A substitution law for B-series vector fields
Philippe Chartier, Ernst Hairer and Gilles Vilmart
Abstract.
In this paper, we derive a new composition law obtained by
substituting a B-series into the vector field appearing in another
B-series. We derive explicit formulas for the computation of this
law and study its algebraic properties. We then focus on the specific
case of Hamiltonian vector fields. It is shown that this new law
allows a convenient derivation of the modified equation occurring
in backward error analysis or in numerical methods based on
generating functions.
Key Words.
Ordinary differential equations, B-series, backward error analysis,
Hamiltonian, symmetric, symplectic, generating function methods.