coupled Stokes-Darcy flows system with the solution transport is studied. The viscosity in the model is assumed to depend on the concentration. The nonconforming piecewise linear Crouzeix-Raviart element and piecewise constant are used to approximate velocity and pressure in the coupled Stokes-Darcy flows system, and conforming piecewise linear finite element is used to approximate concentration in the transport system. The time derivatives are discretized with different step sizes for the partial differential equations in these two systems. The existence and uniqueness of the approximate solution are unconditionally satisfied. A priori error estimates are established, which also provides a guidance on the ratio of time step sizes with respect to the ratio of the physical parameters. Numerical examples are presented to verify the theoretical results.</p></abstract>

Accelerating Ab Initio Quantum Mechanical and Molecular Mechanical (QM/MM) Molecular Dynamics Simulations with Multiple Time Step Integration and a Recalibrated Semiempirical QM/MM Hamiltonian.

Molecular dynamics (MD) simulations employing ab initio quantum mechanical and molecular mechanical (ai-QM/MM) potentials are considered to be the state of the art, but the high computational cost associated with the ai-QM calculations remains a theoretical challenge for their routine application. Here, we present a modified protocol of the multiple time step (MTS) method for accelerating ai-QM/MM MD simulations of condensed-phase reactions. Within a previous MTS protocol [Nam J. Chem. Theory Comput. 2014, 10, 4175], reference forces are evaluated using a low-level (semiempirical QM/MM) Hamiltonian and employed at inner time steps to propagate the nuclear motions. Correction forces, which arise from the force differences between high-level (ai-QM/MM) and low-level Hamiltonians, are applied at outer time steps, where the MTS algorithm allows the time-reversible integration of the correction forces. To increase the outer step size, which is bound by the highest-frequency component in the correction forces, the semiempirical QM Hamiltonian is recalibrated in this work to minimize the magnitude of the correction forces. The remaining high-frequency modes, which are mainly bond stretches involving hydrogen atoms, are then removed from the correction forces. When combined with a Langevin or SIN(R) thermostat, the modified MTS-QM/MM scheme remains robust with an up to 8 (with Langevin) or 10 fs (with SIN(R)) outer time step (with 1 fs inner time steps) for the chorismate mutase system. This leads to an over 5-fold speedup over standard ai-QM/MM simulations, without sacrificing the accuracy in the predicted free energy profile of the reaction.

Acceleration of phase-field lattice Boltzmann simulation of dendrite growth with thermosolutal convection by the multi-GPUs parallel computation with multiple mesh and time step method

Thermosolutal convection inevitably occurs during the solidification of alloys owing to the nonuniform distribution of temperature and/or solute concentration, and this can drastically alter the resulting solidification microstructures. In this study, we present a large-scale simulation scheme for the phase-field lattice Boltzmann model, which can express dendrite growth upon considering the solute, heat transport, and liquid flow. A multiple mesh and time step method was employed to reduce computational costs, where different mesh sizes and time steps are used to solve the phase-field equation, the advection–diffusion equations of heat and solute, and the lattice Boltzmann equations for fluid flow. Furthermore, we implemented parallel computations using multiple graphics processing units (GPUs) to accelerate the large-scale simulation. Through the application of the multiple mesh and time step method, the computation was accelerated by approximately one hundred times compared to the case using a constant mesh and time step for all equations. Moreover, we confirmed that the developed parallel-GPU computation combined with the multiple mesh and time step method could achieve good acceleration and scaling through increasing the number of GPUs. We also confirmed that the developed method could simulate multiple dendrite growth with thermosolutal convection.

A Versatile Multiple Time Step Scheme for Efficient ab Initio Molecular Dynamics Simulations.

We present here our implementation of a time-reversible, multiple time step (MTS) method for full QM and hybrid QM/MM Born-Oppenheimer molecular dynamics simulations. The method relies on a fully flexible combination of electronic structure methods, from density functional theory to wave function-based quantum chemistry methods, to evaluate the nuclear forces in the reference and in the correction steps. The possibility of combining different electronic structure methods is based on the observation that exchange and correlation terms only contribute to low frequency modes of nuclear forces. We show how a pair of low/high level electronic structure methods that individually would lead to very different system properties can be efficiently combined in the reference and correction steps of this MTS scheme. The current MTS implementation makes it possible to perform highly accurate ab initio molecular dynamics simulations at reduced computational cost. Stable and accurate trajectories were obtained with time steps of several femtoseconds, similar to and even exceeding the ones usually adopted in classical molecular dynamics, in particular when using a generalized Langevin stochastic thermostat. Compared to the standard Velocity Verlet integration, the present MTS scheme allows for a 5- to 6-fold overall speedup, at an unaltered level of accuracy.

Molecular Dynamics with Multiple Time Scales: How to Avoid Pitfalls.

Multiple time scale methodologies have gained widespread use in molecular dynamics simulations and are implemented in a variety of ways across numerous packages. However, performance of the algorithms depends upon the details of the implementation. This is particularly important in the way in which the nonbonded interactions are partitioned. In this work, we show why some previous implementations give rise to energy drifts, and how this can be corrected. We also provide a recipe for using multiple time step methods to generate stable trajectories in large scale biomolecular simulations, where long trajectories are needed.

Prediction of viscosity for molecular fluids at experimentally accessible shear rates using the transient time correlation function formalism

Nonequilibrium molecular dynamics (NEMD) simulations were performed and the transient time correlation function (TTCF) method applied to calculate the shear viscosity of n-decane. Using the TTCF method we were able to calculate the viscosity at shear rate orders of magnitude lower than is possible by direct NEMD simulation alone. For the first time for a molecular fluid, we were able to simulate shear rates accessible by experimental measurements, which are typically performed at shear rates well below those accessible by NEMD simulation. The TTCF method allows us to close the gap between the lowest shear rates accessible by MD simulation and the highest shear rates possible in experimental studies. Additionally a multiple time step method for Gaussian thermostatted SLLOD equations of motion was developed following earlier work [G. A. Pan et al., J. Chem. Phys. 122, 4114 (2005)] for atomic fluids.

An asynchronous time-stepping (ATS) integrator for atmospheric applications: Aerosol dynamics

Mixed time integration methods, also known as multiple time-step methods (MTS), have been proposed and implemented successfully to reduce the stiffness of problems in molecular dynamics and solid mechanics, where different time scales are used to integrate different motions or meshes. We apply the similar concept and develop an asynchronous time-stepping (ATS) integrator for atmospheric application. This paper first presents ATS as a generalized ordinary differential equation solver and describes its key components including: (1) global and local time management; (2) dynamic time-scale ordering; and (3) integration and iteration methods. Then we apply ATS to solve stiff problems in aerosol dynamics, where we further introduce several techniques such as grouping particulate species by particle sections to speed up computations. The comparisons between ATS and standard ODE solvers including LSODE, LSODES and VODE in aerosol dynamics simulation are presented, which show that ATS is robust and able to achieve a significant digit average (SDA) value of 2.0 to 3.0 while its speed is around 10 to 100 times faster than LSODE with equivalent accuracy. ATS has been implemented in CMAQ-UCD, a sectional, dynamic aerosol model built on USEPA's CMAQ platform.

Multiple time step molecular dynamics simulation for interaction between dislocations and grain boundaries

A multiple time step algorithm, called reversible reference system propagator algorithm, is introduced for the long time molecular dynamics simulation. In contrast to the conventional algorithms, the multiple time method has better convergence, stability and efficiency. The method is validated by simulating free relaxation and the hypervelocity impact of nano-clusters. The time efficiency of the multiple time step method enables us to investigate the long time interaction between lattice dislocations and low-angle grain boundaries.

Free energy landscapes of two model peptides: α-helical and β-hairpin peptides explored with Brownian dynamics simulation

We applied an atomistic Brownian dynamics (BD) simulation with multiple time step method for the folding simulation of a 13-mer α-helical peptide and a 12-mer β-hairpin peptide, giving successful folding simulations. In this model, the driving energy contribution towards folding came from both electrostatic and van der Waals interactions for the α-helical peptide and from van der Waals interactions for the β-hairpin peptide. Although, many non-native structures having the same or lower energy than that of native structure were observed, the folded states formed the most populated cluster when the structures obtained by the BD simulations were subjected to the cluster analysis based on distance-based root mean square deviation of side-chains between different structures. This result indicates that we can predict the native structures from conformations sampled by BD simulation.

Multiple Time Step Brownian Dynamics for Long Time Simulation of Biomolecules

We report a multiple time step algorithm applied to an atomistic Brownian dynamics simulation for simulating the long time scale dynamics of biomolecules. The algorithm was based on the original multiple time step method; a short time step was used to keep faster motions in local equilibrium. When applied to a 28-mer # # ! folded peptide, the simulation gave stable trajectories and the computation time was reduced by a factor of 160 compared to a conventional molecular dynamics simulation using explicit water molecules. We applied it for the folding simulation of a 13-mer ! -helical peptide, giving a successful folding simulation. These results indicate that the Brownian dynamics with the multiple time step algorithm is useful for studies of biomolecular motions by long time simulation.

Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives. We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives. We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.

A cell multipole based domain decomposition algorithm for molecular dynamics simulation of systems of arbitrary shape

A domain decomposition algorithm for molecular dynamics simulation of atomic and molecular systems with arbitrary shape and non-periodic boundary conditions is described. The molecular dynamics program uses cell multipole method for efficient calculation of long range electrostatic interactions and a multiple time step method to facilitate bigger time steps. The system is enclosed in a cube and the cube is divided into a hierarchy of cells. The deepest level cells are assigned to processors such that each processor has contiguous cells and static load balancing is achieved by redistributing the cells so that each processor has approximately same number of atoms. The resulting domains have irregular shape and may have more than 26 neighbors. Atoms constituting bond angles and torsion angles may straddle more than two processors. An efficient strategy is devised for initial assignment and subsequent reassignment of such multiple-atom potentials to processors. At each step, computation is overlapped with communication greatly reducing the effect of communication overhead on parallel performance. The algorithm is tested on a spherical cluster of water molecules, a hexasaccharide and an enzyme both solvated by a spherical cluster of water molecules. In each case a spherical boundary containing oxygen atoms with only repulsive interactions is used to prevent evaporation of water molecules. The algorithm shows excellent parallel efficiency even for small number of cells/atoms per processor.

Efficient multiple time step method for use with Ewald and particle mesh Ewald for large biomolecular systems

The particle–particle particle–mesh (P3M) method for calculating long-range electrostatic forces in molecular simulations is modified and combined with the reversible reference system propagator algorithm (RESPA) for treating the multiple time scale problems in the molecular dynamics of complex systems with multiple time scales and long-range forces. The resulting particle–particle particle–mesh Ewald RESPA (P3ME/RESPA) method provides a fast and accurate representation of the long-range electrostatic interactions for biomolecular systems such as protein solutions. The method presented here uses a different breakup of the electrostatic forces than was used by other authors when they combined the Particle Mesh Ewald method with RESPA. The usual breakup is inefficient because it treats the reciprocal space forces in an outer loop even though they contain a part that changes rapidly in time. This does not allow use of a large time step for the outer loop. Here, we capture the short-range contributions in the reciprocal space forces and include them in the inner loop, thereby allowing for larger outer loop time steps and thus for a much more efficient RESPA implementation. The new approach has been applied to both regular Ewald and P3ME. The timings of Ewald/RESPA and P3ME/RESPA are compared in detail with the previous approach for protein water solutions as a function of number of atoms in the system, and significant speedups are reported.

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Ewald summation and multiple time step methods for molecular dynamics simulation of biological molecules

Methods by which to determine conditions for a molecular dynamics (MD) simulation of biological molecules were investigated. Derivation of the optimal parameters of the Ewald summation was described so as to give same precision to the real space, the reciprocal space summations and the van der Waals interaction. Later, the procedure by which to determine the condition of the multiple time step method by RESPA (REference System Propagator Algorithm; Tuckerman et al., 1992, J. Chem. Phys., 97, 1990) was described as exemplified by MD simulations of a solvated β-sheet peptide. The conservation of the total energy in a microcanonical ensemble was measured to investigate the stability of the simulation conditions. The most feasible respective combinations of the time steps were: 0.25 fs for bond, angle and torsion interactions; 2 fs for van der Waals interaction and Ewald real-space summation; and 4 fs for Ewald reciprocal-space summation. Though it retained an acceptable accuracy, this condition accelerated the simulation ten-fold compared to that in which a simple velocity-Verlet method with a time step of 0.25 fs was used. The update of the correction term due to excluded neighbors was then investigated. Better results were obtained when the correction was updated with the real-space than when it was updated with the reciprocal-space summation. Finally, an MD simulation as long as 50 ps performed under the optimal Ewald and RESPA parameters was thus determined. The trajectory showed a good stability, indicating the feasibility of the parameters.

Application of a Multiple Time Step Algorithm to Biomolecular Systems

Herein we describe the implementation of the multiple time step method, in conjunction with a reversible integrator and the Nosé−Hoover chain method for temperature and pressure control, into ROAR 1.0. We have extensively tested the MTS method on three systems: an antifreeze peptide in water, the organic solvent dimethylformamide (DMF), and a dimyristoylphosphatidylcholine (DMPC)-based lipid bilayer. From these test simulations, we observe that the MTS method was capable of producing stable trajectories even when a long time step (e.g., 8 fs) is used, while the SHAKE method was unable to do so. The SHAKE method also disturbs the bond vibrational motion while MTS algorithm does not when the time step was smaller than 5 fs. We also observe that we can conservatively obtain a 2.5-fold speed-up using the MTS method over a SHAKE simulation using a 1 fs time step. Overall, the MTS method gives a solid speed-up over the traditional SHAKE method, while simultaneously giving much more stable trajectories.

Extrapolation versus impulse in multiple-timestepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics

Force splitting or multiple timestep (MTS) methods are effective techniques that accelerate biomolecular dynamics simulations by updating the fast and slow forces at different frequencies. Since simple extrapolation formulas for incorporating the slow forces into the discretization produced notable energy drifts, symplectic MTS variants based on periodic impulses became more popular. However, the efficiency gain possible with these impulse approaches is limited by a timestep barrier due to resonance—a numerical artifact occurring when the timestep is related to the period of the fastest motion present in the dynamics. This limitation is lifted substantially for MTS methods based on extrapolation in combination with stochastic dynamics, as demonstrated for the LN method in the companion paper for protein dynamics. To explain our observations on those complex nonlinear systems, we examine here the stability of extrapolation and impulses to force-splitting in Newtonian and Langevin dynamics. We analyze for a simple linear test system the energy drift of the former and the resonance-related artifacts of the latter technique. We show that two-class impulse methods are generally stable except at integer multiples of half the period of the fastest motion, with the severity of the instability worse at larger timesteps. Extrapolation methods are generally unstable for the Newtonian model problem, but the instability is bounded for increasing timesteps. This boundedness ensures good long-timestep behavior of extrapolation methods for Langevin dynamics with moderate values of the collision parameter. We thus advocate extrapolation methods for efficient integration of the stochastic Langevin equations of motion, as in the LN method described in paper I.

Structure of binary polymer blends: Multiple time step hybrid Monte Carlo simulations and self-consistent integral-equation theory

A newly developed self-consistent formulation of the polymer reference interaction site model (PRISM) theory is used to predict the structure of binary polymer blends. Theoretical radial distribution functions are compared to those obtained from hybrid Monte Carlo simulations of mixtures of Lennard-Jones chains. A multiple time step method is implemented to increase the efficiency of the simulations. We examine both the cases of atomic and molecular closures and consider both conventional and self-consistent PRISM. We find that, overall, theoretical distribution functions are in good agreement with simulation.

Simulations of Macromolecules by Multiple Time-Step Methods

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTSimulations of Macromolecules by Multiple Time-Step MethodsMasakatsu Watanabe and Martin KarplusCite this: J. Phys. Chem. 1995, 99, 15, 5680–5697Publication Date (Print):April 1, 1995Publication History Published online1 May 2002Published inissue 1 April 1995https://doi.org/10.1021/j100015a061RIGHTS & PERMISSIONSArticle Views190Altmetric-Citations65LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (3 MB) Get e-Alerts

Dynamics of molecules with internal degrees of freedom by multiple time-step methods

In molecular dynamics simulations of flexible molecules, the treatment of motions that take place on different time scales (e.g., bond stretching and bond angle bending vs dihedral angle librations and overall translation and rotation) or of forces that vary differently with distance (e.g., bonding forces vs electrostatic forces) is a major consideration. The most rapidly varying quantity limits the integration time step, while the most slowly varying quantity usually determines the simulation time required. The reversible multiple time-step methods introduced by Tuckerman, Berne, and Martyna [J. Chem. Phys. 97, 1990 (1992)] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast motions most effectively. As a first step toward extending multiple time-step methods to macromolecules, we investigate their utility for determining the dynamics of n-alkanes as neat liquids and in aqueous solution. Because the dynamics is done in Cartesian rather than internal coordinates, the different time steps are implemented in terms of hard and soft forces acting on individual atoms; e.g., hard forces are contributed by bond stretching (and bond angle bending) terms and soft forces by the remaining intramolecular and intermolecular (van der Waals and electrostatic) terms in the additive empirical potential function used to describe these systems. Integrations with and without a Nose–Hoover thermostat are performed. Comparisons are made with ordinary single time step molecular dynamics and dynamics with constraints applied to the bonds. The stability of the integration methods and the dynamic properties are considered. It is shown that stable integrations that yield agreement with standard dynamics can be achieved with a saving of up to a factor of 5 in computer time.