2023Activity reportProject-TeamMINGUS

2.1 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.

The objective of MINGuS is twofold: the construction and the analysis of numerical schemes 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.

During the last period, some results have been obtained by the members of the team 1, 2, 3, 4, 5.

3.1 Dissipative problems

In the dissipative context, the asymptotic analysis is quite well understood in the deterministic case and multiscale numerical methods have been developed in the last decades. Indeed, the so-called Asymptotic-Preserving schemes has retained a lot of attention, in particular in the context of collisional kinetic equations. But, there is still a lot to do if one is interested in deriving high order asymptotic models, which enable to capture the original solution for all time. Moreover, this analysis is still misunderstood when more complex systems are considered, involving non homogeneous relaxation rates or stochastic terms for instance. Following the methodology we aim at using, we first address the mathematical analysis before deriving multiscale efficient numerical methods.

A simple model of dissipative systems is governed by the following differential equation

for given initial condition $(x_0,y_0)\in {ℝ}^2$ and given smooth functions $𝒢,\mathscr{H}$ which possibly involve stochastic terms.

3.2 Highly-oscillatory problems

As a generic model for highly-oscillatory systems, we will consider the equation

for a given $u_0$ and a given periodic function $ℱ$ (of period $P$ w.r.t. its first variable) which possibly involves stochastic terms. Solution $u^{\epsilon }$ exhibits high-oscillations in time superimposed to a slow dynamics. Asymptotic techniques -resorting in the present context to averaging theory 47- allow to decompose

into a fast solution component, the $\epsilon P$-periodic change of variable ${\Phi }_{t/\epsilon }$, and a slow component, the flow ${\Psi }_t$ of a non-stiff averaged differential equation. Although equation (5) can be satisfied only up to a small remainder, various methods have been recently introduced in situations where (4) is posed in ${ℝ}^n$ or for the Schrödinger equation (1).

In the asymptotic behavior $\epsilon \to 0$, it can be advantageous to replace the original singularly perturbed model (for instance (1) or (2)) by an approximate model which does not contain stiffness any longer. Such reduced models can be derived using asymptotic analysis, namely averaging methods in the case of highly-oscillatory problems. In this project, we also plan to go beyond the mere derivation of limit models, by searching for better approximations of the original problem. This step is of mathematical interest per se but it also paves the way of the construction of multiscale numerical methods.

4.1 Application domains

The MINGuS project aims at applying the new numerical methods on realistic problems arising for instance in physics of nanotechnology and physics of plasmas. Therefore, in addition to efforts devoted to the design and the analysis of numerical methods, the inherent large size of the problems at hand requires advanced mathematical and computational methods which are hard to implement. Another application is concerned with population dynamics for which the main goal is to understand how the spatial propagation phenomena affects the demography of a population (plankton, parasite fungi, ...). Our activity is mostly at an early stage in the process of transfer to industry. However, all the models we use are physically relevant and all have applications in many areas (ITER, Bose-Einstein condensate, wave turbulence, optical tomography, transport phenomena, population dynamics, $\cdots$). As a consequence, our research aims at reaching theoretical physicists or computational scientists in various fields who have strong links with industrial applications. In order to tackle as realistic physical problems as possible, a fundamental aspect will consist in working on the realization of numerical methods and algorithms which are able to make an efficient use of a large number of processors. Then, it is essential for the numerical methods developed in the MINGuS project to be thought through this prism. We will benefit from the strong expertise of P. Navaro in scientific computing and more precisely on the Selalib software library (see description below). Below, we detail our main applications: first, the modeling and numerical approximation of magnetized plasmas is our major application and will require important efforts in terms of software developments to scale-up our multiscale methods; second, the transport of charged particles in nanostructures has very interesting applications (like graphene material), for which our contributions will mainly focus on dedicated problems; lastly, applications on population dynamics will be dedicated to mathematical modeling and some numerical validations.

4.2 Plasmas problems

The SeLaLib (Semi-Lagrangian Library) software library is a modular library for kinetic and gyrokinetic simulations of plasmas in fusion energy devices. Selalib is a collection of fortran modules aimed at facilitating the development of kinetic simulations, particularly in the study of turbulence in fusion plasmas. Selalib offers basic capabilities and modules to help parallelization (both MPI and OpenMP), as well as pre-packaged simulations. Its main objective is to develop a documented library implementing several numerical methods for the numerical approximation of kinetic models. Another objective of the library is to provide physicists with easy-to-use gyrokinetic solvers. It has been originally developed by E. Sonnendrücker and his collaborators in the past CALVI Inria project, and has played an important role in the activities of the IPL FRATRES. P. Navaro is one of the main software engineer of this library and as such he played an important daily role in its development and its portability on supercomputers. Though Selalib has reached a certain maturity, additional work is needed to make it available to the community. There are currently discussions for a possible evolution of Selalib, namely the writing of a new release which will be available for free download. At the scientific level, Selalib is of great interest for us since it provides a powerful tool with which we can test, validate and compare our new methods and algorithms (user level). Besides numerical algorithms the library provides low-level utilities, input-output modules as well as parallelization strategies dedicated to kinetic problems. Moreover, a collection of simulations for typical test cases (of increasing difficulties) with various discretization schemes supplements the library. This library turns out to be the ideal complement of our activities and it will help us to scale-up our numerical methods to high-dimensional kinetic problems. During the last years, several experiments have been successfully performed in this direction (especially with PhD students) and it is important for us that this approach remains thorough. Then, we intend to integrate several of the numerical methods developed by the team within the Selalib library, with the strong help of P. Navaro (contributor level). This work has important advantages: (i) it will improve our research codes (in terms of efficiency software maintenance); (ii) it will help us to promote our research by making our methods available to the research community.

4.3 Quantum problems

Nowadays, a great challenge consists in the downscaling at the nanometer scale of electronic components in order to improve speed and efficiency of semiconductor materials. In this task, modeling and numerical simulations play an important role in the determination of the limit size of the nanotransistors. At the nanoscale, quantum effects have to be considered and the Schrödinger equation is a prominent equation in this context. In the so-called semiclassical regime or when the transport is strongly confined, the solution endows space-time high oscillations which are very difficult to capture numerically. An important application is the modeling of charged particles transport in graphene. Graphene is a sheet of carbone made of a single layer of molecule, organized in a bidimensional honeycomb crystal. The transport of charged particles in this structure is usually performed by Dirac equation (which is the relativistic counterpart of the Schrödinger equation). Due to the unusual properties of graphene -at room temperature, electrons moving in graphene behave as massless relativistic particles- physicists and compagnies are nowadays actively studying this material. In this context, predicting how the material properties are affected by the uncertainties in the hexagonal lattice structure or in external potentials, is a major issue.

4.4 Population dynamics

The main goal is to characterize how spatial propagation phenomena (diffusion, transport, advection, $\cdots$) affect the time evolution of the demography of a population. In collaboration with Y. Lagadeuc (ECOBIO, Rennes), this question has been studied for plankton. In this context, mathematical models have been proposed and it has been shown that the spatial dynamic (in this context, due to the marine current) which is fast compared to demographic scales, can strongly modify the demographic evolution of the plankton. In collaboration with Ecole d'Agronomie de Rennes, a mathematical study on the demography of a parasite fungi of plants has been performed. In this context, the demography is specific: the fungi can proliferate through sexual reproduction or through parthenogenesis. These two ways of reproduction give rise mathematically to quadratic and linear growth rates with respect to the population variable. The demography is then coupled with transport (transport of fungi spore by wind). Here, the goal is to characterize the propagation of the fungi population by finding travelling waves solutions which are well adapted to describe the evolution of invasive fronts. Moreover, this approach enables to recover with a good agreement realistic examples (infection of ash or banana tree) for which experimental data are available. In these contexts, mathematical models are a powerful tool for biologists since measurements are very complicated to obtain and laboratory experiments hardly reproduce reality. The models derived are multiscale due to the nature of the underlying phenomena and the next step is to provide efficient numerical schemes.

5.1 Footprint of research activities

A group called ECO-IRMAR has been created in the IRMAR laboratory to inform about the footprint of research activities at the level of the laboratory. The members of the team follow the advices proposed by this group.

7.1 New software

8.1 Multiscale numerical schemes

Participants: G. Beck, N. Crouseilles, E. Faou, Y. Le Hénaff, D. Prel.

Multiscale (ie highly oscillatory or dissipative) ordinary differential equations (ODEs) have a long history since they are ubiquitous to describe dynamical multiscale physical phenomena in physics or chemistry. They can be obtained by appropriate spatial discretization of the partial differential equation or can directly describe the behavior of dynamical quantities. In addition to the standard difficulties coming from their numerical resolution, multiscale ODEs involve a stiffness (characterized by a parameter $\epsilon \in ]0,1]$). Creating strong gradients in the solution. Hence, to capture these small scales, conventional methods have to consider a time step smaller than $\epsilon$ leading to unacceptable computational cost. The team members proposed several strategies to overcome this stiffness.

In 18, high order asymptotic preserving schemes are constructed and analysed for kinetic equations under a diffusive scaling. The framework enables to consider different cases: the diffusion equation, the advection-diffusion equation and the presence of inflow boundary conditions. Starting from the micro-macro reformulation of the original kinetic equation, high order time integrators are introduced. This class of numerical schemes enjoys the Asymptotic Preserving (AP) property for arbitrary initial data and degenerates when $\epsilon$ goes to zero into a high order scheme which is implicit for the diffusion term, which makes it free from the usual diffusion stability condition. The space discretization is also discussed and high order methods are also proposed based on classical finite differences schemes. The Asymptotic Preserving property is analysed and numerical results are presented to illustrate the properties of the proposed schemes in different regimes.

In 21, we deal with wave propagation into a coaxial cable, which can be modeled by the 3D Maxwell equations or 1D simplified models. The usual one, called the telegrapher's model, is a 1D wave equation on the electrical voltage and current. We derived a more accurate model from the Maxwell equations that takes into account dispersive effects. These two models aim to be a good approximation of the 3D electromagnetic fields in the case where the thinness of the cable is small. We perform some numerical simulations of the 3D Maxwell equations and of the 1D simplified models in order to validate the usual model and the new one. Moreover, we show that while the usual telegrapher model is of order one with respect to the thinness of the cable, the dispersive 1D model is of order two.

The goal of this work 22 is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.

The aim of this paper 25 is to investigate a deterministic particle method for a model containing a Fokker-Planck collision operator in velocity and strong oscillations (characterized by a small parameter ε) induced by a space and velocity transport operator. First, we investigate the properties (collisional invariants and equilibrium) of the asymptotic model obtained when ε → 0. Second a numerical method is developed to approximate the solution of the multiscale Fokker-Planck model. To do so, a deterministic particle method (recently introduced for the Landau equation in [8]) is proposed for Fokker-Planck type operators. This particle method consists in reformulating the collision operator in an advective form and in regularizing the advection field in such a way that it conserves the geometric bracket structure. In the Fokker-Planck homogeneous case, the properties of the resulting method are analysed. In the non homogeneous case, the particle method is coupled with a uniformly accurate time discretization in ε that enables to capture numerically the solution of the asymptotic model. Numerous numerical results are displayed, illustrating the behavior of the method.

Based on recent ideas, stemming from the use of bubbles, we discuss in 29 an algorithm for the numerical simulation of the cubic nonlinear Schrödinger equation with harmonic potential in any dimension, which could be easily extended to other polynomial nonlinearities. For the linear part of the equation, the algorithm consists in discretizing the initial function as a sum of modulated complex functions, each one having its own set of parameters, and then updating the parameters exactly so that the modulated function remains a solution to the equation. When cubic interactions are introduced, the Dirac-Frenkel-MacLachlan principle is used to approximate the time evolution of parameters. We then obtain a grid-free algorithm in any dimension, and it is compared to a spectral method on numerical examples.

Phenotypic plasticity has important ecological and evolutionary consequences. In particular, behavioural phenotypic plasticity such as adaptive foraging (AF) by consumers, may enhance community stability. Yet little is known about the ecological conditions that favor the evolution of AF, and how the evolutionary dynamics of AF may modulate its effects on community stability. In order to address these questions, we constructed in 33 an eco-evolutionary model in which resource and consumer niche traits underwent evolutionary diversification. Consumers could either forage randomly, only as a function of resources abundance, or adaptatively, as a function of resource abundance, suitability and consumption by competitors. AF evolved when the niche breadth of consumers with respect to resource use was large enough and when the ecological conditions allowed substantial functional diversification. In turn, AF promoted further diversification of the niche traits in both guilds. This suggests that phenotypic plasticity can influence the evolutionary dynamics at the community-level. Faced with a sudden environmental change, AF promoted community stability directly and also indirectly through its effects on functional diversity. However, other disturbances such as persistent environmental change and increases in mortality, caused the evolutionary regression of the AF behaviour, due to its costs. The causal relationships between AF, community stability and diversity are therefore intricate, and their outcome depends on the nature of the environmental disturbance, in contrast to simpler models claiming a direct positive relationship between AF and stability.

In 10, we study the construction of multiscale numerical schemes efficient in the finite Larmor radius approximation of the collisional Vlasov equation. Following the paper of Bostan and Finot (2019), the system involves two different regimes, a highly oscillatory and a dissipative regimes, whose asymptotic limits do not commute. In this work, we consider a Particle-In-Cell discretization of the collisional Vlasov system which enables to deal with the multiscale characteristics equations. Different multiscale time integrators are then constructed and analysed. We prove asymptotic properties of these schemes in the highly oscillatory regime and in the collisional regime. In particular, the asymptotic preserving property towards the modified equilibrium of the averaged collision operator is recovered. Numerical experiments are then shown to illustrate the properties of the numerical schemes.

8.2 Geometric numerical schemes

Participants: I. Almuslimani, N. Crouseilles, X. Hong, A. Laurent.

The MINGuS team has a long history in the design and study of numerical schemes for Hamiltonian PDEs. The main examples are Schroedinger or Vlasov equations.

In 6, a novel second order family of explicit stabilized Runge-Kutta-Chebyshev methods for advection-diffusion-reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. An adaptive algorithm to implement the new scheme is proposed. This algorithm is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new algorithm are presented.

In 11, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov-Fokker-Planck system coupled with Poisson or Ampère equation. Splitting methods are used where the linear terms in space are treated by spectral or semi-Lagrangian methods and the nonlinear diffusion in velocity in the collision operator is treated using a stabilized Runge–Kutta–Chebyshev (RKC) integrator, a powerful alternative of implicit schemes. The new schemes are shown to exactly preserve mass and momentum. The conservation of total energy is obtained using a suitable approximation of the electric field. An $H$-theorem is proved in the semi-discrete case, while the entropy decay is illustrated numerically for the fully discretized problem. Numerical experiments that include investigation of Landau damping phenomenon and bump-on-tail instability are performed to illustrate the efficiency of the new schemes.

Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work 15, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge-Kutta method cannot preserve volume.

The aromatic bicomplex is an algebraic tool based on aromatic Butcher trees and used in particular for the explicit description of volume-preserving affine-equivariant numerical integrators. The present work 16 defines new tools inspired from variational calculus such as the Lie derivative, different concepts of symmetries, and Noether's theory in the context of aromatic forests. The approach allows to draw a correspondence between aromatic volumepreserving methods and symmetries on the Euler-Lagrange complex, to write Noether's theorem in the aromatic context, and to describe the aromatic B-series of volume-preserving methods explicitly with the Lie derivative.

Exotic aromatic B-series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in ${ℝ}^d$ and on manifolds. We prove in 32 that exotic aromatic B-series satisfy a universal geometric property, namely that they are characterised by locality and orthogonal-equivariance. This characterisation confirms that exotic aromatic B-series are a fundamental geometric object that naturally generalises aromatic B-series and B-series, as they share similar equivariance properties. In addition, we classify with stronger equivariance properties the main subsets of the exotic aromatic B-series, in particular the exotic B-series. Along the analysis, we present a generalised definition of exotic aromatic trees, dual vector fields, and we explore the impact of degeneracies on the classification.

In 31, we propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for linear hyperbolic system. The method is designed based on the EL DG method for transport problems [J. Comput. Phy. 446: 110632, 2021], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability with respect to the classical RK DG method. Considering each characteristic family, a straightforward application of EL DG for hyperbolic system will be to transform to the characteristic variables, evolve them on associated characteristic related space-time regions, and transform them back to the original variables. However, the conservation could not be guaranteed in a general setting. In this paper, we formulate a conservative semi-discrete EL DG method by decomposing each variable into two parts, each of them associated with a different characteristic family. As a result, four different quantities are evolved in EL fashion and recombined to update the solution. The fully discrete scheme is formulated by using method-of-lines RK methods, with intermediate RK solutions updated on the background mesh. Numerical results for 1D and 2D wave equations are presented to demonstrate the performance of the proposed ELDG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and conservative property.

In 17, we study a grid-free particle method based on following the evolution of the characteristics of the Vlasov-Poisson system, and we show that it converges for smooth enough initial data. This method is built as a combination of well-studied building blocks-mainly time integration and integral quadratures-, hence allows to obtain arbitrarily high orders. By making use of the Non-Uniform Fast Fourier Transform (NUFFT), the overall computational complexity is $𝒪(P+K^{d}log{K}^d)$, where $P$ is the total number of particles and where we only keep the Fourier modes $k\in {\mathbb{Z}}^d$ such that $k_1^2+\cdots +k_d^2\le K^2$. Some numerical results are given for the Vlasov-Poisson system in the one-dimensional case.

In 20, a two-species Vlasov-Poisson model is described together with some numerical simulations, permitting to exhibit the formation of a plasma sheath. The numerical simulations are performed with two different methods: a first order classical finite difference (FD) scheme and a high order semi-Lagrangian (SL) scheme with Strang splitting; for the latter one, the implementation of (non-periodic) boundary conditions is discussed. The codes are first evaluated on a one-species case, where an analytical solution is known. For the two-species case, cross comparisons and the influence of the numerical parameters for the SL method are performed in order to have an idea of a reference numerical simulation.

In 12, we present a numerical method to solve the Vlasov-Maxwell equations for spin-1/2 particles, in a semiclassical approximation where the orbital motion is treated classically while the spin variable is fully quantum. Unlike the spinless case, the phase-space distribution function is a $2\times 2$ matrix, which can also be represented, in the Pauli basis, as one scalar function $f_0\in ℝ$ and one three-component vector function $\overrightarrow{f}\in {ℝ}^3$. The relationship between this "vectorial" representation and the fully scalar representation on an extended phase space first proposed by Brodin et al. [Phys. Rev. Lett. 101, 245002 (2008)] is analyzed in detail. By means of suitable approximations and symmetries, the vectorial spin-Vlasov-Maxwell model can be reduced to two-dimensions in the phase space, which is amenable to numerical solutions using a high-order grid-based Eulerian method. The vectorial model enjoys a Poisson structure that paves the way to accurate Hamiltonian split-time integrators. As an example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, and compare the results to those obtained earlier with the scalar spin-Vlasov-Maxwell model and a particle-in-cell code.

In 13, an exponential Discontinuous Galerkin (DG) method is proposed to solve numerically Vlasov type equations. The DG method is used for space discretization which is combined exponential Lawson Runge-Kutta method for time discretization to get high order accuracy in time and space. In addition to get high order accuracy in time, the use of Lawson methods enables to overcome the stringent condition on the time step induced by the linear part of the system. Moreover, it can be proved that a discrete Poisson equation is preserved. Numerical results on Vlasov-Poisson and Vlasov Maxwell equations are presented to illustrate the good behavior of the exponential DG method.

8.3 Analysis of PDEs and SPDEs

Participants: G. Beck, F. Castella, A. Debussche, E. Faou.

In view of the construction of efficient multiscale numerical schemes, the study and analysis of PDEs or SPDEs is of great importance. Below is a list of some contributions of the team on this aspect.

In 8, we study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajectories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier-Stokes-Fourier system with an external force from the Boltzmann equation. Neither smallness, nor time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cutoff Boltzmann kernels. As a by-product the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.

In 7, we characterize geometrically the regularizing effects of the semigroups generated by accretive non-selfadjoint quadratic differential operators. As a byproduct, we establish the subelliptic estimates enjoyed by these operators, being expected to be optimal. These results prove conjectures by M. Hitrik, K. Pravda-Starov and J. Viola. The proof relies on a new representation of the polar decomposition of these semigroups. In particular, we identify the selfadjoint part as the evolution operator generated by the Weyl quantization of a time-dependent real-valued nonnegative quadratic form for which we prove a sharp anisotropic lower bound.

In 9, we analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This protocol relies on successive pull operations involving $k$ different nodes, with $k\ge 2$, and called $k$-pull operations. Specifically during a $k$-pull operation, an uninformed node a contacts $k-1$ other nodes at random in the network, and if at least one of them knows the rumor, then node a learns it. We perform a detailed study in continuous-time of the total time ${\Theta }_{k,n}$ needed for all the $n$ nodes to learn the rumor. These results extend those obtained in a previous paper which dealt with the discrete-time case. We obtain the mean value, the variance and the distribution of ${\Theta }_{k,n}$ together with their asymptotic behavior when the number of nodes $n$ tends to infinity.

In 14, we prove that the stochastic Nonlinear Schrödinger (NLS) equation is the limit of NLS equation with random potential with vanishing correlation length. We generalize the perturbed test function method to the context of dispersive equations. Apart from the difficulty of working in infinite dimension, we treat the case of random perturbations which are not assumed uniformly bounded.

In 23, we consider the moments and the distribution of the hitting and cover times of a random walk in the complete graph. We study both the time needed to reach any subset of states and the time needed to visit all the states of a subset at least once. We obtain recurrence relations for the moments of all orders and we use these relations to analyze the asymptotic behavior of the hitting and cover times distributions when the number of states tends to infinity.

In 24, we consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction. Subsequently, we study the validity of the so-called Smoluchowski-Kramers diffusion approximation. We show that, under more stringent conditions on the friction, in the small-mass limit the solution of the system of stochastic damped wave equations converges to the solution of a system of stochastic quasi-linear parabolic equations. In this convergence, an additional drift emerges as a result of the interaction between the noise and the state-dependent friction. The identification of this limit is achieved by using a suitable generalization of the classical method of perturbed test functions, tailored to the current infinite dimensional setting.

In 26, we are concerned with a slow-fast system of coupled three dimensional Navier-Stokes equations where the fast component is perturbed by an additive Brownian noise. By means of the rough path theory we establish the convergence in law of the slow component towards a Navier-Stokes system with an Itô-Stokes drift and a rough path driven transport noise. This gives an alternative, more general and direct proof to Debussche, Pappalettera, 2022. In particular, the limit driving rough path is identified as a geometric rough path which does not necessarily coincide with the Stratonovich lift of the Brownian motion.

In 27, we consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (2019) with a sub-quadratic nonlinearity.

In 28, we study slow-fast systems of coupled equations from fluid dynamics, where the fast component is perturbed by additive noise. We prove that, under a suitable limit of infinite separation of scales, the slow component of the system converges in law to a solution of the initial equation perturbed with transport noise, and subject to the influence of an additional Itô-Stokes drift. The obtained limit equation is very similar to turbulent models derived heuristically. Our results apply to the Navier-Stokes equations in dimension $d=2,3$ ; the Surface Quasi-Geostrophic equations in dimension $d=2$ ; and the Primitive equations in dimension $d=2,3$.

In 30, we start from the remark that in wave turbulence theory, exemplified by the cubic twodimensional Schrödinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converge towards the kinetic operator but with different form of convergence.

In 34, we consider the moments and the distribution of hitting times on the lollipop graph which is the graph exhibiting the maximum expected hitting time among all the graphs having the same number of nodes. We obtain recurrence relations for the moments of all order and we use these relations to analyze the asymptotic behavior of the hitting time distribution when the number of nodes tends to infinity.

Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. In hydrodynamic turbulence, when the Reynolds number is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation remains bounded from below even if the fluid viscosity tends to zero. A mathematical description of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov in hydrodynamic turbulence and that of Zakharov in wave turbulence. In both cases, these turbulent cascade mechanisms imply power-law behaviors of several statistical quantities such as power spectral densities. For a long time, these cascades were believed to be associated with nonlinear interactions, but recent works have shown that they can also take place in a dynamics governed by a linear equation with a differential operator of degree 0. In this spirit, we construct in 19 a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a linear transport equation, which corresponds to an operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time.

9.1 Bilateral contracts with industry

Participants: E. Faou.

• Contract with the Cailabs compagny.

  A long standing collaboration has emerged between MINGuS and the company CAILABS whose main aim is the conception and construction of optical fibers. Most of the main objectives of this collaboration are strictly confidential. However they have strong common point with the scientific goals of the MINGuS project, for instance the development of efficient numerical methods for quantum simulation and many aspects of mathematical and physical analysis of quantum systems. The impact of this collaboration are very important both from the transfer of technological points of view and from the interaction with a very active startup providing very practical problems that are often very close to hot academic topics. We believe that this interaction will last long and continue to feed the scientific activity of the whole project with problem directly coming from the industrial and economical worlds.

• Agence Lebesgue.

  Participants: E. Faou.

  Since 2019 and up to August 2023, E. Faou was head of the Agence Lebesgue pour les mathématiques whose role is precisely to increase the role of mathematics in the socio-economic world by facilitating contact between mathematicians and companies or institute working in distant sector of activity.

  An important activity of the Agence Lebesgue is formation where mathematicians go to industries, companies of the private sector or other institutes to organize some crash course in some hot topics in mathematics, or on demand depending on the requirement of the partners.

• CIFRE Phd with Naval Group: Tom Laborde defended his PhD in december 2023. The advisors were members of the team but they left in 2021.

10.1 International initiatives

10.2 International research visitors

10.3 International initiatives

E. Faou is PI of the Simons collaboration on wave turbulence gathering mathematicians and physicists from New-York University, ENS Paris, ENS Lyon and Torino.

10.4 National initiatives

• 2018-2023: participation IPL SURF headed by A. Vidard (Airsea team).

  Participants: Arnaud Debussche, Erwan Faou.

  This project aims at the modelling and simulation of coastal and littoral ocean circulation problems, including quantification. This project involves 7 Inria teams and Ifremer, BRGM and SHOM.

• 2023: project funded by Fédération de Recherche Fusion par Confinement Magnétique, headed by N. Crouseilles. 5000 euros.

  Participants: Nicolas Crouseilles.

  This project is focused on the design of numerical schemes for tokamak plasmas and involve members of the team but also colleagues from university of Nantes.

• 2019-2023: A. Debussche is the local coordinator of the ANR project ADA, headed by J. Vovelle (ENS Lyon). 160000 euros

  Participants: Arnaud Debussche.

  This project focuses on multiscale models which are both infinite-dimensional and stochastic with a theoretic and computational approach. The project involved a group in Lyon and MINGuS members.

• 2019-2024 GdR TRAG on rough path theory.

  Participants: Arnaud Debussche.

  The goal of the TRAG GDR is to gather french mathematicians who work on the rough path theory. GDR TRAG.

• 2023-2027: E. Faou is the PI of the ANR project KEN (Kinetic, PDE and Numerics). The partners are R. Krikorian (Ecole Polytechnique) and B. Grébert (Univeristy Nantes). Budget total 391000 euros

  Participants: Nicolas Crouseilles, Erwan Faou.

  The project involved a group in Nantes, Ecole Polytechnique and some MINGuS members. It gathers people from the Dynamical system community, specialists of the analysis of Partial Differential Equations, as well as people coming from the numerical analysis and scientific computing worlds.

• 2023-2027: G. Beck is a member of the ANR project BOURGEONS, headed by A.-L. Dalibard (Sorbonne university).

  Participants: Geoffrey Beck.

  This project focuses on some theoretical challenges and environmental applications on boundary, congestion and vorticity in fluids.

10.5 Regional initiatives

• X. Hong was granted by the Région Bretagne council. '

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

• G. Beck gave a talk at "5min Lebesgue": Turbulence I : les cascades de Kolmogorov comme théorie du ruissellement des "échelles".
• N. Crouseilles gave a conference for the master students of the university of Nantes (October 2023).
• N. Crouseilles collaborated with T. Menuet to create an enigma combining mathematics and music.
• Y. Le Hénaff participated to the event on the interaction between music and mathematic proposed by T. Menuet.
• Y. Le Hénaff participated to C2+.
• A. Laurent participated to the event "Rennes en Science" (November 2023).
• P. Navaro organized the following workshops at IRMAR: Julia (for statistics, optimization, create packages), git, PLM.
• D. Prel participated to Science Dating pour Nuit Blanche des Chercheurs.
• D. Prel participated to "Fête de la science" at Nantes.

12.1 Major publications

• 1 articleF.Fernando Casas, N.Nicolas Crouseilles, E.Erwan Faou and M.Michel Mehrenberger. High-order Hamiltonian splitting for Vlasov-Poisson equations.Numerische Mathematik13532017, 769-801HALDOIback to text
• 2 articleP.Philippe Chartier, N.Nicolas Crouseilles, M.Mohammed Lemou, F.Florian Méhats and X.Xiaofei Zhao. Uniformly accurate methods for three dimensional Vlasov equations under strong magnetic field with varying direction.SIAM Journal on Scientific Computing4222020, B520--B547HALDOIback to text
• 3 articleN.Nicolas Crouseilles, P.-A.Paul-Antoine Hervieux, Y.Yingzhe Li, G.Giovanni Manfredi and Y.Yajuan Sun. Geometric Particle-in-Cell methods for the Vlasov-Maxwell equations with spin effects.Journal of Plasma Physics873May 2021, article n° 825870301HALDOIback to text
• 4 articleA.Arnaud Debussche and J.Julien Vovelle. Diffusion-approximation in stochastically forced kinetic equations.Tunisian Journal of Mathematics312021, 1-53HALDOIback to text
• 5 articleE.Erwan Faou. Linearized wave turbulence convergence results for three-wave systems.Communications in Mathematical Physics3782September 2020, 807–849HALDOIback to text

12.2 Publications of the year

12.3 Cited publications

• 35 bookC.C. Birdsall and A.A. Langdon. Plasmas physics via computer simulations. New York Taylor and Francis2005back to text
• 36 articleA.A. Brizard and T.T. Hahm. Foundations of nonlinear gyrokinetic theory.Reviews of Modern Physics792007back to text
• 37 articleJ.J. Carr. Applications of Centre Manifold Theory. Applied Mathematical Sciences Series351981back to text
• 38 articleP.P. Chartier, N.N. Crouseilles, M.M. Lemou and F.F. Méhats. Uniformly accurate numerical schemes for highly-oscillatory Klein-Gordon and nonlinear Schrödinger equations. Numer. Math.1292015, 513--536back to textback to text
• 39 articleP.P. Chartier, A.A. Murua and J.J. Sanz-Serna. Higher-order averaging, formal series and numerical integration III: error bounds. Foundation of Comput. Math.152015, 591--612back to text
• 40 articleA.A. Debussche and J.J. Vovelle. Diffusion limit for a stochastic kinetic problem. Commun. Pure Appl. Anal.112012, 2305--2326back to text
• 41 articleE.E. Faou and F.F. Rousset. Landau damping in Sobolev spaces for the Vlasov-HMF model. Arch. Ration. Mech. Anal.2192016, 887--902back to text
• 42 bookE.Ernst Hairer, C.Christian Lubich and G.G. Wanner. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition.Springer Series in Computational Mathematics 31BerlinSpringer2006back to textback to text
• 43 articleS.S. Jin and H.H. Lu. An Asymptotic-Preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings. J. Comp. Phys.3342017, 182--206back to text
• 44 articleM.M. Lemou, F.F. Méhats and P.P. Raphaël. Orbital stability of spherical galactic models.Invent. Math.1872012, 145--194back to text
• 45 articleC.C. Mouhot and C.C. Villani. On Landau damping.Acta Math.2072011, 29--201back to text
• 46 bookS.S. Nazarenko. Wave turbulence. Springer-Verlag2011back to text
• 47 articleL.L. Perko. Higher order averaging and related methods for perturbed periodic and quasi-periodic systems.SIAM J. Appl. Math.171969, 698--724back to textback to text