Many differential equations of practical interest evolve on Lie groups or manifolds acted upon by groups. The retention Lie-group structure under discretization is often vital in the recovery qualitatively correct geometry and dynamics minimization numerical error. Having introduced requisite elements geometry, this paper surveys novel theory integrators that respect structure, highlighting theory, algorithmic issues a number applications.

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

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker–Campbell–Hausdorff formula recently developed group for integration of differential equations on manifolds. This paper is concerned with complexity optimization such general case where algebra free, i.e. no specific assumptions are made about its structure. It shown how transformations applied to original variables a problem yield elements graded free whose...

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the brackets $\bar{g}$ are related via post-Lie algebra structure. The latter can be extended to enveloping $U(g)$. This permits us define another associative product on $U(g)$, which gives rise Hopf isomorphism between $U(\bar{g})$ new assembled from $U(g)$ with product. For free these constructions provide refined understanding fundamental appearing in theory numerical integration methods for...

Associated to a symmetric space there is canonical connection with zero torsion and parallel curvature. This acts as binary operator on the vector of smooth sections tangent bundle, it linear respect real numbers. Thus section bundle together form an algebra we call algebra. The constraints constant curvature makes into Lie admissible triple type that generalises pre-Lie algebras, can be embedded post-Lie in way embedding systems algebras. free described by incorporating triple-brackets...

Understanding the algebraic structure underlying a manifold with general affine connection is natural problem. In this context, A. V. Gavrilov introduced notion of framed Lie algebra, consisting bracket (the usual Jacobi vector fields) and magmatic product without any compatibility relations between them. work, we will show that an curvature torsion always gives rise to post-Lie algebra as well D -algebra. The notions together Gavrilov's special polynomials double exponential are revisited...

We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations the space, its Lie triple system (LTS), and exponential LTS to space. Examples of spaces $ n $-spheres Grassmann manifolds, space positive definite matrices, groups with a product, elliptic hyperbolic constant sectional curvatures. illustrate abstract algorithm concrete examples. In particular $-sphere $-dimensional...

Abstract The exotic aromatic Butcher series were originally introduced for the calculation of order conditions high numerical integration ergodic stochastic differential equations in $$\mathbb {R} ^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and on manifolds. We prove this paper that B-series satisfy a universal geometric property, namely they are characterised by locality...

In this paper we describe the use of theory 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 property that, if $Z \in {\frak{g}}$, Lie algebra matrices, then approximation for exp(Z) resides in G, group ${\frak{g}}$. This is very relevant when solving Lie-group ODEs not usually fulfilled by standard approximations We propose based on splitting Z...