21 research outputs found
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive.publishedVersio
Detecting and determining preserved measures and integrals of rational maps
In this paper we use the method of discrete Darboux polynomials to calculate
preserved measures and integrals of rational maps. The approach is based on the
use of cofactors and Darboux polynomials and relies on the use of symbolic
algebra tools. Given sufficient computing power, most, if not all, rational
preserved integrals can be found (and even some non-rational ones).
We show, in a number of examples, how it is possible to use this method to
both determine and detect preserved measures and integrals of the considered
rational maps. Many of the examples arise from the Kahan-Hirota-Kimura
discretization of completely integrable systems of ordinary differential
equations
Geometric Integration & Kahan's Method
The first half of this talk presents a general overview of Geometric Numerical Integration of differential equations. The second half discusses geometric integrability properties of Kahan¹s
method.Non UBCUnreviewedAuthor affiliation: La Trobe UniversityFacult