Resonances as a computational tool
Frédéric Rousset, Katharina Schratz
TL;DR
This article surveys resonance-based numerical schemes for nonlinear dispersive equations, addressing the limitations of classical methods at low regularity where oscillatory dynamics are poorly captured. It explains how embedding the leading nonlinear resonance structure into discretizations, starting with the dominant part of the resonance in cubic NLS, yields first-order schemes with improved robustness to rough data and reduced derivative requirements. The authors extend the approach to a general abstract setting, discuss error analysis in Bourgain spaces, and present higher-order extensions via iterative Duhamel expansions and refined oscillatory integral approximations, including explicit second-order schemes. They also highlight open problems, particularly around symmetry and structure preservation, and emphasize the potential for resonance-based methods to provide stable, accurate simulations for rough data across a broad class of PDEs.
Abstract
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this article we review a new class of resonance-based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong properties at low regularity.