On Conjugate B-series and Their Geometric Structure

Abstract: The characterizations of B-series of symplectic and energy preserving integrators
are well-known. The graded Lie algebra of B-series of modified vector fields include the
Hamiltonian and energy preserving cases as Lie subalgebras, these spaces are relatively
well understood. However, two other important classes are the integrators which are conjugate
to Hamiltonian and energy preserving methods respectively. The modified vector
fields of such methods do not form linear subspaces and the notion of a grading must be
reconsidered. We suggest to study these spaces as filtrations, and viewing each element
of the filtraton as a vector bundle whose typical fiber replaces the graded homogeneous
components. In particular, we shall study properties of these fibers, a particular result is
that, in the energy preserving case, the fiber of degree n is a direct sum of the nth graded
component of the Hamiltonian and energy preserving space. We also give formulas for the
dimension of each fiber, thereby providing insight into the range of integrators which are
conjugate to symplectic or energy preserving.

 

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