Contexte et atouts du poste
The PhD will be in the team ALPINES, a joint research group between Inria and J.L. Lions Laboratory, Sorbonne University, which focuses on scientific computing.
The focus of the research in ALPINES is on the development of novel parallel numerical algorithms and tools appropriate for state-of-the-art mathematical models used in complex scientific applications, and in particular numerical simulations.
We plan to work in collaboration with Laura Grigori (Inria, ALPINES team), an expert in applied mathematics and computer science.
The general context : The modelling and the numerical simulation of plasma (a gas of charged particles) is of great importance from a physical and mathematical point of view.
In this context, the Vlasov-Maxwell equations provide a kinetic modelling approach of the dynamics of charged particles under the influence of an electro-magnetic field.
Difficulties in solving numerically such equations come from the existence of several scales in space and time of the solutions.
Proposed work and implementation :
The specific problem to be treated is the Vlasov-Poisson system with an additional strong external magnetic field, which has several applications in plasma physics, for example the confinement of particles.
The multi-scale behaviour due to high frequency oscillations in time imposes tiny time steps to the discretizations and therefore, the computational cost of long time simulations is prohibitive.
A solution for avoiding this problem is to use reduced models, based on averaging, whose solutions are free of oscillations.
An example is the two-scale limit model ( 5 ).
Nevertheless, in some applications this model does not cover the general situation where the rapid motion around the magnetic field line is not periodic.
In addition, the model only incorporates a two scale behaviour, whereas realistic phenomena may contain more than two scales.
Task 1 :
A first direction of research will be the development of new reduced models for Vlasov-Poisson problems. More precisely, our first aim is to improve existing results on homogenization (see 5 ) to a broader framework, which is free of periodicity and can deal with three time scales or more.
Such general results exist in the literature ( 3, 4 ) only for diffusion equations and do not seem to be derived for Vlasov-like equations.
Secondly, we plan to develop first-order homogenized models to increase accuracy, following the strategy developed in 5 for standard models.
Task 2 :
The objective is to efficiently implement the previous models in order to perform simulations for several applications in plasma physics.
We therefore aim to develop, analyze and implement a parareal method (see 2 ) for solving the previous equations. This algorithm is an efficient method which performs real time simulations by means of parallel-in-time integration.
We will follow a strategy where a different (reduced) model than the original one is used for the coarse solver. This method was successfully applied in 1 for solving highly oscillatory differential equations with plasma physics applications.
Reduced model-based parareal simulations of oscillatory singularly
perturbed ordinary differential equations'', Journal of Computational Physics,
vol. 436, 110282, 2021.
of PDE's'', Comptes Rendus de l’Académie des Sciences - Series I - Mathematics,
332, 661 668, 2001.
Proceedings of the Royal Society of Edinburgh Section A : Mathematics 126.2,
Analysis and its Applications, vol. 23, 483 508, 2004.
perturbed convection equation'', J. Math. Pures Appl., vol. 80, 815 843, 200
The profile is in applied mathematics / scientific computing.
Knowledge in mathematical modelling and numerical methods for partial differential equations.
Programming in Fortran or C / C++.