The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods
Résumé
Aromatic Butcher series were successfully introduced for the study and design of numeri- cal integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras and pre-Hopf algebroids struc- tures, and aromatic trees were shown to form the free tracial pre-Lie-Rinehart algebra. In this paper, we present the generalisation of aromatic trees for the study of divergence-free integrators on manifolds. We introduce planar aromatic trees, show that they span the free tracial post-Lie-Rinehart algebra, and apply them for deriving new Lie-group methods that preserve geometric divergence-free features up to a high order of accuracy.
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