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

Abstract

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations.

On Hamiltonian flows whose orbits are straight lines

“…Assuming M f = 0, the equationq = f (M q) defines a flow on R n that is linear in time: Φ t (q 0 ) = q 0 + tf (M q 0 ). This is similar to the flow considered in[11, Lemma 5]. It extends to a Hamiltonian flow on R 2n , as does every flow on R n .…”

Explicit volume-preserving splitting methods for polynomial divergence-free vector fields

“…In the sequel, we will often use the short-hand notation ẋ = F j (x) (6) for (5). Each of these divergence-free vector fields is associated to a monomial basis element, and will be called an elementary divergence-free vector field (in short, EDFVF).…”

“…The volume-preserving case, which involves n polynomials subject to the divergence-free condition, is even harder, although there is a conjecture by [6] that they can be expressed as a sum of n + d shears, each a function of n − 1 variables. The case of linear and quadratic divergence-free vector fields was studied in detail in [5], where several explicit volume-preserving splitting methods were introduced. In that paper, two main classes of methods were considered: a) methods that distinguish the diagonal and off-diagonal part and b) methods that do not.…”

Foreword

“…His FoCM colleague, Peter Olver, writes on algorithms for differential invariants [8]. We [12] and Blanes, Casas, and Murua [5] write on splitting and composition methods; Arieh first wrote on composition methods in 1982 [1]. His former students, Moan and Niesen, write on the Magnus series [13] and his former postdoc, Celledoni, writes on exponential integrators [7]; Arieh first studied integrators using exponentials and Lie brackets in 1984 [2].…”