272 research outputs found
Geometric Numerical Integration Applied to The Elastic Pendulum at Higher Order Resonance
In this paper we study the performance of a symplectic numerical integrator
based on the splitting method. This method is applied to a subtle problem i.e.
higher order resonance of the elastic pendulum. In order to numerically study
the phase space of the elastic pendulum at higher order resonance, a numerical
integrator which preserves qualitative features after long integration times is
needed. We show by means of an example that our symplectic method offers a
relatively cheap and accurate numerical integrator.Comment: 15 pages, 6 figure
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
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