In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrodinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L(2)-norm and/or the energy of the system.

Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi-Pasta-Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discrelizations.

Umeå University, Faculty of Science and Technology, Department of Computing Science. School of Science and Technology, Örebro University, Örebro, Sweden.

Johansson, Stefan

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Kågström, Bo

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Van Dooren, Paul

Department of Mathematical Engineering, Université catholique de Louvain, Louvain-la-Neuve, Belgium.

Geometry of Matrix Polynomial Spaces2019In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383Article in journal (Refereed)

Abstract [en]

We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials' orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Aromatic Butcher series2016In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 16, no 1, p. 183-215Article in journal (Refereed)

Abstract [en]

We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series), which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge-Kutta methods, that extends the class of Runge-Kutta methods and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produce equivariant homogeneous space integrators. Moreover, we show that the RKMK, Crouch-Grossman, or commutator-free methods are equivariant. To show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection.