The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

Abstract Aromatic B-series were introduced as an extension of standard Butcher-series for the study volume-preserving integrators. It was proven with their help that only method is exact flow differential equation. The question raised whether there exists a integrator can be expanded aromatic B-series. In this work, we introduce new algebraic tool, called bicomplex, similar to variational bicomplex in calculus. We prove exactness and use it describe explicitly key object integrators: forms vanishing divergence. analysis provides us handful tools B-series, gives insights on process integration by parts trees, allows integrator. particular, conclude Runge–Kutta cannot preserve volume.