2019-01360 - Post-Doctoral Research Visit F/M Large-time behavior of numerical schemes for kinetic equations

Contract type : Public service fixed-term contract

Level of qualifications required : PhD or equivalent

Fonction : Post-Doctoral Research Visit

About the research centre or Inria department

The Inria Lille - Nord Europe Research Centre was founded in 2008 and employs a staff of 360, including 300 scientists working in sixteen research teams. Recognised for its outstanding contribution to the socio-economic development of the Nord - Pas-de-Calais Region, the Inria Lille - Nord Europe Research Centre undertakes research in the field of computer science in collaboration with a range of academic, institutional and industrial partners.

The strategy of the Centre is to develop an internationally renowned centre of excellence with a significant impact on the City of Lille and its surrounding area. It works to achieve this by pursuing a range of ambitious research projects in such fields of computer science as the intelligence of data and adaptive software systems. Building on the synergies between research and industry, Inria is a major contributor to skills and technology transfer in the field of computer science.

Context

In the context of a collaboration with the region Haut-de-France, Inria offers a postdoctoral position held at the Inria Lille - Nord Europe research center. The successful candidate will be integrated in the project-team RAPSODI (Reliable numerical approximations of dissipative systems). The activity of the team is devoted to the design, the analysis, and efficient implementation of numerical schemes for dissipative models arising in physics or biology. 

Assignment

Kinetic partial differential equations are mathematical models that describe the evolution of systems of many interacting particles in a wide variety of applications (plasma physics, rarefied gas dynamics, biology, social sciences...). The conception of reliable numerical methods for these high dimensional and multiscale models is a very active and challenging research topic.

The goal of the postdoc is the design and asymptotic analysis of numerical schemes for kinetic equations. The main focus will be put on the numerical analysis of the large-time behavior of numerical solutions. 

Main activities

Recently, several papers (see [1, 4] and references therein) succesfully adapted hypocoercivity techniques (see [3, 7] and references therein) to the discrete setting in order to obtain precise quantitative estimates on the large-time behavior of numerical solutions.

Thanks to these novel methods, several research directions will be explored by the hired researcher. The first research direction concerns the long-time analysis of exponential-fitting schemes proposed in [5]. These schemes converge to the celebrated Scharfetter-Gummel / Il’in numerical schemes in the diffusive limit, which are well-known for their accurate large-time behavior. The second research direction is the design of numerical schemes for specific kinetic models with hypocoercive structure such as those proposed in [2] and [6]. Once again the main focus will concern the obtention of an accurate large-time behavior at the discrete level. 

[1] Marianne Bessemoulin-Chatard, Maxime Herda, and Thomas Rey. Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations. arXiv preprint arXiv:1812.05967, 2018.
[2] Jean Dolbeault, Axel Klar, Clément Mouhot, and Christian Schmeiser. Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes. Applied Mathematics Research eXpress, 2013(2):165–175, 2012.
[3] Jean Dolbeault, Clément Mouhot, and Christian Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Transactions of the American Mathematical Society, 367(6):3807–3828, 2015.
[4] Guillaume Dujardin, Frédéric Hérau, and Pauline Lafitte. Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations. arXiv preprint arXiv:1802.02173, 2018.
[5] Laurent Gosse and Nicolas Vauchelet. Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting. Numerische Mathematik, pages 1–54, 2018.
[6] Lukas Neumann and Christian Schmeiser. A kinetic reaction model: decay to equilibrium and macroscopic limit. arXiv preprint arXiv:1503.05745, 2015.
[7] Cédric Villani. Hypocoercivity. Number 949-951. American Mathematical Soc., 2009.

Benefits package

  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours)
  • Social security coverage

Remuneration

Net monthly salary (after taxes) : 2132.97€