Numerical simulation of Boltzmann-Nordheim equation

Project reference: 2122

Since twenty years, fast spectral methods have been developed for simulating rarefied gases. Such physical phenomenon can be modelled with Boltzmann-type equations that are written in 1D+nD+mD in the time-phase space (n = 1,2,3 and m=2,3).

We are interested in the Boltzmann-Nordheim model where the collision term is trilinear and includes quantum effects of the collisions when the temperature of the gas is low. In such case, Bose-Einstein condensate can occur, which leads to a degenerescence of the distribution function into a combination of a Dirac  delta distribution and a singular Maxwellian distribution.

The aim of this project is the numerical investigation of such phenomenon. Since the Boltzmann-Nordheim model is potentially written in 7D, the numerical methods for discretizing it must be chosen carefully. In 2012, the homogeneous 2D case has been studied by Filbet & al. thanks to fast spectral methods. They also presented some numerical investigations of the approximation of this kinetic model by its macroscopic limit. At present time, we are about to finish the extension to the 3D space homogeneous (n=0) case. With this new step, we may have the first Bose-Einstein condensate simulations through kinetic modelling.

The aim of the internship(s) is to extend the current work to non-homogeneous cases by discretizing the transport term and extending the numerical methods and codes to 1D+nD+3D with n = 1,2,3 in the phase space. For this purpose, strong skills in numerical analysis, code parallelization and high performance scientific computing are welcome to address the incoming challenges.

Project Mentor: Alexandre Mouton 

Project Co-mentor: Thomas Rey

Site Co-ordinator: Benoît Fresse

Participants: David Knapp, Artem Mavilutov

Learning Outcomes:
At the end of the internship, the student would be able to manage parallelized codes with MPI or OpenMP (or even with hybrid techniques). In addition, the student will be trained to usual numerical methods for discretizing Boltzmann-type kinetic models.

Student Prerequisites (compulsory):
Mathematics : good knowledge of PDEs, numerical methods for ODEs and PDEs, discrete Fourier transform
Computer Science : strong skills of C and Python languages, at ease with Unix systems
Scientific computing : skills in parallel computing (at least with MPI and OpenMP), data visualization

Student Prerequisites (desirable):
Skills in Cmake and with kinetic and/or fluid modelling of rarefied gases is welcome.

Training Materials:
Additional informations about the internship offer can be found on  KINEBEC webpage:
https://sites.google.com/site/moutonalexandre/research/kinebec

Workplan:

Week 1 : training sessions with all students of SoHPC.
Weeks 2, 3, 4 : development of numerical methods, getting started with the code and the HPC environment
Weeks 5, 6, 7 : implementation and numerical tests
Week 7, 8 : numerical tests, report writing
Week 8 : numerical tests, report submission, preparation for the defense

Final Product Description:
The main goal of the internship is to provide a simulation of a Bose-Einstein condensate with the non-homogeneous Boltzmann-Nordheim model at least with 1D dimensionality in physical space. Higher dimensionality in space is the final goal but requires a preliminary work in numerical analysis before any implementation.

Adapting the Project: Increasing the Difficulty:
If the 1D+3D model is successfully discretized and investigated, we may think about the extension to 2D+3D model.

Adapting the Project: Decreasing the Difficulty:
If some difficulties remain, the student can work on alternative parallelization methods for discretizing the collision operator. At present time, MPI techniques are used so developing alternative version of these routines by using OpenMP or Cuda is also interesting.

Resources:
Laboratoire Paul Painlevé owns 3 nodes dedicated to developers in scientific computing. The Computing Central Service also provide larger HPC ressources for any lab members who works in Lille University research units. These accesses are for free and will be activated at the beginning of the internship.

Organisation:
LPP-Laboratoire Paul Painlevé (Université de Lille & CNRS)