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28
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 63 (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.
Practical Symplectic Partitioned RungeKutta and RungeKuttaNyström Methods
, 2000
"... We present new symmetric fourth and sixthorder symplectic Partitioned Runge Kutta and RungeKuttaNystrom methods. We studied compositions using several extra stages, optimising the efficiency. An effective error, E f , is defined and an extensive search is carried out using the extra parameter ..."
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Cited by 39 (7 self)
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We present new symmetric fourth and sixthorder symplectic Partitioned Runge Kutta and RungeKuttaNystrom methods. We studied compositions using several extra stages, optimising the efficiency. An effective error, E f , is defined and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods found in the literature. When applied to several examples they perform up to two orders of magnitude better than previously known method, which is in very good agreement with the values of E f .
On the necessity of negative coefficients for operator splitting schemes of order higher than two,”Applied
 Numerical Mathematics,
, 2005
"... Abstract In this paper we analyse numerical integration methods applied to differential equations which are separable in solvable parts. These methods are compositions of flows associated with each part of the system. We propose an elementary proof of the necessary existence of negative coefficient ..."
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Cited by 24 (4 self)
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Abstract In this paper we analyse numerical integration methods applied to differential equations which are separable in solvable parts. These methods are compositions of flows associated with each part of the system. We propose an elementary proof of the necessary existence of negative coefficients if the schemes are of order, or effective order, p 3 and provide additional information about the distribution of these negative coefficients. It is shown that if the methods involve flows associated with more general terms this result does not necessarily apply and in some cases it is possible to build higher order schemes with positive coefficients.
Splitting and composition methods in the numerical integration of differential equations
, 2008
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Numerical integrators that preserve symmetries and reversing symmetries
 SIAM J. Numer. Anal
, 1998
"... Abstract. We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framewo ..."
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Cited by 21 (14 self)
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Abstract. We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framework unifies several known constructions. We also use the concept of “covariance” of a numerical method with respect to a group of coordinate transformations. The main application is to explore the relationship between spatial symmetries, reversing symmetries, and time symmetry of flows and numerical integrators.
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 18 (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
Families of highorder composition methods
, 2001
"... We propose a new rule of thumb for designing highorder composition methods for ODEs: instead of minimizing (some norm of) the principal error coefficients, simply set all the outer stages equal. This rule automatically produces families of minimum error 4th order and corrected 6th order methods, an ..."
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Cited by 9 (1 self)
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We propose a new rule of thumb for designing highorder composition methods for ODEs: instead of minimizing (some norm of) the principal error coefficients, simply set all the outer stages equal. This rule automatically produces families of minimum error 4th order and corrected 6th order methods, and very good standard 6th order methods, parameterized by the number of stages. Intriguingly, the most accurate methods (evaluated with the total work held fixed) have a very large number of stages.
Cheap Implicit Symplectic Integrators
 Appl. Num. Math
, 1997
"... The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems, by say, the leapfrog method, the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic m ..."
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Cited by 7 (0 self)
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The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems, by say, the leapfrog method, the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic methods like implicit midpoint have less severe stability restrictions but the cost of solving large nonlinear systems with dense Jacobian matrices is probably too high to make them worthwhile. To bring down the cost of implicit methods, we have designed (i) mixed implicitexplicit and (ii) linearly implicit methods that retain the property of being symplectic. Keywords: linearly implicit method, stiff oscillatory, symplectic integrator, Hamiltonian systems, implicit midpoint. AMS(MOS) Subject Classifications: 65L20 CR Subject Classifications: G.1.3 1
Almost symplectic RungeKutta schemes for Hamiltonian systems
 J. Comput. Phys
, 2005
"... Abstract. Symplectic RungeKutta schemes for integration of general Hamiltonian systems are implicit. In practice the implicit equations are often approximately solved based on the Contraction Mapping Principle, in which case the resulting integration scheme is no longer symplectic. In this note we ..."
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Cited by 4 (0 self)
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Abstract. Symplectic RungeKutta schemes for integration of general Hamiltonian systems are implicit. In practice the implicit equations are often approximately solved based on the Contraction Mapping Principle, in which case the resulting integration scheme is no longer symplectic. In this note we prove that, under suitable conditions, the integration scheme based on an nstep successive approximation is O(δn+2) away from a symplectic scheme with δ ∈ (0, 1). Therefore, this scheme is “almost ” symplectic when n is large.