Research

Research interests

I am interested in the analysis and the conception of structure-preserving numerical schemes for partial differential equations.
More specifically, I am working on discretisations of anisotropic advection-diffusion equations on general meshes. One of my main preoccupation is the preservation, at the discrete level, of the positivity (or bounds) on the computed solutions. I also focus on the asymptotic preserving features of the methods, and especially the long-time behaviour of the schemes, that is to say the asymptotics of solutions as time tends to infinity. To do so, I design nonlinear entropic schemes that mimic the entropy structure of the continuous problems. This structure is then used as a starting point to analyse the schemes. I also implement the methods in C++ to test them and assert the good behaviour of the schemes on some practical cases. I also like to compare my methods with other ones.
The ultimate goal of my PhD was to design high-order skeletal methods enjoying these features. The schemes I design are based on the Hybrid Finite Volume (low order) and Hybrid High-Order methods.

The main applications I work on are related to semiconductor models.
In this context, I also work with physicists on Two-Points Flux Approximation Finite Volumes methods for models with very irregular advection fields. These models are used to describe random alloy fluctuation on some devices, especially LEDs.

I am currently investigating the use of such entropic schemes on cross-diffusion systems.

My codes are available on my Gitlab page.

Publications and preprints

Numerical analysis 

• Structure preservation in high-order hybrid discretisations of advection-diffusion equations: linear and nonlinear approaches - Mathematics in Engineering, 2024
  with S. Lemaire

• A structure preserving hybrid finite volume scheme for semi-conductor models with magnetic field on general meshes - ESAIM: Mathematical Modelling and Numerical, 2023

• Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches - Numerische Mathematik, 2022
  with C. Chainais-Hillairet, M. Herda and S. Lemaire

Proceedings

• Structure-preserving schemes for drift-diffusion systems on general meshes: DDFV vs HFV - FVCA X - Volume 1, Elliptic and Parabolic Problems, 2023
  with S. Krell

• A skeletal high-order structure preserving scheme for advection-diffusion equations - FVCA X - Volume 1, Elliptic and Parabolic Problems, 2023

Collaborations with physicists (journals and proceedings)

• Impact of random alloy fluctuations on the carrier distribution in multi-color (In,Ga)N/GaN quantum well systems  - Physical Review Applied, 2024
  with M. O'Donovan, P. Farrell, T. Streckenbach, T. Koprucki and S. Schulz

• Theoretical investigation of carrier transport and recombination processes for deep UV (Al,Ga)N light emitters  - 2023 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 2023
  with R. Finn, M. O'Donovan, P. Farrell, T. Streckenbach, T. Koprucki and S. Schulz     [preprint of the proceeding]

• Importance of satisfying thermodynamic consistency in in optoelectronic device simulations for high carrier densities - Optical and Quantum Electronics, 2023
  with P. Farrell,  M. O'Donovan, S. Schulz and  T. Koprucki

• Theoretical study of the impact of alloy disorder on carrier transport and recombination processes in deep UV (Al,Ga)N light emitters  - Applied Physics Letters, 2023
  with R. Finn, M. O'Donovan, P. Farrell, T. Streckenbach, T. Koprucki and S. Schulz

• Comparison of flux discretizations for varying band edge energies  - 2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 2022
  with P. Farrell      [preprint of the proceeding  |  blog article associated]

PhD thesis 

I defended my PhD thesis the 26th September 2023, in Lille.
The thesis is entitled "Development and analysis of high order schemes for convection-diffusion models, study of their long time behavior"

Talks

• "Reliable numerical approximation: from continuous structures to discrete ones", Séminaire compréhensible, Institut Fourier, Grenoble, Feb. 2022

• "Approximation numérique fiable : des structures continues aux propriétés discrètes", PhD seminar, laboratoire Paul Painlevé, Lille, Feb. 2022

• "A structure preserving hybrid finite volume scheme for semi-conductor models on general meshes", Seminar Numerische Mathematik, WIAS, Berlin, Jun. 2022

• "Comparison of flux discretizations for varying band-edge energies", NUSOD 2022, Torino, Sep. 2022  

• "Reliable numerical approximation: from continuous structures to discrete ones", PhD seminar, Institut de Mathématiques de Bourgogne, Dijon, Dec. 2022

• "A structure preserving hybrid finite volume scheme for semi-conductor models with magnetic fields on general meshes", POEMS 2022, Milano, Dec. 2022

• "A high-order scheme for advection-diffusion preserving positivity and long-time behaviour", ABPDE 5, Lille, Jun. 2023

• "A skeletal high-order structure preserving scheme for advection-diffusion equations", FVCA X, Strasbourg, Nov. 2023

• "Schémas volumes finis préservant la structure pour des modèles de semi-conducteurs anisotropes", Seminar Calcul Scientifique et Modélisation, IMB, Bordeaux, Feb. 2024

• "An arbitrary-order entropic method for structure-preserving approximations of advection-diffusion", ALGORITMY 2024, Podbanské, Mar. 2024

• "High-efficiency and reliable schemes for drift-diffusion systems", AMaSiS 2024, Berlin, Sep. 2024

Poster presentations 

• Long-time behaviour of a hybrid finite volume scheme for the Drift-Diffusion model with magnetic field, presented at AMaSiS 2021 ONLINE (WIAS, Berlin, 2021)

• Long-time behaviour of hybrid finite volumes schemes for advection-diffusion models, presented at the CJC-MA (École Polytechnique, Palaiseau, 2021)  and at ABPDE 4 (Lille, 2021)

• An entropic scheme for drift-diffusion systems on general meshes, presented at the closure conference of ANR MOHYCON (Pornichet, 2022)

• High-order and structure preserving scheme for advection-diffusion, presented at the conference New Trends in the Numerical Analysis of PDEs (Lille, 2024)