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62
Mechanical Integrators Derived from a Discrete Variational Principle
"... Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical system ..."
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Cited by 86 (12 self)
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Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct timestepping algorithms that approximate the flow of continuous ODE's for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We girst present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing mechanical integrators for continuous Lagrangian systems. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.
Geometric numerical integration illustrated by the StörmerVerlet method
, 2003
"... The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric nume ..."
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Cited by 65 (6 self)
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The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a crosssection of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent longtime behaviour of the method: longtime energy conservation, linear error growth and preservation of invariant tori in nearintegrable systems, a discrete virial theorem, and preservation of adiabatic invariants.
Symplectic Integration Of Constrained Hamiltonian Systems
"... . A Hamiltonian system in potential form (H(q; p) = p t M \Gamma1 p=2 + F (q)) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in R n . In this paper, methods which reduce "Hamiltonian differentialalgebra ..."
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Cited by 53 (10 self)
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. A Hamiltonian system in potential form (H(q; p) = p t M \Gamma1 p=2 + F (q)) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in R n . In this paper, methods which reduce "Hamiltonian differentialalgebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraintinvariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically. Key words. differentialalgebraic equations, constrained Hamiltonian systems, canonical discretization schemes, symplectic methods AMS(MOS) subj...
Algorithmic challenges in computational molecular biophysics
 Journal of Computational Physics
, 1999
"... A perspective of biomolecular simulations today is given, with illustrative applications and an emphasis on algorithmic challenges, as reflected by the work of a multidisciplinary team of investigators from five institutions. Included are overviews and recent descriptions of algorithmic work in long ..."
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Cited by 37 (3 self)
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A perspective of biomolecular simulations today is given, with illustrative applications and an emphasis on algorithmic challenges, as reflected by the work of a multidisciplinary team of investigators from five institutions. Included are overviews and recent descriptions of algorithmic work in longtime integration for molecular dynamics; fast electrostatic evaluation; crystallographic refinement approaches; and implementation of large, computationintensive programs on modern architectures. Expected future developments of the field are also discussed. c ○ 1999 Academic Press Key Words: biomolecular simulations; molecular dynamics; longtime integration; fast electrostatics; crystallographic refinement; highperformance platforms.
Nonholonomic integrators
, 2001
"... Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide ..."
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Cited by 31 (0 self)
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Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide an explanation for the good performance of the proposed method. This is tested on two examples: a nonholonomic particle with a quadratic potential and a mobile robot with fixed orientation.
Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation
 ANNU. REV. BIOPHYS. BIOMOL. STRUCT
, 1997
"... Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stab ..."
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Cited by 28 (10 self)
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Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stability. This timestep problem is challenging if strictly faster methods with the same allatom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multipletimescale contexts—where the fast motions are rapidly decaying or largely decoupled from others—have not been as successful for biomolecules, where vibrational coupling is strong. This review examines general issues that limit the timestep and describes available methods (constrained, reducedvariable, implicit, symplectic, multipletimestep, and normalmodebased schemes). A section compares results of selected integrators for a model dipeptide, assessing physical and numerical performance. Included is our dual timestep method LN, which relies on an approximate linearization of the equations of motion every �t interval (5 fs or less), the solution of which is obtained by explicit integration at the inner timestep �τ (e.g., 0.5 fs). LN is computationally competitive, providing 4–5 speedup factors, and results are in good agreement, in comparison to 0.5 fs trajectories. These collective algorithmic efforts help fill the gap between the time range that can be simulated and the timespans of major biological interest (milliseconds and longer). Still, only a hierarchy of models and methods, along with
Equivariant constrained symplectic integration
 J. Nonlinear Sci
, 1995
"... We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting ..."
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Cited by 27 (3 self)
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We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for LiePoisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum. 1.
Structure Preservation For Constrained Dynamics With Super Partitioned Additive RungeKutta Methods
 SIAM J. Sci. Comput
, 1998
"... A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is ..."
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Cited by 20 (9 self)
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A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the EulerLagrange equations is presented. A new class of integrators is defined: the super partitioned additive RungeKutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structurepreservation, the class of sstage Lobatto IIIABCC* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.
Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems
 Physica D
, 1995
"... We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential e ..."
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Cited by 19 (8 self)
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We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small stepsize, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bondangle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion...
Integration schemes for molecular dynamics and related applications. In The Graduate Student’s Guide to Numerical Analysis
 Series on Computer Mathematics
, 1999
"... Presented are a variety of modern practical techniques for the derivation of integration schemes that are useful for molecular dynamics and a variety of related applications. In particular, the emphasis is on Hamiltonian systems, including those with constraints, and to a lesser extent stochastic d ..."
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Cited by 17 (4 self)
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Presented are a variety of modern practical techniques for the derivation of integration schemes that are useful for molecular dynamics and a variety of related applications. In particular, the emphasis is on Hamiltonian systems, including those with constraints, and to a lesser extent stochastic differential equations. Among the techniques discussed are operator splitting, multiple time stepping, and accuracy enhancement through “postprocessing. ” Attention is also given to analytical tools for selecting among different integration schemes, for example, smalltimestep analysis of the backward error, linear analysis, and smallenergy analysis. 1