Symmetric resonance based integrators and forest formulae

In the present work we introduce a unified framework that allows for the very first systematic construction of symmetric resonance-based integrators to approximate a wide class of nonlinear dispersive equations at low-regularity. The inclusion of symmetries in the construction of resonance-based schemes presents serious challenges and induces a need for a significant extension of prior approaches to allow for sufficient number of degrees of freedom in the resulting schemes while preserving the favorable low-regularity convergence properties of prior constructions. Motivated by recent work arXiv:2005.01649, we achieve this by introducing a novel formalism based on forest formulae that allows us to encode a wider range of possibilities of iterating Duhamel's formula and interpolatory approximations of lower order parts in the construction of these time-stepping methods. The forest formulae allow for a simple characterisation of symmetric schemes and provides a fascinating algebraic structure in its own right which echo those used in Quantum Field Theory for renormalising Feynman diagrams and those used for the renormalisation of singular SPDEs via the theory of Regularity Structures. Our constructions lead to the development of several new symmetric low regularity integrators that exhibit remarkable structure preservation and convergence properties which are witnessed in numerical experiments.