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95
Computations in a Free Lie Algebra
, 1998
"... Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity ..."
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Cited by 62 (16 self)
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Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witts formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides accurate bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new...
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 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...
Geometric Integration Using Discrete Gradients
, 1998
"... This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODE's). Given an ODE and one or more first integrals (i.e., constants of the motion) and/or Lyapunov functions, ..."
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Cited by 54 (18 self)
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This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODE's). Given an ODE and one or more first integrals (i.e., constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a `lineargradient system.' Discrete gradients are used to construct discrete approximations to the ODE which preserve the first integrals and Lyapunov functions exactly. The method applies to all Hamiltonian, Poisson, and gradient systems, and also to many dissipative systems (those with a known first integral or Lyapunov function).
MultiSymplectic RungeKutta Collocation Methods for Hamiltonian Wave Equations
 J. Comput. Phys
, 1999
"... A number of conservative PDEs, like various wave equations, allow for a multisymplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that GaussLegendre collocation in space and time leads to multisymplectic integrators, i.e., to nume ..."
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Cited by 42 (7 self)
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A number of conservative PDEs, like various wave equations, allow for a multisymplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that GaussLegendre collocation in space and time leads to multisymplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for Hamiltonian ODEs. We also discuss the issue of conservation of energy and momentum. Since time discretization by a GaussLegendre method is computational rather expensive, we suggest several semiexplicit multisymplectic methods based on GaussLegendre collocation in space and explicit or linearly implicit symplectic discretizations in time. 1 Introduction The scalar wave equation @ tt u = @ xx u \Gamma V 0 (u); (x; t) 2 U ae R 2 ; (1) V : R ! R some smooth function, is an example of a multisymplectic Hamiltonian PDE [3] of type M@ t z +K@ x z = r z S...
Approximating the exponential from a Lie algebra to a Lie group
 Math. Comp
, 1998
"... Abstract. Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ GandA: R + × G → g, wheregis a Lie algebra of the matricial Lie group G. Every B ∈ g canbemappedtoGbythematrixexponentialmap exp (tB) witht∈R. Most numerical methods for solving ordinary differential equations (ODEs) on ..."
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Cited by 41 (8 self)
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Abstract. Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ GandA: R + × G → g, wheregis a Lie algebra of the matricial Lie group G. Every B ∈ g canbemappedtoGbythematrixexponentialmap exp (tB) witht∈R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(tn), tn ∈ R +, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y0. This ensures that yn ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper. 1.
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 .
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, ..."
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Cited by 37 (6 self)
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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 ..."
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Cited by 28 (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.
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.