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Clémentine Courtès

Publications and source records attributed to Clémentine Courtès.

9 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

Neural non-canonical Hamiltonian dynamics for long-time simulations

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

cs.LG↗
arXiv2025

The Spear and the Ring: Emergent Structures in Magnetic Colloidal Suspensions

We study from a mathematical point of view the nanoparticle model of a magnetic colloid, presented by G. Klughertz. Our objective is to obtain properties of stable stationary structures that arise in the long-time limit for the magnetic nanoparticles dynamics following this model. In this article, we present a detailed study of two specific structures using techniques from the calculus of variations. The first, called the spear, consists of a chain of aligned particles interacting via a Lennard-Jones potential. We establish existence and uniqueness results, derive bounds on the distances between neighboring particles, and provide a sharp asymptotic description as the number of particles tends to infinity. The second structure, the ring, features particles uniformly distributed along a circle. We prove its existence and uniqueness and derive an explicit formula for its radius.

math-ph↗
arXiv2024

Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires

In this article, we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.

math.AP↗
arXiv2024

Micromagnetic simulations of the size dependence of the Curie temperature in ferromagnetic nanowires and nanolayers

We solve the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. By rescaling the temperature proportionally to the computational cell size $Δx$ ($T \to T\,Δx/a_{\text{eff}}$, where $a_{\text{eff}}$ is the lattice constant) [M. B. Hahn, J. Phys. Comm., 3:075009, 2019], we obtain Curie temperatures $T_{\text{C}}$ that are in line with the experimental values for cobalt, iron and nickel. For finite-sized objects such as nanowires (1D) and nanolayers (2D), the Curie temperature varies with the smallest size $d$ of the system. We show that the difference between the computed finite-size $T_{\text{C}}$ and the bulk $T_{\text{C}}$ follows a power-law of the type: $(ξ_0/d)^λ$, where $ξ_0$ is the correlation length at zero temperature, and $λ$ is a critical exponent. We obtain values of $ξ_0$ in the nanometer range, also in accordance with other simulations and experiments. The computed critical exponent is close to $λ=2$ for all considered materials and geometries. This is the expected result for a mean-field approach, but slightly larger than the values observed experimentally.

cond-mat.mes-hall↗
arXiv2023

Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples

In this paper, we consider a ferromagnetic material of ellipsoidal shape. The associated magnetic moment then has two asymptotically stable opposite equilibria, of the form $\pm\overline{m}$. In order to use these materials for memory storage purposes, it is necessary to know how to control the magnetic moment. We use as a control variable a spatially uniform external magnetic field and consider the question of flipping the magnetic moment, i.e., changing it from the $+\overline{m}$ configuration to the $-\overline{m}$ one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic field used. We therefore include a constraint on the $L^\infty$ norm of the controls, assumed to be less than a threshold value $U$. We show that, generically with respect to the dimensions of the ellipsoid, there is a minimal value of $U$ for this problem to have a solution. We then characterize it precisely. Finally, we investigate some particular configurations associated to geometries enjoying symmetries properties and show that in this case the magnetic moment can be controlled in minimal time without imposing a threshold condition on $U$.

math.AP↗
arXiv2018

Error estimates of finite difference schemes for the Korteweg-de Vries equation

This article deals with the numerical analysis of the Cauchy problem for the Korteweg-de Vries equation with a finite difference scheme. We consider the Rusanov scheme for the hyperbolic flux term and a 4-points $θ$-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant-Friedrichs-Lewy condition when $θ\geq \frac{1}{2}$ and under an "Airy" Courant-Friedrichs-Lewy condition when $θ<\frac{1}{2}$. More precisely, we get the first order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the non-smooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$ , to the price of a loss in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq3$.

math.NA↗
arXiv2018

Discrete energy estimates for the abcd-systems

In this article, we propose finite volume schemes for the $abcd$-systems and we establish stability and error estimates. The order of accuracy depends on the so-called BBM-type dispersion coefficients $b$ and $d$. If $bd>0$, the numerical schemes are $O(Δt+(Δx)^2)$ accurate, while if $bd=0$, we obtain an $O(Δt+Δx)$ -order of convergence. The analysis covers a broad range of the parameters $a,b,c,d$. In the second part of the paper, numerical experiments validating the theoretical results as well as head-on collision of traveling waves are investigated.

math.NA↗