68 research outputs found
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction
In this paper we focus on WENO-based methods for the simulation of the 1D
Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser
wave interacts with and heats a plasma by penetrating into it. We propose
several non-oscillatory methods based on either Runge--Kutta (explicit) or
Time-Splitting (implicit) time discretizations. We then show preliminary
numerical experiments
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
On algebraic structures of numerical integration on vector spaces and manifolds
Numerical analysis of time-integration algorithms has been applying advanced
algebraic techniques for more than fourty years. An explicit description of the
group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared
in Butcher's work on composition of integration methods in 1972. In more recent
years, the analysis of structure preserving algorithms, geometric integration
techniques and integration algorithms on manifolds have motivated the
incorporation of other algebraic structures in numerical analysis. In this
paper we will survey structures that have found applications within these
areas. This includes pre-Lie structures for the geometry of flat and torsion
free connections appearing in the analysis of numerical flows on vector spaces.
The much more recent post-Lie and D-algebras appear in the analysis of flows on
manifolds with flat connections with constant torsion. Dynkin and Eulerian
idempotents appear in the analysis of non-autonomous flows and in backward
error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di
Bruno Hopf algebra are other examples of structures appearing naturally in the
numerical analysis of integration on manifolds.Comment: 42 pages, final versio
Lie Group integrators for mechanical systems
Since they were introduced in the 1990s, Lie group integrators have become a
method of choice in many application areas. These include multibody dynamics,
shape analysis, data science, image registration and biophysical simulations.
Two important classes of intrinsic Lie group integrators are the
Runge--Kutta--Munthe--Kaas methods and the commutator free Lie group
integrators.
We give a short introduction to these classes of methods. The Hamiltonian
framework is attractive for many mechanical problems, and in particular we
shall consider Lie group integrators for problems on cotangent bundles of Lie
groups where a number of different formulations are possible. There is a
natural symplectic structure on such manifolds and through variational
principles one may derive symplectic Lie group integrators. We also consider
the practical aspects of the implementation of Lie group integrators, such as
adaptive time stepping. The theory is illustrated by applying the methods to
two nontrivial applications in mechanics. One is the N-fold spherical pendulum
where we introduce the restriction of the adjoint action of the group
to , the tangent bundle of the two-dimensional sphere. Finally, we show
how Lie group integrators can be applied to model the controlled path of a
payload being transported by two rotors. This problem is modeled on
and put in a format where Lie group integrators can be applied.Comment: 35 page
Casimir preserving stochastic Lie-Poisson integrators
Casimir preserving integrators for stochastic Lie-Poisson equations with Stratonovich noise are developed extending Runge-Kutta Munthe-Kaas methods. The underlying Lie-Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie-Poisson dynamics by means of the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid
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