Publications and preprints

2024

Journal article

A Composite Finite Volume Scheme for the Euler Equations with Source Term on Unstructured Meshes

Mohamed Boujoudar - Emmanuel Franck - Philippe Hoch - Clément Lasuen - YLH - Paul Paragot

In this work we focus on an adaptation of the method described in [1] in order to deal with source term in the 2D Euler equations. This method extends classical 1D solvers (such as VFFC, Roe, Rusanov) to the two-dimensional case on unstructured meshes. The resulting schemes are said to be composite as they can be written as a convex combination of a purely node-based scheme and a purely edge-based scheme. We combine this extension with the ideas developed by Alouges, Ghidaglia and Tajchman in an unpublished work [2] -- focused mainly on the 1D case -- and we propose two attempts at discretizing the source term of the Euler equations in order to better preserve stationary solutions. We compare these discretizations with the ``usual'' centered discretization on several numerical examples.

2024

Preprint

A Generalized Spectral Concentration Problem and the Varying Masks Algorithm

Erwan Faou - YLH

In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.

2023

Journal article

Grid-Free Weighted Particle Method Applied to the Vlasov--Poisson Equation

YLH

We study a grid-free particle method based on following the evolution of the characteristics of the Vlasov--Poisson system, and we show that it converges for smooth enough initial data. This method is built as a combination of well-studied building blocks -- mainly time integration and integral quadratures -- and allows to obtain arbitrarily high orders. By making use of the Non-Uniform Fast Fourier Transform, the overall computational complexity is $\mathcal{O}(P \log P + K d \log K d )$, where P is the total number of particles and where we only keep the Fourier modes $k\in (Z^d)^*$ such that $k_1^2 + \dots + k_d^2 \leq K^2$. Some numerical results are given for the Vlasov--Poisson system in the one-dimensional case.

2023

Preprint

Modulation Algorithm for the Nonlinear Schrödinger Equation

Erwan Faou - YLH - Pierre Raphaël

Based on recent ideas, stemming from the use of bubbles, we discuss an algorithm for the numerical simulation of the cubic nonlinear Schrödinger equation with harmonic potential in any dimension, which could be easily extended to other polynomial nonlinearities. For the linear part of the equation, the algorithm consists in discretizing the initial function as a sum of modulated complex functions, each one having its own set of parameters, and then updating the parameters exactly so that the modulated function remains a solution to the equation. When cubic interactions are introduced, the Dirac-Frenkel-MacLachlan principle is used to approximate the time evolution of parameters. We then obtain a grid-free algorithm in any dimension, and it is compared to a spectral method on numerical examples.