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A substitution law for B-series vector fields

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Philippe Chartier, Ernst Hairer and Gilles Vilmart

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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.

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Key Words. Ordinary differential equations, B-series, backward error analysis, Hamiltonian, symmetric, symplectic, generating function methods.

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