Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes

TL;DR

This paper introduces Runge-Kutta resonance-based methods that achieve low-regularity solutions for dispersive PDEs like KdV and NLSE while preserving their geometric structure, bridging a key gap in numerical analysis.

Contribution

It develops a new class of symplectic resonance-based Runge-Kutta methods that combine low-regularity convergence with structure preservation for Hamiltonian dispersive equations.

Findings

Successfully characterizes symplectic resonance-based methods for KdV and NLSE.

Achieves low-regularity approximations that preserve geometric properties.

Bridges the gap between low regularity and structure preservation in numerical schemes.

Abstract

Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric properties of the flow. This is particularly drastic in the case of the Korteweg-de Vries (KdV) equation and the nonlinear Schr\"odinger equation (NLSE) which are fundamental models in the broad field of dispersive infinite-dimensional Hamiltonian systems, possessing infinitely many conserved quantities, an important property which we wish to capture - at least up to some degree - also on the discrete level. Nowadays, a wide range of structure preserving integrators for Hamiltonian systems are available, however, typically these existing algorithms can only approximate highly regular solutions efficiently. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. In this work we introduce a novel framework, so-called Runge-Kutta resonance-based methods, for a large class of dispersive nonlinear equations which incorporate a much larger amount of degrees of freedom than prior resonance-based schemes while featuring similarly favourable low-regularity convergence properties. In particular, for the KdV and NLSE case, we are able to bridge the gap between low regularity and structure preservation by characterising a large class of symplectic (in the Hamiltonian picture) resonance-based methods for both equations that allow for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.