Explicit Geometric Integration of Polynomial Vector Fields

Abstract

Dedicated to Syvert Nørsett on the occasion of his 60th birthday. AbstractWe present a unified framework in which to study splitting methods for polynomial vector fields in R n . The vector field is to be represented as a sum of shears, each of which can be integrated exactly, and each of which is a function of k < n variables. Each shear must also inherit the structure of the original vector field: we consider Hamiltonian, Poisson, and volumepreserving cases. Each case then leads to the problem of finding an optimal distribution of points on an appropriate homogeneous space, generally the Grassmannians of k-planes or (in the Hamiltonian case) isotropic k-planes in R n . These optimization problems have the same structure as those of constructing optimal experimental designs in statistics.

Explicit Volume-Preserving Splitting Methods for Linear and Quadratic Divergence-Free Vector Fields

“…The cases studied in [11] indicate that the vectors a i which are on average best for all matrices should be chosen to be as widely or regularly spaced as possible. In this case, a maximally symmetric configuration is possible, namely, the lines through the origin to the vertices of the n-simplex in R n inscribed in S n−1 .…”

“…By a direct count of free parameters, McLachlan and Quispel [11] conjecture that a d-degree divergence-free polynomial vector field can be split in n + d terms, each of them a function of n − 1 variables (n − 1 planes in R n ). In coordinates, one chooses an orthonormal basis a 1 , .…”

“…Divergence-free vector fields occur naturally in incompressible fluid dynamics, and preservation of phase-space volume is also a crucial ingredient in many if not all ergodic theorems. An earlier paper on splitting polynomial vector fields is [11]. That paper had some discussion of the divergence-free case, but mainly dealt with the Hamiltonian case.…”

“…Investigations of the Hamiltonian case, which involves expressing a scalar polynomial of degree d in n variables as a sum of functions of fewer variables, have shown that good splitting methods exist, but that finding and analyzing them (especially for general n and d) is very difficult [4,11,14]. The volume-preserving case, which involves n polynomials subject to the divergence-free condition, is even harder, although there is a conjecture [11] that they can be expressed as a sum of n + d shears, each a function of n − 1 variables.…”

“…The volume-preserving case, which involves n polynomials subject to the divergence-free condition, is even harder, although there is a conjecture [11] that they can be expressed as a sum of n + d shears, each a function of n − 1 variables. Therefore, in the present paper we undertake a study of several possible approaches to splitting linear and quadratic divergence-free vector fields in any dimension n, with a view to understanding the structure of the problem and identifying promising approaches for the general case.…”