Clémentine Courtès

Clémentine Courtès, Frédéric Lagoutière, Frédéric Rousset

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 $s\geq3$.

Cosmin Burtea, Clémentine Courtès

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.

Clémentine Courtès

We compute the rate of convergence of forward, backward and central finite difference $θ$-schemes for linear PDEs with an arbitrary odd order spatial derivative term. We prove convergence of the first or second order for smooth and less smooth initial data.

Clémentine Courtès, Emmanuel Franck, Michael Kraus et al.

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.