Derivation of resonance-based schemes via normal forms
Yvain Bruned
TL;DR
The article develops a systematic derivation of resonance-based numerical schemes for dispersive PDEs by marrying Poincaré-Dulac normal form ideas with combinatorial Hopf-algebraic tools. It encodes oscillatory integrals from Duhamel expansions using Fourier decorated trees, an arborification map, and a forest formula derived from a Butcher-Connes-Kreimer coproduct to produce explicit, low-regularity schemes with provable local error bounds. The core innovation is the dominant/low phase splitting and a Taylor-expansion strategy applied to iterated integrals, enabling explicit coefficient formulas and a robust discretisation framework. Under a mild Assumption 1, these normal-form schemes are expected to share the local error behavior of prior low-regularity methods, bridging normal forms and resonance-based discretisations with explicit algebraic structure.
Abstract
In this work, we propose a systematic derivation of resonance-based schemes via normal forms. The main idea is to use an arborification map on decorated trees together with a Butcher-Connes-Kreimer type coproduct and lower-dominant parts decompositions of the Fourier operator coming from the nonlinear interactions. This new family of low regularity schemes has explicit formulae for its coefficients and its local error. Under a mild assumption, one could expect these schemes to have a similar local error as the low regularity schemes proposed in arXiv:2005.01649.