arXiv subjects

Yvonne Alama Bronsard

Publications and source records attributed to Yvonne Alama Bronsard.

8 recordsLinked to original sources

arXiv2026

Splitting methods for Intermediate Long Wave and perturbed Benjamin--Ono models

Recently, novel integrability techniques, and most notably Birkhoff coordinates followed by an explicit formula, have been introduced to solve the Benjamin--Ono (BO) equation, leading to major theoretical and computational advances. We propose to perturb this explicit BO formula via a novel splitting method for numerically solving a class of PDEs, which includes the Intermediate Long Wave (ILW) equation, as well as other quasilinear equations that are not necessarily integrable. Using the Birkhoff coordinates of BO we establish first-order convergence in the $H^s$-norm ($s \ge 0$) while requiring only one additional derivative on the initial data $u_0 \in H^{s+1}_{0}$, significantly improving upon the regularity requirements of typical splitting methods for nonlinear dispersive equations. Furthermore, we prove that in the deep-water limit, the scheme for the ILW equation converges to the BO solution. Computational advantages of these schemes are shown in simulations: unlike classical splitting methods, our approach does not require a restrictive quadratic time step condition to nearly preserve energy over long time scales, thus rendering efficient and accurate long-time simulations feasible. As an application, we numerically explore the soliton resolution conjecture for both the ILW and a KdV--BO equation.

math.NA↗

arXiv2026

On the convergence of explicit formulas for $L^2$ solutions to the Benjamin-Ono and continuum Calogero-Moser equations

By developing discrete counterparts to recent advances in nonlinear integrability, and in particular to the discovery of explicit formulas, we design and analyze fully-discrete approximations to the Benjamin-Ono (BO) and continuum Calogero-Moser (CCM) equations on the torus. We build on the key observation that discretizing such explicit formulas yields schemes that are exact in time (requiring only spatial discretization) and have a computational cost independent of the final time $T$. In this work, we first generalize the fully-discrete schemes of arXiv:2412.13480 to include numerical approximations with better structure preservation properties, including the conservation of mass and momentum in the case of the (BO) equation. Secondly, building on recent analyses of the corresponding Lax operators, we extend the convergence results to this class of schemes for rough solutions $u(t)$ merely belonging to $L^2(\mathbb{T})$ for (BO) and $L^2_{+}(\mathbb{T})$ for (CCM), the latter of which is precisely the scaling-critical regularity. Our main theorem states that the $L^2(\mathbb{T})$-norm of the error goes to zero as the truncation parameters go to infinity, uniformly on any bounded time interval $[-T,T]$. As an example, we apply our scheme to the (BO) equation with a square-wave initial profile, and obtain the first numerical evidence of the Talbot effect for (BO) supported by a rigorous convergence result.

math.NA↗

arXiv2025

Symmetric resonance based integrators and forest formulae

In the present work we introduce a unified framework that allows for the very first systematic construction of symmetric resonance-based integrators to approximate a wide class of nonlinear dispersive equations at low-regularity. The inclusion of symmetries in the construction of resonance-based schemes presents serious challenges and induces a need for a significant extension of prior approaches to allow for sufficient number of degrees of freedom in the resulting schemes while preserving the favorable low-regularity convergence properties of prior constructions. Motivated by recent work arXiv:2005.01649, we achieve this by introducing a novel formalism based on forest formulae that allows us to encode a wider range of possibilities of iterating Duhamel's formula and interpolatory approximations of lower order parts in the construction of these time-stepping methods. The forest formulae allow for a simple characterisation of symmetric schemes and provides a fascinating algebraic structure in its own right which echo those used in Quantum Field Theory for renormalising Feynman diagrams and those used for the renormalisation of singular SPDEs via the theory of Regularity Structures. Our constructions lead to the development of several new symmetric low regularity integrators that exhibit remarkable structure preservation and convergence properties which are witnessed in numerical experiments.

math.NA↗

arXiv2025

Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas

We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szegő equations on the torus, which are $\textit{exact in time}$ with $\textit{spectral accuracy}$ in space. We prove spectral convergence for the first two equations, of order $K^{-s+1}$ in $L^2$ norm for initial data in $H^s(\mathbb T)$, $s>1$, with an error constant depending $\textit{linearly}$ on the final time instead of exponentially. These schemes are based on $\textit{explicit formulas}$, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations.

math.NA↗

arXiv2023

A symmetric low-regularity integrator for the nonlinear Schrödinger equation

We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schrödinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb{T}^d$ and on a smooth bounded domain $Ω\subset \mathbb{R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e. under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes.

math.NA↗

arXiv2023

Approximations of dispersive PDEs in the presence of low-regularity randomness

We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(τ, v^η)|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the dispersive equation and $ v^η(x) $ the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance based discretisation (see arXiv:2005.01649) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via Regularity Structures. In contrast to classical approaches, we do not discretize the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level.

math.PR↗

arXiv2023

Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity

We analyse a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $Ω\subset \mathbb{R}^d$, $d \le 3$. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space $H^{r}$, $r \ge 0$, beyond the more typical $L^2$ or $H^σ(σ>\frac{d}{2}$) -error analysis. Numerical experiments illustrate our results.

math.NA↗

arXiv2022

Low regularity integrators via decorated trees

We introduce a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as dispersive equations, up to arbitrary high order on general domains. The structure of the local error of the new schemes is driven by nested commutators which in general require (much) lower regularity assumptions than classical methods do. Our main idea lies in embedding the central oscillations of the nonlinear PDE into the numerical discretisation. The latter is achieved by a novel decorated tree formalism inspired by singular SPDEs with Regularity Structures and allows us to control the nonlinear interactions in the system up to arbitrary high order on the infinite dimensional (continuous) as well as finite dimensional (discrete) level.

math.NA↗