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36
Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 93 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
RungeKutta methods on Lie groups
, 1997
"... . We construct generalized Runge#Kutta methods for integration of di#erential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolveonthe correct manifold. Our methods must satisfy two di#erent ..."
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Cited by 71 (14 self)
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. We construct generalized Runge#Kutta methods for integration of di#erential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolveonthe correct manifold. Our methods must satisfy two di#erent criteria to achieve a given order: # Coe#cients A i;j and b j must satisfy the classical order conditions. This is done by picking the coe#cients of any classical RK scheme of the given order. # Wemust construct functions to correct for certain non#commutative e#ects to the given order. These tasks are completely independent, so once correction functions are found to the given order, we can turn any classical RK scheme into a RK method of the same order on any Lie group. The theory in this paper shows the tight connections between the algebraic structure of the order conditions of RK methods and the algebraic structure of the so called `universal enveloping algebra' of Lie algebras. This m...
High order RungeKutta methods on manifolds
 APPL. NUMER. MATH
, 1999
"... This paper presents a family of RungeKutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical RungeKutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a fa ..."
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Cited by 52 (10 self)
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This paper presents a family of RungeKutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical RungeKutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and present a family of algorithms that are relatively simple to implement.
Numerical integration of differential equations on homogeneous manifolds
 Foundations of Computational Mathematics
, 1997
"... We present anoverview of intrinsic integration schemes for differential equations evolving on manifolds, paying particular attention to homogeneous spaces. Various examples of applications are introduced, RungeKutta methods. We argue that homogeneous spaces are the natural structures for the stud ..."
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Cited by 38 (14 self)
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We present anoverview of intrinsic integration schemes for differential equations evolving on manifolds, paying particular attention to homogeneous spaces. Various examples of applications are introduced, RungeKutta methods. We argue that homogeneous spaces are the natural structures for the study and the analysis of these methods.
RungeKutta methods and renormalization
, 1999
"... A connection between the algebra of rooted trees used in renormalization theory and RungeKutta methods is pointed out. Butcher’s group and Bseries are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally Bseries are used to sol ..."
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Cited by 34 (0 self)
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A connection between the algebra of rooted trees used in renormalization theory and RungeKutta methods is pointed out. Butcher’s group and Bseries are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally Bseries are used to solve a class of nonlinear partial differential equations. 1
Multistep Methods on Manifolds
, 1997
"... We explore the retention of invariants by a class of timestepping discretization methods, inclusive of multistep methods and truncated Taylor expansions. Our main ..."
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Cited by 27 (11 self)
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We explore the retention of invariants by a class of timestepping discretization methods, inclusive of multistep methods and truncated Taylor expansions. Our main
The Newton Iteration on Lie Groups
, 1996
"... We de#ne the Newton iteration for solving the equation f#y# = 0, where f is a map from a Lie group to its corresponding Lie algebra. Twoversions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, an ..."
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Cited by 27 (4 self)
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We de#ne the Newton iteration for solving the equation f#y# = 0, where f is a map from a Lie group to its corresponding Lie algebra. Twoversions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f , the proposed method converges quadratically.We illustrate the techniques by solving a #xedpoint problem arising from the numerical integration of a Lietype initial value problem via implicit Euler. # This work was in part sponsored by The Norwegian Research Council under contract no. 111038#410, through the SYNODE project. WWW: http:##www.imf.unit.no#num#synode y Email: Brynjulf.Owren@imf.unit.no, WWW: http:##www.imf.unit.no#~bryn z Email: bdw@math.la.asu.edu, WWW: http:##math.la.asu.edu#~bdw 1 1 Motivation Recently, there has been an increased intere...
RungeKutta Methods Adapted to Manifolds and Based on Rigid Frames
 BIT
, 1999
"... We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E1 , ..., En on the manifold. The numerical approximation is obtained by composing flows of certain vector fields ..."
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Cited by 24 (15 self)
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We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E1 , ..., En on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E1 , ..., En that are tangent to the differential system at various points. The methods reduce to traditional RungeKutta methods if the frame vector fields are chosen as the standard basis of euclidean R n . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.
The method of iterated commutators for ordinary differential equations on Lie groups
, 1996
"... We construct numerical methods to integrate ordinary differential equations that evolve on Lie groups. These schemes are based on exponentials and iterated commutators, they are explicit and their order analysis is relatively simple. Thus we can construct groupinvariant integrators of arbitraril ..."
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Cited by 22 (5 self)
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We construct numerical methods to integrate ordinary differential equations that evolve on Lie groups. These schemes are based on exponentials and iterated commutators, they are explicit and their order analysis is relatively simple. Thus we can construct groupinvariant integrators of arbitrarily high order. Among other applications we show that this approach can be used to obtain new symplectic schemes when applied to Hamiltonian problems. Some numerical experiments are presented.
Integration Methods Based on Rigid Frames
, 1997
"... We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E 1 ; : : : ; En on the manifold. The numerical approximation is obtained by composition of flows of vector fields ..."
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Cited by 17 (5 self)
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We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E 1 ; : : : ; En on the manifold. The numerical approximation is obtained by composition of flows of vector fields in the linear span of E 1 ; : : : ; En . The methods reduce to traditional RungeKutta methods if the frame vector fields are chosen as the standard basis of euclidean R n . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results. AMS Subject Classification: 65L06, 34A50 Key Words: Geometric integration, Numerical integration of ordinary differential equations on manifolds, RungeKutta methods 1 Introduction Consider the initial value problem y = f(t; y); y(0) = y 0 ; f : R \Theta R n ! R n : (1) Such and similar openings have been fairly common in pa...