Hans Z. Munthe-Kaas

Lie-group methods

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group 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 Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

RUNGE-KUTTA METHODS ON LIE GROUPS

AbstractWe 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.• CoefficientsAi,j andbj 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. 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.

High order Runge-Kutta methods on manifolds

We present a family of Runge-Kutta type integration schemes of arbitrarily high order for differential equations evolving on manifolds. We prove that any classical Runge-Kutta 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. These are defined in a general abstract framework, based on a Lie algebra acting on the manifold. The general framework gives rise to a wide range of different concrete applications; we present some examples.

Computations in a free Lie algebra

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker–Campbell–Hausdorff 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 good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new MATLAB toolbox ‘DIFFMAN’

Lie-Butcher theory for Runge-Kutta methods

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 Butchers celebrated theory of order conditions on the other. In Butchers 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.

Topics in structure-preserving discretization

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

Generalized Polar Decompositions for the Approximation of the Matrix Exponential

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297--324] to approximate a matrix exponential. The algorithms presented have the property that, if

On the Hopf Algebraic Structure of Lie Group Integrators

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On Post-Lie Algebras, Lie–Butcher Series and Moving Frames

, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of

Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms

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