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 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 $P$ 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 45- 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, A. Laurent, 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 7, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler-Poisson (EP) equations in the quasineutral limit is presented. As the quasineutral limit is singular for the governing equations, the time discretisation is tantamount to achieving an accurate numerical method. To this end, the EP system is viewed as a differential algebraic equation system (DAEs) via the method of lines. As a consequence of this vantage point, high order linearly semi-implicit (SI) time discretisation are realised by employing a novel combination of the direct approach used for implicit discretisation of DAEs and, two different classes of IMEX-RK schemes: the additive and the multiplicative. For both the time discretisation strategies, in order to account for rapid plasma oscillations in quasineutral regimes, the nonlinear Euler fluxes are split into two different combinations of stiff and non-stiff components. The high order scheme resulting from the additive approach is designated as a classical scheme while the one generated by the multiplicative approach possesses the asymptotic preserving (AP) property. Time discretisations for the classical and the AP schemes are performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the stiff terms are treated implicitly and the non-stiff ones explicitly. In order to discretise in space a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. AP property is also established for the space-time fully-discrete scheme obtained using the multiplicative approach. Results of numerical experiments are presented, which confirm that the high order schemes based on the SI-IMEX-RK time discretisation achieve uniform second order convergence with respect to the Debye length and are AP in the quasineutral limit.

In 28, we focus on developing a new class of numerical methods able to both handle quasineutrality and charge separation in plasmas. At large temporal and spatial scales, plasmas tend to be quasineutral, meaning that the local net charge density is nearly zero. However, when the scale at which one observes the plasma dynamics is smaller than the characteristic distance over which the electric field and charges are typically screened, then quasineutrality breaks down. In such regimes, standard numerical methods face severe stability constraints, rendering them practically unusable. To address this issue, in this work, we introduce and analyze a new class of finite volume penalized-IMEX Runge-Kutta methods for the Euler-Poisson (EP) system, specifically designed to handle the quasineutral limit. We show that, these proposed schemes are uniformly stable with respect to the Debye length and degenerate into high order methods as the quasineutral limit is approached. Several numerical tests confirm that this new class of methods exhibits the desired properties.

This work 10 deals with wave propagation into a coaxial cable, which can be modelled by the 3D Maxwell equations or 1D simplified models. The usual model, called the telegrapher's model, is a 1D wave equation of 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 thickness 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 thickness of the cable, the dispersive 1D model is of order two.

In 13, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov-Maxwell examples are presented to illustrate the good behavior of the approach.

In 15, 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.

In 16, we propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for a linear hyperbolic system. The method is designed based on the ELDG method for transport problems (J. Comput. Phys. 446:110,632, 2021), which tracks solutions along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability with respect to the classical RKDG method. Considering each characteristic family, a straightforward application of ELDG for the 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 ELDG 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.

This work 22 deals with the numerical approximation of plasmas which are confined by the effect of a fast oscillating magnetic field (see [12]) 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.

The main concern of the paper 29 is the mathematical modelling and numerical simulation of thermonuclear fusion plasmas, constituted of two different kinds of particles, namely the thermal electron/ion bulk and an energetic α-particle population, created by the fusion reaction at very high speeds (3.5 MeV). This $\alpha$-particle population behaves differently than the thermal plasma bulk, and due to its high energy contain, can have a considerable impact on the stability and thus confinement of the plasma bulk. An adequate modelling of this $\alpha$-population is therefore crucial, and the present paper intends to make use of kinetic Fokker-Planck models to describe the evolution of these energetic particles experiencing Coulomb collisions. The main mathematical and numerical difficulties encountered in this study are related to the multi-scale nature of the overall problem, involving different types of particles and their multiscale dynamics.

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 8, we construct a mean-field model that describes the nonlinear dynamics of a spinpolarized 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 identied, which depend essentially on the electron spin polarization rate $\eta$ 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 rst developed in a previous work. 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 $\left({10}^{29}{m}^{-3}\right)$ and temperatures of the order of $10eV$ may be relevant to the warm dense matter regime appearing in some inertial fusion experiments.

24 Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x}=f_0\left(x\right)+f_1\left(x\right)$ where $f_0$ encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}=f_0\left(x\left(t\right)\right)+u\left(t\right)f_1\left(x\left(t\right)\right)$, associated with a control $u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0+u\left(t\right)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper 25 the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.

In 14, we consider a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the almost second-order convergence of these discrete functions towards Hermite functions, uniformly for large numbers of modes. We then describe an efficient way to simulate these eigenfunctions and the corresponding transformation. We finally show some numerical experiments corroborating our different results.

In 32, we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.

In 12, we focus on an adaptation of the method described in a previous in order to deal with source term in the 2D Euler equations. This method extends classical 1D solvers (such as VFFC, Roe, Rusanov) to the two-dimensional case on unstructured meshes. The resulting schemes are said to be composite as they can be written as a convex combination of a purely node-based scheme and a purely edge-based scheme. We combine this extension with the ideas developed by Alouges, Ghidaglia and Tajchman in an unpublished work – focused mainly on the 1D case – and we propose two attempts at discretizing the source term of the Euler equations in order to better preserve stationary solutions. We compare these discretizations with the “usual” centered discretization on several numerical examples.

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 17defines 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.

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 9, 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 19, we study a model describing the interaction between waves at the surface of a sea and a freely floating partially immersed cylinder in the Boussinesq regime. This wave-interaction system can be reduced to an initial boundary value problem for the Boussinesq equations in an exterior domain. The boundary condition depends on the motion of the floating cylinder which is in turn determined by a nonlinear ODE with forcing terms coming from the exterior wave field.

In 21, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic mode.

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics was recently proposed that can generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear partial differential equation stirred by an additive random forcing term that is $\delta$-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space that aims to transfer energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We present in this work 11 some simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudospectral simulations. We show that our algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two, and three spatial dimensions, and we display the solutions both in physical and Fourier space. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities.

In 20, we consider a stochastic nonlinear formulation of classical coastal waves models under location uncertainty (LU). In the formal setting investigated here, stochastic versions of the Serre-Green- Nagdi, Boussinesq and classical shallow water wave models are obtained through an asymptotic expansion, which is similar to the one operated in the deterministic setting. However, modified advection terms emerge, together with advection noise terms. These terms are well-known features arising from the LU formalism, based on momentum conservation principle.

In 23, 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.

In 30, 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 to a former paper. 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.

In 31, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measure, after suitable truncation in the focusing case. The set and the measure are invariant under the flow. The main ingredients used are Strichartz estimates on periodic NLS with distributional potential to obtain local well-posedness for low regularity initial data.

In 26, 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 27, 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.

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.

10.1 International initiatives

10.2 International research visitors

The MINGuS members welcame and visited several international laboratories.

10.3 National initiatives

• 2024: 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.

• 2019-2024 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 (University 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: 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.

  Participants: Francois Castella.

  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.

• 2023-2027: A. Debussche is a member of the CNRS-MITI project SpatialBioNet.

  Participants: Arnaud Debussche.

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

10.4 Regional initiatives

G. Beck obtained an AIS grant from Britanny region council.

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

• G. Beck, N. Crouseilles, A. Laurent, P. Navaro collaborated with Thomas Menuet (musician, who spent two years at the IRMAR laboratory) to create an enigma combining music and mathematics. More informations can be found Résidence Thomas Menuet but also in OuestFrance and in canalb
• A. Laurent gave a talk at the 5' Lebesgue
• A. Laurent participated at 'la fête de la science' in Rennes.
• A. Laurent participated at the 'club Rennes et Maths'
• A. Laurent participated at the livestorm on the scientific outreach dedicated to PhD students
• A. Laurent participated to the event 'les Marmottes'

12.1 Major publications

• 1 articleJ.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 Mathematik13532017, 769-801HALDOIback 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 Computing4222020, B520--B547HALDOIback 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 Physics873May 2021, article n° 825870301HALDOIback to text
• 5 articleA.Arnaud Debussche and J.Julien Vovelle. Diffusion-approximation in stochastically forced kinetic equations.Tunisian Journal of Mathematics312021, 1-53HALDOIback to text
• 6 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

• 33 bookC.C. Birdsall and A.A. Langdon. Plasmas physics via computer simulations. New York Taylor and Francis2005back to text
• 34 articleA.A. Brizard and T.T. Hahm. Foundations of nonlinear gyrokinetic theory.Reviews of Modern Physics792007back to text
• 35 articleJ.J. Carr. Applications of Centre Manifold Theory. Applied Mathematical Sciences Series351981back to text
• 36 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
• 37 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
• 38 articleA.A. Debussche and J.J. Vovelle. Diffusion limit for a stochastic kinetic problem. Commun. Pure Appl. Anal.112012, 2305--2326back to text
• 39 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
• 40 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
• 41 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
• 42 articleM.M. Lemou, F.F. Méhats and P.P. Raphaël. Orbital stability of spherical galactic models.Invent. Math.1872012, 145--194back to text
• 43 articleC.C. Mouhot and C.C. Villani. On Landau damping.Acta Math.2072011, 29--201back to text
• 44 bookS.S. Nazarenko. Wave turbulence. Springer-Verlag2011back to text
• 45 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