Results 1  10
of
40
On the numerical integration of ordinary differential equations by symmetric composition methods
 SIAM J. Sci. Comput
, 1995
"... Abstract. Differential equations of the form ˙x = X = A + B are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. ..."
Abstract

Cited by 95 (11 self)
 Add to MetaCart
(Show Context)
Abstract. Differential equations of the form ˙x = X = A + B are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary firstorder integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given. 1. Composition methods. Composition methods are particularly useful for numerically integrating differential equations when the equations have some special structure which it is advantageous to preserve. They tend to have larger local truncation errors than standard (RungeKutta, multistep) methods [4,5], but this defect can be more than compensated for by their superior conservation properties. Capital letters such as X will denote vector fields on some space with coordinates x, with flows exp(tX), i.e., ˙x = X(x) ⇒ x(t) = exp(tX)(x(0)). The vector field X is given and is to be integrated numerically with fixed time step t. Composition methods apply when one can write X = A + B in such a way that exp(tA), exp(tB) can both be calculated explicitly. Then the most elementary such method is the map (essentially the “LieTrotter ” formula [26]) ϕ: x ↦ → x ′ = exp(tA) exp(tB)(x) = x(t) + O(t 2). (1.1) The advantage of composing exact solutions in this way is that many geometric properties of the true flow exp(tX) are preserved: group properties in particular. If X, A, and B are Hamiltonian vector fields then both exp(tX) and the map ϕ
Geometric Integrators for ODEs
 J. Phys. A
, 2006
"... Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, ..."
Abstract

Cited by 37 (6 self)
 Add to MetaCart
(Show Context)
Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, timereversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.
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 ..."
Abstract

Cited by 28 (3 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
(Show Context)
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.
Geometric Integration and Its Applications
 in Handbook of numerical analysis
, 2000
"... This paper aims to give an introduction to the relatively new eld of geometric integration. ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
(Show Context)
This paper aims to give an introduction to the relatively new eld of geometric integration.
Lie Group Methods for Rigid Body Dynamics and Time Integration on Manifolds
 Computer Methods in Applied Mechanics and Engineering
, 1999
"... Recently there has been an increasing interest in time integrators for ordinary dierential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods l ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
(Show Context)
Recently there has been an increasing interest in time integrators for ordinary dierential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general. An important part of the Lie group methods involves the computation of a coordinate map and its derivative. Various options are available, and they vary in cost, accuracy and ability to approximately conserve invariants. We discuss how the computation of these maps can be optimized for the rigid body case, and we provide numerical experiments which give an idea of the performance of Lie group methods compared to other known integration schemes. AMS Subject Classication: 65L05 Key Words: time integration, geometric integration, numerical integration of ordinary dierential equati...
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 ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
(Show Context)
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
Explicit variable stepsize and timereversible integration
 Appl. Numer. Math
, 1999
"... In [9], a variable stepsize, semiexplicit variant of the explicit StormerVerlet method has been proposed for the timereversible integration of Newton's equations of motion. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
In [9], a variable stepsize, semiexplicit variant of the explicit StormerVerlet method has been proposed for the timereversible integration of Newton's equations of motion. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for general timereversible differential equations. As applications, we discuss the variable stepsize, timereversible, and fully explicit integration of rigid body motion and reversible Nos'eHoover dynamics.
Statistical mechanics of Arakawa’s discretizations
, 2007
"... The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. This is illustrated for quasigeostrophic flow with topographic forcing, for which a well esta ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. This is illustrated for quasigeostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa (1966) which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.
Symplectic Integrators for Systems of Rigid Bodies
 Integration Algorithms and Classical Mechanics, volume 10 of Fields Institute Communications. AMS
, 1996
"... Recent work reported in the literature suggest that for the longterm integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. This has especially been shown for the numerical treatment of conservative manyparticle systems as they arise ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Recent work reported in the literature suggest that for the longterm integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow. This has especially been shown for the numerical treatment of conservative manyparticle systems as they arise, e.g., in molecular dynamics and astronomy. In this paper we use recent results by McLachlan [10] and Reich [13] on the symplectic integration of LiePoisson systems to derive an explicit symplectic integrator for rigid bodies moving under the influence of external forces. In case that rigid bodies are interconnected by joints, the resulting holonomic constraints can be treated in the same way as described by Reich [12] for general Hamiltonian systems with holonomic constraints. Keywords: Hamiltonian systems, symplectic discretization, multibody systems AMS(MOS) Subject Classifications: 65L05, 7008, 70F20 1 Introduction Much recent research has gone into developing numerical discretizat...