Section: Overall Objectives

Presentation

In applications involving complex physics, such as plasmas and nanotechnologies, numerical simulations serve as a prediction tool supplementing real experiments and are largely endorsed by engineers or researchers. Their performances rely not only on computational power, but also on the efficiency of the underlying numerical method and the complexity of the underlying models. The contribution of applied mathematics is then required, on the one hand for a better understanding of qualitative properties and a better identification of the different regimes present in the model, and on the other hand, for a more sounded construction of new models based on asymptotic analysis. This mathematical analysis is expected to greatly impact the design of multiscale numerical schemes.

The proposed research group MINGuS will be dedicated to the mathematical and numerical analysis of (possibly stochastic) partial differential equations (PDEs), originating from plasma physics and nanotechnologies, with emphasis on multiscale phenomena either of highly-oscillatory, of dissipative or stochastic types. These equations can be also encountered in applications to rarefied gas dynamics, radiative transfer, population dynamics or laser propagation, for which the multiscale character is modelled by a scale physical parameter $\epsilon$.

Producing accurate solutions of multiscale equations is extremely challenging owing to severe restrictions to the numerical methods imposed by fast (or stiff) dynamics. Ad-hoc numerical methods should aim at capturing the slow dynamics solely, instead of resolving finely the stiff dynamics at a formidable computational cost. At the other end of the spectrum, the separation of scales -as required for numerical efficiency- is envisaged in asymptotic techniques, whose purpose is to describe the model in the limit where the small parameter $\epsilon$ tends to zero. MINGuS aspires to accommodate sophisticated tools of mathematical analysis and heuristic numerical methods in order to produce simultaneously rich asymptotic models and efficient numerical methods.

To be more specific, MINGuS aims at finding, implementing and analysing new multiscale numerical schemes for the following physically relevant multiscale problems:

• Highly-oscillatory Schrödinger equation for nanoscale physics: In quantum mechanics, the Schrödinger equation describes how the quantum state of some physical system changes with time. Its mathematical and numerical study is of paramount importance to fundamental and applied physics in general. We wish to specifically contribute to the mathematical modeling and the numerical simulation of confined quantum mechanical systems (in one or more space dimensions) possibly involving stochastic terms. Such systems are involved in quantum semi-conductors or atom-chips, as well as in cold atom physics (Bose-Einstein condensates) or laser propagation in optical fibers.

  The prototypical equation is written

  $\begin{array}{|c|}\hline i\epsilon {\partial }_t{\psi }^{\epsilon }=\frac{{\epsilon }^2}{\beta }\Delta {\psi }^{\epsilon }+{\left|{\psi }^{\epsilon }\right|}^2{\psi }^{\epsilon }+{\psi }^{\epsilon }\xi \\ \hline\end{array}$

  (1)

  where the function ${\psi }^{\epsilon }={\psi }^{\epsilon }(t,x)\in ℂ$ depends on time $t\ge 0$ and position $x\in {ℝ}^3$, $\xi =\xi (x,t)$ is a white noise and where the small parameter $\epsilon$ is the Planck's constant describing the microscopic/macroscopic ratio. The limit $\epsilon \to 0$ is referred to as the semi-classical limit. The regime $\epsilon =1$ and $\beta \to 0$ (this can be for instance the relative length of the optical fiber) is highly-oscillatory. The noise $\xi$ acts as a potential, it may represent several external perturbations. For instance temperature effects in Bose-Einstein condensation or amplification in optical fibers. The highly oscillatory regime combined with noise introduces new challenges in the design of efficient schemes.

• Highly-oscillatory or highly-dissipative kinetic equations: Plasma is sometimes considered as the fourth state of matter, obtained for example by bringing a gas to a very high temperature. A globally neutral gas of neutral and charged particles, called plasma, is then obtained and is described by a kinetic equation as soon as collective effects dominate as compared to binary collisions. A situation of major importance is magnetic fusion in which collisions are not predominant. In order to confine such a plasma in devices like tokamaks (ITER project) or stellarators, a large magnetic field is used to endow the charged particles with a cyclotronic motion around field lines. Note that kinetic models are also widely used for modeling plasmas in earth magnetosphere or in rarefied gas dynamics.

  Denoting $f^{\epsilon }=f^{\epsilon }(t,x,v)\in {ℝ}^{+}$ the distribution function of charged particles at time $t\ge 0$, position $x\in {ℝ}^3$ and velocity $v\in {ℝ}^3$, a typical kinetic equation for $f^{\epsilon }$ reads

  $\begin{array}{|c|}\hline {\partial }_t{f}^{\epsilon }+v·{\nabla }_x{f}^{\epsilon }+\left(E+\frac{1}{\epsilon }(v\times B)\right)·{\nabla }_v{f}^{\epsilon }=\frac{1}{\beta }Q\left(f^{\epsilon }\right)+f^{\epsilon }{m}^{\epsilon }\\ \hline\end{array}$

  (2)

  where $(E,B)$ is the electro-magnetic field (which may itself depend on $f$ through Maxwell's equations), $m^{\epsilon }$ is a random process (which may describe absorption or creation of particles) and $Q$ is a collision operator. The dimensionless parameters $\epsilon ,\beta$ are related to the cyclotronic frequency and the mean free path. Limits $\epsilon \to 0$ and $\beta \to 0$ do not share the same character (the former is oscillatory and the latter is dissipative) and lead respectively to gyrokinetic and hydrodynamic models. The noise term $m^{\epsilon }$ is correlated in space and time. At the limit $\epsilon \to 0$, it converges formally to a white noise and stochastic PDEs are obtained.

MINGuS project is the follow-up of IPSO , ending in december in 2017. IPSO original aim was to extend the analysis of geometric schemes from ODEs to PDEs. During the last evaluation period, IPSO also considered the numerical analysis of geometric schemes for (S)PDEs, possibly including multiscale phenomena. Breakthrough results [36], [38], [39], [42] have been recently obtained which deserve to be deepened and extended. It thus appears quite natural to build the MINGuS team upon these foundations.

The objective of MINGuS is twofold: the construction and the analysis of numerical schemes (such as “Uniformly Accurate numerical schemes", introduced by members of the IPSO project) for multiscale (S)PDEs originating from physics. In turn, this requires $\left(i\right)$ a deep mathematical understanding of the (S)PDEs under consideration and $\left(ii\right)$ a strong involvement into increasingly realistic problems, possibly resorting to parallel computing. For this aspect, we intend to benefit from the Inria Selalib software library which turns out to be the ideal complement of our activities.