Journal ArticleDOI
Runge-Kutta methods on Lie groups
TLDR
In this paper, generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group are presented. But these methods must satisfy two different criteria to achieve a given order.Abstract:
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.read more
Citations
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Solving Ordinary Differential Equations
TL;DR: The variable-order Adams method (SIVA/DIVA) package as discussed by the authors is a collection of subroutines for solution of non-stiff ODEs.
Journal ArticleDOI
The Magnus expansion and some of its applications
TL;DR: Magnusson expansion as discussed by the authors provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory (TEPT).
Journal ArticleDOI
Lie-group methods
TL;DR: A survey of numerical integrators that respect Lie-group structure is given in this paper, highlighting theory, algorithmic issues, and a number of applications in the field of Lie group discretization.
Journal ArticleDOI
On the solution of linear differential equations in lie groups
Arieh Iserles,Syvert P. Nørsett +1 more
TL;DR: The solution of the linear differential equation y′ = a(t)y, y(0) = y0 is represented as an infinite series whose terms are indexed by binary trees, which leads both to a convergence proof and to a constructive computational algorithm.
Journal ArticleDOI
High order Runge-Kutta methods on manifolds
TL;DR: It is proved that any classical Runge-Kutta method can be turned into an invariant method of the same order on a general homogeneous manifold, and a family of algorithms that are relatively simple to implement are presented.
References
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Book
Foundations of Differentiable Manifolds and Lie Groups
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
Solving Ordinary Differential Equations
TL;DR: The variable-order Adams method (SIVA/DIVA) package as discussed by the authors is a collection of subroutines for solution of non-stiff ODEs.
Book
Manifolds, tensor analysis, and applications
TL;DR: In this paper, the authors provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists, including manifolds, dynamical systems, tensors, and differential forms.
Book
Lie groups, Lie algebras, and their representations
TL;DR: In this article, differentiable and analytic manifolds and Lie Groups and Lie Algebras have been studied in the context of structure theory and representation theory, and complex semisimple Lie Algebraic structures have been proposed.