Derivation of resonance-based schemes via normal forms

TL;DR

This paper introduces a systematic method for deriving resonance-based numerical schemes using normal forms, arborification maps, and coproduct structures, resulting in explicit formulas and potential low regularity advantages.

Contribution

It presents a novel systematic derivation framework for resonance-based schemes employing algebraic structures and normal forms, with explicit coefficient formulas and error analysis.

Findings

New family of low regularity schemes with explicit coefficients

Potential for schemes with similar local error to existing low regularity methods

Use of arborification maps and coproducts in scheme derivation

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.