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89
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. ..."
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Cited by 95 (11 self)
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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 first-order 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 (Runge-Kutta, 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 “Lie-Trotter ” 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
Fourth-order time stepping for stiff PDEs
- SIAM J. SCI. COMPUT
, 2005
"... A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made ..."
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Cited by 94 (3 self)
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A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competing methods of implicit-explicit differencing, integrating factors, time-splitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
Geometric numerical integration illustrated by the Störmer-Verlet 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 long-time 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 long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section 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 long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.
Symplectic Numerical Integrators in Constrained Hamiltonian Systems
, 1994
"... : Recent work reported in the literature suggests that for the long-time integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic (or canonical) structure of the flow. Here we investigate the symplecticness of numerical integrators for constrained dynamics, su ..."
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Cited by 61 (8 self)
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: Recent work reported in the literature suggests that for the long-time integration of Hamiltonian dynamical systems one should use methods that preserve the symplectic (or canonical) structure of the flow. Here we investigate the symplecticness of numerical integrators for constrained dynamics, such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies. This leads to a constrained Hamiltonian system of smaller dimension. Previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space. Here it is shown that the very popular Verlet method with SHAKE-type constraints is equivalent to the same method with RATTLE-type constraints and that the latter is symplectic and time reversible. (This assumes that the iteration is carried to convergence.) We also demonstrate the global convergence of the Verlet scheme in the presence of SHAKE--type and RATTLE-...
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 54 (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 constraint-invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically. Key words. differential-algebraic equations, constrained Hamiltonian systems, canonical discretization schemes, symplectic methods AMS(MOS) subj...
High order symplectic integrators for perturbed Hamiltonian systems
, 2001
"... Abstract. We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB. We give a constructive proof that for all integer p, there exists an integrator with positive steps with a remainder of order O(τ p ε + τ 2 ε 2), where τ is the ..."
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Cited by 41 (4 self)
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Abstract. We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB. We give a constructive proof that for all integer p, there exists an integrator with positive steps with a remainder of order O(τ p ε + τ 2 ε 2), where τ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes O(τ p ε + τ 4 ε 2). The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.
Explicit Lie–Poisson Integration and the Euler Equations
"... Abstract. We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D ..."
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Cited by 40 (9 self)
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Abstract. We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D Euler equations. Hamiltonian systems are fundamental, and symplectic integrators (SI’s) have been increasingly used to do useful extremely-long-time numerical integrations of them. Wisdom [17] has used fast SI’s to integrate the solar system far more efficiently than with standard methods; there are numerous examples illustrating the
The Adaptive Verlet Method
- SIAM J. Sci. Comput
, 1997
"... We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symme ..."
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Cited by 37 (11 self)
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We discuss the integration of autonomous Hamiltonian systems via dynamical rescaling of the vector field (reparameterization of time). Appropriate rescalings (e.g., based on normalization of the vector field or on minimum particle separation in an N-body problem) do not alter the time-reversal symmetry of the flow, and it is desirable to maintain this symmetry under discretization. For standard form mechanical systems without rescaling, this can be achieved by using the explicit leapfrog-Verlet method; we show that explicit time-reversible integration of the reparameterized equations is also possible if the parameterization depends on positions or velocities only. For general rescalings, a scalar nonlinear equation must be solved at each step, but only one force evaluation is needed. The new method also conserves the angular momentum for an N-body problem. The use of reversible schemes, together with a step control based on normalization of the vector field (arclength reparameterization), is demonstrated in several numerical experiments, including a double pendulum, the Kepler problem, and a three-body problem.
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structure-preserving time-integrators for rigid body-type mechanical ..."
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Cited by 36 (7 self)
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Abstract In this paper, structure-preserving time-integrators for rigid body-type mechanical
The accuracy of symplectic integrators,
- Nonlinearity
, 1992
"... Abstract. We judge symplectic integrators by the accuracy with which they represent the Hamiltonian function. This accuracy is computed, compared and tested far several different methods. We develop new, highly accurate explicit fourth-and fifth-order methods valid when the Hamiltonian is separable ..."
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Cited by 28 (2 self)
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Abstract. We judge symplectic integrators by the accuracy with which they represent the Hamiltonian function. This accuracy is computed, compared and tested far several different methods. We develop new, highly accurate explicit fourth-and fifth-order methods valid when the Hamiltonian is separable with quadratic kinetic energy. For the near-integrable case, we confirm several of their properties expected from KAM theory; convergence of some of the characteristics of chaotic motions are also demonstrated. We paint out cases in which long-time stability is intrinsically lost.
