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

TL;DR

The paper introduces Runge-Kutta resonance-based schemes that bridge low-regularity convergence and geometric structure preservation for dispersive PDEs such as KdV and NLSE. By combining a twisted-variable formulation, symplectic kernel approximations, and multistage Runge–Kutta updates, the authors obtain implicit, structure-preserving integrators that handle rough data efficiently. They provide a comprehensive convergence analysis in $H^2$ and $H^1$ and demonstrate favorable numerical performance, including invariant preservation and competitive convergence rates against existing methods. This framework extends to a broad class of dispersive Hamiltonian systems and lays groundwork for higher-order, symplectic, low-regularity time integrators in infinite dimensions.

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 Schrödinger 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.

Figures (12)

• Figure 1: Order plot measured in $H^1$.
• Figure 2: Order plot measured in $H^1$, initial data $u_0\in C^\infty$, as per \ref{['eqn:def_smooth_initial_data_numerics']}.
• Figure 3: Relative error in the $L^2$-norm with $u_0\in H^2$, as per \ref{['eqn:law_for_random_initial_conditions']} with $\vartheta=2$.
• Figure 4: Relative error in the $L^2$-norm with $u_0\in C^\infty$, as per \ref{['eqn:def_smooth_initial_data_numerics']}.
• Figure 5: Error in the Hamiltonian of the numerical solution, for $u_0\in C^\infty(\mathbb{T})$ as per \ref{['eqn:def_smooth_initial_data_numerics']}.

Theorems & Definitions (86)

• Lemma 2.1
• Remark 2.3
• Example 2.4
• Example 2.5
• Remark 2.7
• Example 2.8
• Example 2.9: Cubic nonlinear Schrödinger equation
• Example 2.10: Low-regularity kernel approximation for cubic NLSE
• Example 2.11: First order method introduced in ostermann2018low
• Example 2.12: Symmetric method from AlamaBronsard23banicamaierhoferschratz22