2025Activity‌ 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

  $$\begin{array}{|c|}\hline i\epsilon {\mathrm{\partial }}_t{\psi }^{\epsilon }=‌\frac{{\epsilon }^2}{\beta }\mathrm{\Delta ‌}{\psi }^{\epsilon }+{|‌‌{\psi }^{\epsilon }|}^{\mathrm{2}}{\psi }^{\epsilon }+{\mathrm{\psi ‌}}^{\epsilon }\xi \\ \hline\end{array}$$ 1

  where‌ the function ${\psi }^{\mathrm{\epsilon ‌‌}}={\psi }^{\epsilon }(t,x)‌\in ℂ$ depends on‌ time $t\ge \mathrm{0‌‌}$ and position $x\in {ℝ}^3$, $\mathrm{\xi ‌}=\xi (\mathrm{x‌},t)$ is‌​‌ a white noise and​​ where the small parameter​​​‌ $\epsilon$ is the Planck's‌ constant describing the microscopic/macroscopic‌​‌ ratio. The limit $\mathrm{\epsilon }\to 0$ is referred​​​‌ to as the semi-classical‌ limit. The regime $\mathrm{\epsilon ‌‌}=1$ and $\mathrm{\beta }\to 0$ (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 ${\mathrm{f‌}}^{\epsilon }=f^{\mathrm{\epsilon }}(t,\mathrm{x‌},v)\in {ℝ}^{+}$ the distribution​‌ function of charged particles​​ at time $t\ge ‌0$, position $\mathrm{x}\in {ℝ}^3$ and​‌ velocity $v\in {\mathrm{ℝ}}^3$, a typical​​​‌ kinetic equation for ${\mathrm{f}}^{\epsilon }$ reads

  $$\begin{array}{|c|}\hline {\partial }_{\mathrm{t‌}}{f}^{\epsilon }+\mathrm{v}·{\nabla }_x{\mathrm{f‌}}^{\epsilon }+\left(E+\frac{1}{\epsilon }(\mathrm{v‌}\times B)\right)·{\nabla }_v{f}^{\mathrm{\epsilon ‌}}=\frac{1}{\beta }\mathrm{Q}(f^{\epsilon })‌+f^{\epsilon }{\mathrm{m}}^{\epsilon }\\ \hline\end{array}$$ 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 $\mathrm{\epsilon ‌}\to 0$ and $\mathrm{\beta }\to 0$ 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 $\mathrm{\epsilon }\to 0$, 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 $(i)$ a deep​​​‌ mathematical understanding of the​ (S)PDEs under consideration and​‌ $(ii)$ 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 2​​​‌, 3, 4​, 5, 6​‌.

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 $\mathrm{P}$ w.r.t. its first variable)​​​‌ which possibly involves stochastic​ terms. Solution $u^{\mathrm{\epsilon ‌}}$ exhibits high-oscillations in time​​ superimposed to a slow​​​‌ dynamics. Asymptotic techniques -resorting​ in the present context​‌ to averaging theory 48​​- allow to decompose​​​‌

into a fast solution​​ component, the $\epsilon \mathrm{P‌}$-periodic change of variable​ ${\Phi }_{t/\mathrm{\epsilon ‌}}$, and a slow​​ component, the flow ${\mathrm{\Psi ‌}}_t$ 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.​​

6.1 Awards

No​ highlight this year.

7.1 Latest​​​‌ software developments

7.2 Open‌ data

8.1 Multiscale‌​‌ numerical schemes

Participants: M.​​ Badsi, A. Busnot​​​‌ Laurent, N. Crouseilles‌, A. Debussche.‌​‌

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 $\epsilon \in ‌]0,\mathrm{1‌}]$).

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.​​

This work 7 deals​​​‌ with the numerical approximation‌ of plasmas which are‌​‌ confined by the effect​​ of a fast oscillating​​​‌ magnetic field in the‌ Vlasov model. The presence‌​‌ of this magnetic field​​ induces oscillations (in time)​​​‌ to the solution of‌ the characteristic equations. Due‌​‌ to its multiscale character,​​ a standard time discretization​​​‌ would lead to an‌ inefficient solver. In this‌​‌ work, time integrators are​​ derived and analyzed for​​​‌ a class of highly‌ oscillatory differential systems. We‌​‌ prove the uniform accuracy​​ property of these time​​​‌ integrators, meaning that the‌ accuracy does not depend‌​‌ on the small parameter​​ ε. Moreover, we construct​​​‌ an extension of the‌ scheme which degenerates towards‌​‌ an energy preserving numerical​​ scheme for the averaged​​​‌ model, when $\epsilon \to ‌0$. Several numerical‌​‌ results illustrate the capabilities​​ of the method.

In​​​‌ 18, we propose‌ and study a fully‌​‌ implicit finite volume scheme​​ for the pressureless Euler-Poisson-Boltzmann​​​‌ equations on the one‌ dimensional torus. Especially, we‌​‌ design a consistent and​​ dissipative discretization of the​​​‌ force term which yields‌ an unconditional energy decay.‌​‌ In addition, we establish​​​‌ a discrete analogue of​ the modulated energy estimate​‌ around constant states with​​ a small velocity. Numerical​​​‌ experiments are carried to​ illustrate our theoretical results​‌ and to assess the​​ accuracy of our scheme.​​​‌ A test case of​ the literature is also​‌ illustrated.

In 8,​​ we construct and study​​​‌ a mean-field model that​ describes the nonlinear dynamics​‌ of a spin-polarized electron​​ gas interacting with fixed,​​​‌ positively-charged ions possessing a​ magnetic moment that evolves​‌ in time. The mobile​​ electrons are modeled by​​​‌ a four-component distribution function​ in the two-dimensional phase​‌ space $(x,v)$, obeying​​​‌ a Vlasov-Poisson set of​ equations. The ions are​‌ modeled by a Landau-Lifshitz​​ equation for their spin​​​‌ density, which contains ion-ion​ and electron-ion magnetic exchange​‌ terms. We perform a​​ linear response study of​​​‌ the coupled Vlasov-Poisson-Landau-Lifshitz (VPLL)​ equations for the case​‌ of a Maxwell-Boltzmann equilibrium,​​ focussing in particular on​​​‌ the spin dispersion relation.​ Condition of stability or​‌ instability for the spin​​ modes are identified, which​​​‌ essentially depend on the​ electron spin polarization rate​‌ η and the electron-ion​​ magnetic coupling constant K.​​​‌ We also develop an​ Eulerian grid-based computational code​‌ for the fully nonlinear​​ VPLL equations, based on​​​‌ the geometric Hamiltonian method​ recently developed previously. This​‌ technique allows us to​​ achieve great accuracy for​​​‌ the conserved quantities, such​ as the modulus of​‌ the ion spin vector​​ and the total energy.​​​‌ Numerical tests in the​ linear regime are in​‌ accordance with the estimations​​ of the linear response​​​‌ theory. For two-stream equilibria,​ we study the interplay​‌ of instabilities occurring in​​ both the charge and​​​‌ the spin sectors. The​ set of parameters used​‌ in the simulations, with​​ densities close to those​​​‌ of solids ($\approx {10}^{29}{m}^{-‌3}$) and temperatures​​ of the order of​​​‌ 10 eV, may be​ relevant to the warm​‌ dense matter regime appearing​​ in some inertial fusion​​​‌ experiments.

In 19,​ we study the interplay​‌ between spin waves (magnons)​​ and plasma waves (plasmons)​​​‌ in a ferromagnetic material,​ using an augmented Vlasov-Poisson​‌ model that includes the​​ electron spin dynamics. The​​​‌ ions are fixed, but​ their spins can evolve​‌ in time on the​​ unit sphere according to​​​‌ the Landau-Lifshitz equation, which​ includes nearest-neighbor magnetic interactions.​‌ The two components interact​​ not only through the​​​‌ electrostatic Coulomb force, but​ also via magnetic-exchange interaction​‌ terms. The linear response​​ analysis reveals the existence​​​‌ of a wave-particle resonance​ occurring at the frequency​‌ of the magnons. This​​ resonance gives rise to​​​‌ significant energy exchanges between​ the magnons and the​‌ electrons, resulting in a​​ rapid loss of the​​​‌ localized magnetism akin to​ the ultrafast demagnetisation observed​‌ in experiments on thin​​ ferromagnetic films. Depending on​​​‌ the initial electronic spin​ polarization, the resonance can​‌ lead to either damping​​ or instability of the​​​‌ wave. These results show​ that wave-particle effects, similar​‌ to those frequently encountered​​ in plasmas physics, may​​​‌ play a key role​ in spin-polarized plasmas and​‌ electron beams.

In 9​​, we study the​​ convergence of a Zakharov​​​‌ system driven by a‌ time white noise, colored‌​‌ in space, to a​​ multiplicative stochastic nonlinear Schrödinger​​​‌ equation, as the ion-sound‌ speed tends to infinity.‌​‌ In the absence of​​ noise, the conservation of​​​‌ energy gives bounds on‌ the solutions, but this‌​‌ evolution becomes singular in​​ the presence of the​​​‌ noise. To overcome this‌ difficulty, we show that‌​‌ the problem may be​​ recasted in the diffusion-approximation​​​‌ framework, and make use‌ of the perturbed test-function‌​‌ method. We also obtain​​ convergence in probability. The​​​‌ result is limited to‌ dimension one, to avoid‌​‌ too much technicalities. As​​ a prerequisite, we prove​​​‌ the existence and uniqueness‌ of regular solutions of‌​‌ the stochastic Zakharov system.​​

The unified gas kinetic​​​‌ scheme (UGKS) was initially‌ designed to address multiscale‌​‌ challenges in rarefied gas​​ dynamics and then extended​​​‌ to radiative transfert theory,‌ as described by BGK‌​‌ like relaxation models. In​​ this work 33,​​​‌ we extend its application‌ to linear kinetic models‌​‌ with non isotropic scattering​​ collision operators, as well​​​‌ as Fokker-Planck models .‌ These problems typically exhibit‌​‌ a fully diffusive nature​​ in the optically thick​​​‌ limit (corresponding to a‌ small Knudsen number). It‌​‌ still leads to an​​ asymptotic preserving (AP) property​​​‌ not only in this‌ diffusive regime but also‌​‌ in the free transport​​ limit. A series of​​​‌ numerical experiments confirm the‌ effectiveness of the approach.‌​‌

8.2 Geometric numerical schemes​​

Participants: G. Beck,​​​‌ A. Busnot Laurent,‌ N. Crouseilles, E.‌​‌ Faou, L. Martaud​​.

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.

The‌​‌ goal of this work​​ 10 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.

This work​​ 22 concerns the numerical​​​‌ approximations of the boundary​ conditions for linear hyperbolic​‌ systems. Discrete boundary procedure​​ computations and an artificial​​​‌ viscosity are proposed to​ corrected any three-points finite​‌ volume schemes. The fully​​ discrete stability of the​​​‌ resulting numerical schemes is​ established. Numerical test cases​‌ are performed to illustrate​​ the relevancy of the​​​‌ proposed procedure.

This paper​ 23 presents a simple​‌ and robust method to​​ enforce discrete entropy stability​​​‌ in firstorder well-balanced finite​ volume schemes for systems​‌ of balance laws, including​​ non-conservative terms. By leveraging​​​‌ an artificial viscosity technique​ originally introduced by Tadmor​‌ and extended in recent​​ works, the authors propose​​​‌ an entropy-preserving modification applicable​ to a wide class​‌ of three-point schemes. The​​ core contribution lies in​​​‌ the construction of a​ viscosity coefficient that maintains​‌ the well-balanced property while​​ ensuring entropy dissipation at​​​‌ the discrete level. A​ theoretical framework is developed​‌ to guarantee robustness, well-balancing,​​ and entropy stability under​​​‌ a suitable CFL condition.​ The approach is validated​‌ through numerical experiments, including​​ applications to shallow water​​​‌ systems and two-layer flows,​ demonstrating accuracy and stability.​‌

This work 24 concerns​​ the numerical approximations of​​​‌ the weak solutions of​ scalar hyperbolic conservation laws.​‌ After showing how to​​ bypass the barrier theorems​​​‌ for the linear advection,​ the derivation of a​‌ second-order entropy-satisfying scheme is​​ presented for non-linear equations.​​​‌ The fully discrete stability​ result is established for​‌ regular strictly convex entropy​​ and under a parabolic​​​‌ CFL-like condition. Some numerical​ experiments are done to​‌ assess the accuracy and​​ the stability of the​​​‌ proposed scheme.

For a​ class of ergodic parabolic​‌ semilinear stochastic partial differential​​ equations (SPDEs) with gradient​​​‌ structure, we introduce in​ 25 a preconditioning technique​‌ and design high-order integrators​​ for the approximation of​​​‌ the invariant distribution. The​ preconditioning yields improved temporal​‌ regularity of the dynamics​​ while preserving the invariant​​​‌ distribution and allows the​ application of postprocessed integrators.​‌ For the semilinear heat​​ equation driven by space-time​​​‌ white noise in dimension​ 1, we obtain new​‌ temporal integrators with orders​​ 1 and 2 for​​​‌ sampling the invariant distribution​ with a minor overcost​‌ compared to the standard​​ semilinear implicit Euler method​​​‌ of order $1/2$. Numerical experiments​‌ confirm the theoretical findings​​ and illustrate the efficiency​​​‌ of the approach.

In​ 26, we present​‌ a new class of​​ numerical methods for solving​​​‌ stochastic differential equations with​ additive noise on general​‌ Riemannian manifolds with high​​ weak order of accuracy.​​​‌ In opposition to the​ popular approach with projection​‌ methods, the proposed methods​​ are intrinsic: they only​​​‌ rely on geometric operations​ and avoid coordinates and​‌ embeddings. We provide a​​ robust and general convergence​​​‌ analysis and an algebraic​ formalism of exotic planar​‌ Butcher series for the​​ computation of order conditions​​ at any high order.​​​‌ To illustrate the methodology,‌ an explicit method of‌​‌ second weak order is​​ introduced, and several numerical​​​‌ experiments confirm the theoretical‌ findings and extend the‌​‌ approach for the sampling​​ of the invariant measure​​​‌ of Riemannian Langevin dynamics.‌

In 28, we‌​‌ introduce the notion of​​ post-Hopf algebroids, generalizing the​​​‌ pre-Hopf algebroids introduced recently‌ in the study of‌​‌ exotic aromatic S-series. We​​ construct action post-Hopf algebroids​​​‌ through actions of post-Hopf‌ algebras. We show that‌​‌ the universal enveloping algebra​​ of a post-Lie-Rinehart algebra​​​‌ (post-Lie algebroid) is naturally‌ a post-Hopf algebroid. As‌​‌ a byproduct, we construct​​ the free post-Lie-Rinehart algebra​​​‌ using a magma algebra‌ with a linear map‌​‌ to the derivation Lie​​ algebra of a commutative​​​‌ associative algebra. Applications in‌ geometric numerical integration on‌​‌ manifolds are given.

In​​ 27, we exhibit​​​‌ a new pre-Lie algebra‌ in the framework of‌​‌ symplectic groupoids and, in​​ turn, introduce a pre-Lie​​​‌ formalism of Butcher trees‌ for the approximation of‌​‌ Hamilton-Jacobi solutions on any​​ symplectic groupoid $G\widetilde{\mathrm{N‌}}M$. The‌ impact of this new‌​‌ algebraic approach is twofold.​​ On the geometric side,​​​‌ it yields algebraic operations‌ to approximate Lagrangian bisections‌​‌ of $G$ using the​​ Butcher-Connes-Kreimer Hopf algebra and,​​​‌ in turn, aims at‌ a better understanding of‌​‌ the group of Hamiltonian​​ diffeomorphisms of $M$.​​​‌ On the computational side,‌ we define a new‌​‌ class of Poisson integrators​​ for Hamiltonian dynamics on​​​‌ Poisson manifolds.

In 32‌, a new numerical‌​‌ method to approximate the​​ solution of the Vlasov-Maxwell​​​‌ equations is presented. The‌ method uses a phase‌​‌ space discretization and its​​ main properties are: energy​​​‌ and charge conservation thanks‌ to a semi-implicit treatment‌​‌ of the Maxwell equations,​​ but allowing for an​​​‌ explicit and efficient update‌ of the unknown. One‌​‌ of the main ingredients​​ lies in the introduction​​​‌ of an auxiliary scalar‌ variable inspired from the‌​‌ Scalar Auxiliary Variable (SAV)​​ approach together with a​​​‌ suitable splitting inspired from‌ previous works which enables‌​‌ the use of a​​ semi-Lagrangian method.

The use​​​‌ of symplectic numerical schemes‌ on Hamiltonian systems is‌​‌ widely known to lead​​ to favorable long-time behaviour.​​​‌ While this phenomenon is‌ thoroughly understood in the‌​‌ context of finite-dimensional Hamiltonian​​ systems, much less is​​​‌ known in the context‌ of Hamiltonian PDEs. In‌​‌ this work 35,​​ we provide the first​​​‌ dimension-independent backward error analysis‌ for a Runge-Kutta-type method,‌​‌ the midpoint rule, which​​ shows the existence of​​​‌ a modified energy for‌ this method when applied‌​‌ to nonlinear Schroedinger equations​​ regardless of the level​​​‌ of spatial discretisation. We‌ use this to establish‌​‌ long-time stability of the​​ numerical flow for the​​​‌ midpoint rule.

This work‌ 17 concerns the design‌​‌ of well-balanced entropy-stable numerical​​ schemes for the shallow​​​‌ water equations. The fully‌ discrete entropy inequality is‌​‌ reached by introducing a​​ local entropy condition incorporated​​​‌ in the scheme design.‌ The source term is‌​‌ discretized to preserve both​​ steady states and entropy​​​‌ stability. The method yields‌ explicit schemes which are‌​‌ relevantly illustrated with several​​​‌ test cases.

8.3 Analysis​ of 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 21​‌, we deal with​​ the interactions of waves​​​‌ governed by a non-linear​ dispersive Boussinesq type system​‌ with the vertical displacement​​ of a cylindrical floating​​​‌ structure in an axisymmetric​ without swirl situation. The​‌ Boussinesq regime is a​​ good approximation of free​​​‌ surface Euler's equations when​ the non-linear parameter and​‌ the shallowness parameter are​​ small. The vertical motion​​​‌ of the floating body​ is governed by the​‌ Newton equation. The full​​ coupled wave-structure interaction problem​​​‌ under consideration is reduced​ to a boundary problem.​‌ The boundary condition satisfied​​ by the discharge is​​​‌ given in terms of​ the vertical displacement of​‌ the floating cylinder. The​​ latter is calculated using​​​‌ an ODE, which requires​ knowledge of the trace​‌ of the surface elevation​​ and its second-time derivative.​​​‌ We use the dispersion​ in order to exhibit​‌ a hidden second order​​ ODE on the trace​​​‌ of the surface elevation.​ This finally allows us​‌ to rewrite the waves-structure​​ interaction problem as a​​​‌ system of non-local conservative​ PDEs plus bounded radial​‌ terms with a dispersive​​ boundary layer, combined with​​​‌ an ODE at the​ boundary. This is what​‌ we call the Augmented​​ formulation. Afterwards we showed​​​‌ that this formulation is​ well-posed with two different​‌ methods. In our proof,​​ we have tracked down​​​‌ the explicit dependence on​ the shallowness parameter, which​‌ is the small parameter​​ representing dispersion. The first​​​‌ method gives a continuous​ solution with a small​‌ existence time, i.e. proportional​​ to the product of​​​‌ the inverse of the​ non-linear parameter and the​‌ square of the shallowness​​ parameter, whereas the second​​​‌ one requires higher regularity​ of the solution but​‌ with a larger existence​​ time, i.e. proportional to​​​‌ the product of the​ inverse of the non-linear​‌ parameter and the shallowness​​ parameter. Finally, we study​​​‌ the return to equilibrium​ situation in the linear​‌ regime. In particular, we​​ have improved previous results​​​‌ on the explicit time​ decay. We have shown​‌ that the center mass​​ of the floating body​​​‌ cannot converge to its​ equilibrium faster than $\mathrm{O‌}\left(t^{-\mathrm{1}/2}\right)$ in​​​‌ 2D without viscosity and​ faster than $O(‌t^{-3/2})$ with viscosity.​​​‌

In 20, we​ propose linear dynamics that​‌ can generate a given​​ sea wavenumber spectrum via​​​‌ a linear partial differential​ equation stirred by an​‌ additive random forcing term​​ that is $\delta$-correlated​​​‌ in time. In particular,​ the correlation structure of​‌ the solution to this​​ linear dynamics converges towards​​​‌ the target spectrum as​ time passes. The main​‌ linear mechanism is a​​ transport in Fourier space,​​​‌ which models the transfer​ of energy from large​‌ scales towards small scales.​​ The proposed linear dynamics​​ generalize previous works for​​​‌ power-law spectra in the‌ context of hydrodynamic turbulence‌​‌ to more general spectra,​​ possibly non-radial, including sea​​​‌ wavenumber spectra such as‌ the JONSWAP spectrum. Finally,‌​‌ we present simulations of​​ the correlation structure of​​​‌ the solution to these‌ dynamics in 2D, whose‌​‌ spectrum converges towards the​​ JONSWAP spectrum. These simulations​​​‌ are based on a‌ finite volume method in‌​‌ the Fourier domain and​​ a splitting method in​​​‌ time, following a recently‌ proposed numerical method by‌​‌ the same authors which​​ improves on pseudospectral simulations.​​​‌

In 29, we‌ present an elementary approach‌​‌ to observe frequency cascade​​ on forced nonlinear Schrödinger​​​‌ equations. The forcing term‌ consists of a constant‌​‌ term, perturbed by a​​ modulated Gaussian well. Algebraic​​​‌ computations provide an explicit‌ frequency cascade when time‌​‌ and space derivatives are​​ discarded from the nonlinear​​​‌ Schrödinger equation. We provide‌ stability results, showing that‌​‌ when derivatives are incorporated​​ in the model, the​​​‌ initial algebraic solution may‌ be little affected, possibly‌​‌ over long time intervals.​​ Numerical simulations are provided,​​​‌ which support the analysis.‌

In 12, 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 11​​, 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.​​​‌

In 30, we‌ consider the moments and‌​‌ the distribution of the​​ hitting times and cover​​​‌ time of the star‌ graph composed of $\mathrm{m‌‌}$ arms each of length​​ $n$. We obtain​​​‌ recurrence relations for the‌ distribution and the moments‌​‌ of all order of​​ both the hitting times​​​‌ of the leaves and‌ the cover times for‌​‌ various initial states. We​​ use the recurrence relations​​​‌ on the moments to‌ analyze the asymptotic behavior‌​‌ of the cover times​​ distribution when the number​​​‌ $m$ of arms, which‌ may depend on depends‌​‌ on the length $\mathrm{n}$ of each arm, tends​​​‌ to infinity when n‌ tends to infinity. We‌​‌ call this new graph,​​ composed of an infinite​​​‌ number of arms each‌ of infinite length, the‌​‌ sun graph.

The stochastic​​ ordering is usefull to​​​‌ compare random variables in‌ terms of their probability‌​‌ distribution: if $X$ and​​ $Y$ are two real​​​‌ random variables, we say‌ $X$ is less than‌​‌ $Y$ in the usual​​​‌ stochastic ordering whenever $\mathrm{P}(X>\mathrm{x‌})\le P(Y>x)‌$ for all $x\in ℝ$. We then​‌ write $X{\le }_{\mathrm{s}t}Y$. For​​​‌ instance if $X$ and​ $Y$ are two durations​‌ then $X{\le }_{\mathrm{s}t}Y$ means that​​​‌ for all values of​ $x$, the duration​‌ $X$ has less chances​​ to exceed $x$ than​​​‌ duration $Y$. In​ 31, we consider​‌ the case of sums​​ of independent and exponentially​​​‌ distributed random variables: if​ $X_i$, resp.​‌ $Y_i(\mathrm{i}=1,\cdots ‌,n)$,​ are independent random variables​‌ that are exponentially distributed​​ with rates ${\lambda }_{\mathrm{i‌}}$, resp. ${\mu }_{\mathrm{i}}$, we examine under​‌ which conditions on the​​ ${\lambda }_i$'s and​​​‌ ${\mu }_i$'s one​ can ensure ${\sum }_{\mathrm{i‌}=1}^n{\mathrm{X}}_i{\le }_{s\mathrm{t‌}}{\sum }_{i=\mathrm{1}}^n{Y}_i$.​‌ Adopting a geomeric point​​ of view in the​​​‌ space of the parameters​ ${\lambda }_i$ and ${\mathrm{\mu ‌}}_i$, we come​​ up with close to​​​‌ optimal conditions, improving previous​ results on this question,​‌ as well as shedding​​ a geometric light on​​​‌ known results.

In 13​, 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 16​, 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 $\mathrm{d‌}=2,\mathrm{3}$.

This work 15​​​‌ investigates variational frameworks for​ modeling stochastic dynamics in​‌ incompressible fluids, focusing on​​ large-scale fluid behavior alongside​​​‌ small-scale stochastic processes. The​ authors aim to develop​‌ a coupled system of​​ equations that captures both​​​‌ scales, using a variational​ principle formulated with Lagrangians​‌ defined on the full​​ flow, and incorporating stochastic​​​‌ transport constraints. The approach​ smooths the noise term​‌ along time, leading to​​ stochastic dynamics as a​​ regularization parameter approaches zero.​​​‌ Initially, fixed noise terms‌ are considered, resulting in‌​‌ a generalized stochastic Euler​​ equation, which becomes problematic​​​‌ as the regularization parameter‌ diminishes. The study then‌​‌ examines connections with existing​​ stochastic frameworks and proposes​​​‌ a new variational principle‌ that couples noise dynamics‌​‌ with large-scale fluid motion.​​ This comprehensive framework provides​​​‌ a stochastic representation of‌ large-scale dynamics while accounting‌​‌ for fine-scale components. Our​​ main result is that​​​‌ the evolution of the‌ small-scale velocity component is‌​‌ governed by a linearized​​ Euler equation with random​​​‌ coefficients, influenced by large-scale‌ transport, stretching, and pressure‌​‌ forcing.

In 34,​​ we investigate how weakening​​​‌ the classical hydrostatic balance‌ hypothesis impacts the well-posedness‌​‌ of the stochastic LU​​ primitive equations. The models​​​‌ we consider are intermediate‌ between the incompressible 3D‌​‌ LU Navier-Stokes equations and​​ the LU primitive equations​​​‌ with standard hydrostatic balance.‌ As such, they are‌​‌ expected to be numerically​​ tractable, while accounting well​​​‌ for phenomena within the‌ grey zone between hydrostatic‌​‌ balance and non-hydrostatic processes.​​ Our main result is​​​‌ the well-posedness of a‌ low-pass filtering-based stochastic interpretation‌​‌ of the LU primitive​​ equations, with rigid-lid type​​​‌ boundary conditions, in the‌ limit of "quasi-barotropic" flow.‌​‌ This assumption is linked​​ to the structure assumption​​​‌ proposed previously, which can‌ be related to the‌​‌ dynamical regime where the​​ primitive equations remain valid.​​​‌ Furthermore, we present and‌ study two eddy-(hyper)viscosity-based models.‌​‌

In 14, we​​ address 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 compared to​​ the literature. Notably, the​​​‌ limiting rough path is‌ identified as a geometric‌​‌ rough path, which does​​ not necessarily coincide with​​​‌ the Stratonovich lift of‌ the Brownian motion.

9.1 Bilateral‌ contracts with industry

Participants:‌​‌ Erwan 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.

10.1 International​ initiatives

10.2​ International research visitors

10.3 National​​​‌ initiatives

• Participants: Mehdi Badsi​, Nicolas Crouseilles.​‌

  2025: project funded by​​ Fédération de Recherche Fusion​​​‌ par Confinement Magnétique, headed​ by N. Crouseilles. Budget​‌ 10 keuros. 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.​

• Participants: Nicolas Crouseilles,​‌ Erwan Faou.

  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​ (University Nantes). Budget 391​‌ keuros. 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.

• Participants:​​​‌ Francois Castella.

  2023-2027:‌ F. Castella is a‌​‌ member of the ANR​​ project BEEP (Behavioural epidemiology​​​‌ and evolution of plant‌ pathogens) headed by F.‌​‌ Hamelin. The partners are​​ Inrae Rennes and Sophia-Antipolis,​​​‌ CIRAD Montpellier, Univ. Cambridge‌ and Univ. Osnabrueck. 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.

• Participants: Adrien Busnot‌ Laurent.

  2025-2029: A.‌​‌ Busnot Laurent is the​​ PI of the ANR​​​‌ JCJC MaStoC (Manifolds and‌ Stochastic Computations). Budget 225‌​‌ keuros. This project will​​ allow for the recruitment​​​‌ of one or two‌ postdocs and will fathom‌​‌ local collaborations in the​​ team. The aim of​​​‌ MaStoC is the design‌ of new numerical methods‌​‌ in the spirit of​​ Lie-group methods for solving​​​‌ SDEs on Riemannian manifolds,‌ possibly with multiscale features.‌​‌

• Participants: Nicolas Crouseilles,​​ Ludovic Martaud.

  2025-2029:​​​‌ N. Crouseilles and L.‌ Martaud are members of‌​‌ the ANR project Cookie​​ (COmputing and apprOximating KInetic​​​‌ Equations) headed by F.‌ Filbet (université Toulouse, France).‌​‌

• Participants: Geoffrey Beck.​​

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

• Participants: Arnaud Debussche​​​‌.

  2023-2027: A. Debussche‌ is a member of‌​‌ the CNRS-MITI project SpatialBioNet.​​ This project focuses on​​​‌ some theoretical challenges and‌ environmental applications on boundary,‌​‌ congestion and vorticity in​​ fluids.

10.4 Regional initiatives​​​‌

• Participants: Nicolas Crouseilles.‌

  2025: Actions internationales (université‌​‌ Rennes): between IRMAR and​​ IISER Thiruvanantapuram (India). In​​​‌ this project, we aim‌ at deriving and analysing‌​‌ Asymptotic Preserving schemes for​​ fluid models arising in​​​‌ plasma physics.

• Participants: Adrien‌ Busnot Laurent.

  AIS‌​‌ (selective grant from Rennes​​ metropole for young researcher)​​​‌ obtained by A. Busnot‌ Laurent (2025). Budget 10‌​‌ keuros.

11.1​​ Promoting scientific activities

11.2 Teaching - Supervision‌​‌ - Juries - Educational​​ and pedagogical outreach

11.3 Popularization‌​‌

12.1​‌ Major publications

• 1 article​​J.Joackim Bernier,​​​‌ E.Erwan Faou and​ B.Benoit Grebert.​‌ Long time behavior of​​ the solutions of NLW​​​‌ on the d-dimensional torus​.Forum of Mathematics,​‌ Sigma82020,​​ E12HALDOI
• 2​​​‌ articleF.Fernando Casas​, N.Nicolas Crouseilles​‌, E.Erwan Faou​​ and M.Michel Mehrenberger​​​‌. High-order Hamiltonian splitting​ for Vlasov-Poisson equations.​‌Numerische Mathematik1353​​2017, 769-801HAL​​​‌DOIback to text​
• 3 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 Computing42​​​‌22020, B520--B547​HALDOIback to​‌ text
• 4 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 Physics873​​​‌May 2021, article​ n° 825870301HALDOI​‌back to text
• 5​​ articleA.Arnaud Debussche​​​‌ and J.Julien Vovelle​. Diffusion-approximation in stochastically​‌ forced kinetic equations.​​Tunisian Journal of Mathematics​​​‌312021,​ 1-53HALDOIback​‌ to text
• 6 article​​E.Erwan Faou.​​​‌ Linearized wave turbulence convergence​ results for three-wave systems​‌.Communications in Mathematical​​ Physics3782September​​​‌ 2020, 807–849HAL​DOIback to text​‌

12.2 Publications of the​​ year

12.3 Cited​ publications

• 36 bookC.​‌C. Birdsall and A.​​A. Langdon. Plasmas​​​‌ physics via computer simulations​. New York Taylor​‌ and Francis2005back​​ to text
• 37 article​​A.A. Brizard and​​​‌ T.T. Hahm.‌ Foundations of nonlinear gyrokinetic‌​‌ theory.Reviews of​​ Modern Physics792007​​​‌back to text
• 38‌ articleJ.J. Carr‌​‌. Applications of Centre​​ Manifold Theory. Applied​​​‌ Mathematical Sciences Series35‌1981back to text‌​‌
• 39 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--536​​back to textback​​​‌ to text
• 40 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.15​​​‌2015, 591--612back‌ to text
• 41 article‌​‌A.A. Debussche and​​ J.J. Vovelle.​​​‌ Diffusion limit for a‌ stochastic kinetic problem.‌​‌ Commun. Pure Appl. Anal.​​112012, 2305--2326​​​‌back to text
• 42‌ articleE.E. Faou‌​‌ and F.F. Rousset​​. Landau damping in​​​‌ Sobolev spaces for the‌ Vlasov-HMF model. Arch.‌​‌ Ration. Mech. Anal.219​​2016, 887--902back​​​‌ to text
• 43 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 text‌​‌back to text
• 44​​ 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
• 45 articleM.‌M. Lemou, F.‌​‌F. Méhats and P.​​P. Raphaël. Orbital​​​‌ stability of spherical galactic‌ models.Invent. Math.‌​‌1872012, 145--194​​back to text
• 46​​​‌ articleC.C. Mouhot‌ and C.C. Villani‌​‌. On Landau damping​​.Acta Math.207​​​‌2011, 29--201back‌ to text
• 47 book‌​‌S.S. Nazarenko.​​ Wave turbulence. Springer-Verlag​​​‌2011back to text‌
• 48 articleL.L.‌​‌ Perko. Higher order​​ averaging and related methods​​​‌ for perturbed periodic and‌ quasi-periodic systems.SIAM‌​‌ J. Appl. Math.17​​1969, 698--724back​​​‌ to textback to‌ text