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R. I. McLachlan H. Z. Munthe-Kaas G. R. W. Quispel A. Zanna 《Foundations of Computational Mathematics》2008,8(3):335-355
We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods
can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that
do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal
parts, giving rise to a number of possible combinations.
This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary. 相似文献
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H. Z. Munthe-Kaas G. R. W. Quispel A. Zanna 《Foundations of Computational Mathematics》2001,1(3):297-324
The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix
and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they
induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group.
In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized
polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve
them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion.
Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor
in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G .
September 12, 2000. Final version received: April 16, 2001. 相似文献
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Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted
generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps.
In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres. 相似文献
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Hans Munthe-Kaas 《BIT Numerical Mathematics》1998,38(1):92-111
We construct generalized Runge-Kutta methods for integration of differential 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 evolve on the correct
manifold. Our methods must satisfy two different criteria to achieve a 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 an 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 may give important insight
also into the classical RK theory.
This work is sponsored by NFR under contract no. 111038/410, through the SYNODE project. WWW:http://www.math.ntnu.no/num/synode. 相似文献
– | • CoefficientsA i,j andb j must satisfy the classical order conditions. This is done by picking the coefficients of any classical RK scheme of the given order. |
– | • We must construct functions to correct for certain non-commutative effects to the given order. |
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Lie-Butcher theory for Runge-Kutta methods 总被引:1,自引:0,他引:1
Hans Munthe-Kaas 《BIT Numerical Mathematics》1995,35(4):572-587
Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Although this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper. 相似文献
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In the past few years, a number of Lie-group methods based on Runge—Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge—Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie-group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness.In this paper we discuss Lie-group methods and their dependence on centering coordinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge—Kutta schemes and symmetric coordinates, based on geodesic or on flow midpoint. As result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献