Volume preservation of Butcher series methods from the operad viewpoint

Abstract

We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series.

arXiv comment25 pages, comments are welcome

References in corpus (8)

• M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
• Aromatic Butcher Series
• Lie elements in pre-Lie algebras, trees and cohomology operations
• Free pre-Lie algebras are free as Lie algebras
• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• Deformation theory of Cohomological Field Theories
• Homotopical rigidity of the pre-Lie operad
• Graph insertion operads