TONUS has started in January 2014. It is a team of the Inria Antenna in Strasbourg. It is located in the mathematics institute (IRMA) of the University of Strasbourg.

The International Thermonuclear Experimental Reactor (ITER) is a large-scale scientific experiment that aims to demonstrate that it is possible to produce energy from fusion, by confining a very hot hydrogen plasma inside a toroidal chamber, called tokamak. In addition to physics and technology research, the design of tokamaks also requires mathematical modeling and numerical simulations on supercomputers.

The objective of the TONUS project is to deal with such mathematical and computing issues. We are mainly interested in kinetic, gyrokinetic and fluid simulations of tokamak plasmas. In the TONUS project-team we are working on the development of new numerical methods devoted to such simulations. We investigate several classical plasma models, study new reduced models and new numerical schemes adapted to these models. We implement our methods in two software projects: Selalib and SCHNAPS adapted to recent computer architectures.

We have strong relations with the CEA-IRFM team and participate in the development of their gyrokinetic simulation software GYSELA. We were involved in two Inria Project Labs, respectively devoted to tokamak mathematical modeling and high performance computing. The numerical tools developed from plasma physics can also be applied in other contexts. For instance, we collaborate with a small company in Strasbourg specialized in numerical software for applied electromagnetism. We also study kinetic acoustic models with the CEREMA and multiphase flows with EDF.

Finally, our topics of interest are at the interface between mathematics, computer science, High Performance Computing, physics and practical applications.

The fundamental model for plasma physics is the coupled Vlasov-Maxwell kinetic model: the Vlasov equation describes the distribution function of particles (ions and electrons), while the Maxwell equations describe the electromagnetic field. In some applications, it may be necessary to take relativistic particles into account, which leads to consider the relativistic Vlasov equation, even if in general, tokamak plasmas are supposed to be non-relativistic. The distribution function of particles depends on seven variables (three for space, three for the velocity and one for time), which yields a huge amount of computation. To these equations we must add several types of source terms and boundary conditions for representing the walls of the tokamak, the applied electromagnetic field that confines the plasma, fuel injection, collision effects, etc.

Tokamak plasmas possess particular features, which require developing specialized theoretical and numerical tools.

Because the magnetic field is strong, the particle trajectories have a very fast rotation around the magnetic field lines. A full resolution would require a prohibitive amount of computation. It is necessary to develop reduced models for large magnetic fields in order to obtain tractable calculations. The resulting model is called a gyrokinetic model. It allows us to reduce the dimensions of the problem. Such models are implemented in GYSELA and Selalib.

On the plasma boundary, the collisions can no longer be neglected. Fluid models, such as MagnetoHydroDynamics (MHD), become relevant again. For the good operation of the tokamak, it is necessary to control MHD instabilities that arise at the plasma boundary. Computing these instabilities requires special implicit numerical discretizations with excellent long time behavior.

In addition to theoretical modeling tools, it is necessary to develop numerical schemes adapted to kinetic, gyrokinetic and fluid models. Three kinds of methods are studied in TONUS: Particle-In-Cell (PIC) methods, semi-Lagrangian and fully Eulerian approaches.

In most phenomena where oscillations are present, we can establish a
three-model hierarchy: (i) the model parameterized by the oscillation period,
(ii) the limit model and (iii) the two-scale model,
possibly with its corrector.
In a context where one wishes to simulate such a phenomenon where
the oscillation period is small and the oscillation amplitude
is not small, it is important to have numerical methods based on an
approximation of the two-scale model. If the oscillation period varies
significantly over the domain of simulation, it is important to have
numerical methods that approximate properly and effectively the model
parameterized by the oscillation period and the two-scale model. Implementing
two-scale numerical methods (for instance by Frénod et al.
37) is based on a numerical approximation
of the Two-Scale model. These are called of order 0. A Two-Scale Numerical
Method is called of order 1 if it incorporates information from the
corrector and from the equation of which this corrector is a solution.
If the oscillation period varies between very small values and values
of order 1, it is necessary to have new types of numerical schemes
(Two-Scale Asymptotic Preserving Schemes of order 1 or TSAPS) that preserve
the asymptotics between the model
parameterized by the oscillation period and the Two-Scale model with
its corrector. A first work in this direction has been initiated by
Crouseilles et al. 36.

The TONUS team, and more generally the scientific computing team at IRMA Strasbourg, has a long and recognized experience in numerical methods for Vlasov-type equations. We are specialized in both particle and phase space solvers for the Vlasov equation: Particle-in-Cell (PIC) methods and semi-Lagrangian methods. We also have a long-standing collaboration with CEA Cadarache for the development of the GYSELA software for gyrokinetic tokamak plasmas.

The Vlasov and the gyrokinetic models are partial differential equations that express the transport of the distribution function in the phase space. In the original Vlasov case, the phase space is the six-dimension position-velocity space. For the gyrokinetic model, the phase space is five-dimensional because we consider only the parallel velocity in the direction of the magnetic field and the gyrokinetic angular velocity instead of three velocity components.

A few years ago, Eric Sonnendrücker and his collaborators introduced a new family of methods for solving transport equations in the phase space. This family of methods are the semi-Lagrangian methods. The principle of these methods is to solve the equation on a grid of the phase space. The grid points are transported with the flow of the transport equation for a time step and interpolated back periodically onto the initial grid. The method is then a mix of particle Lagrangian methods and Eulerian methods. The characteristics can be solved forward or backward in time leading to the Forward Semi-Lagrangian (FSL) or Backward Semi-Lagrangian (BSL) schemes. Conservative schemes based on this idea can be developed and are called Conservative Semi-Lagrangian (CSL).

GYSELA is a 5D full gyrokinetic code based on a classical backward semi-Lagrangian scheme (BSL) 42 for the simulation of core turbulence that has been developed at CEA Cadarache in collaboration with our team 38.

More recently, we have started to apply the semi-Lagrangian methods to more general kinetic equations. Indeed, most of the conservation laws of physics can be represented by a kinetic model with a small set of velocities. Compressible fluids or MHD equations have such representations. Semi-Lagrangian methods then become a very appealing and efficient approach for solving these equations.

Historically PIC methods have been very popular for solving the Vlasov equations. They make it possible to solve the equations in phase space at a relatively low cost. The main disadvantage of this approach is that, due to its random aspect, it produces an important numerical noise that has to be controlled in some way, for instance by regularizations of the particles, or by divergence correction techniques in the Maxwell solver. We have a long-standing experience in PIC methods and we started implementing them in Selalib. An important aspect is to adapt the method to new multicore computers. See the work by Crestetto and Helluy 35.

As already said, kinetic plasmas computer simulations are very intensive, because of the gyrokinetic turbulence. In some situations, it is possible to make assumptions on the shape of the distribution function that simplify the model. We obtain in this way a family of fluid models or reduced models.

Assuming that the distribution function has a Maxwellian shape, for instance, we obtain the MagnetoHydroDynamic (MHD) model. It is physically valid only in some parts of the tokamak (at the edges for instance). The fluid model is generally obtained from the hypothesis that the collisions between particles are strong.

However, the reduction is not necessarily a consequence of collisional effects. Indeed, even without collisions, the plasma may still relax to an equilibrium state over sufficiently long time scales (Landau damping effect).

In the fluid or reduced-kinetic regions, the approximation of the distribution function could require fewer data while still achieving a good representation, even in the collisionless regime.

Therefore, a fluid or a reduced model is
a model where the explicit dependency on the velocity variable is
removed. In a more mathematical way, we consider that in some regions
of the plasma, it is possible to exhibit a (preferably small) set
of parameters

In this case it is sufficient to solve for

Another way to reduce the model is to try to find an abstract kinetic representation with a set of kinetic velocities as small as possible. The kinetic approach has then only a mathematical meaning. It allows solving very efficiently many equations of physics.

An efficient method for solving the reduced models is the Discontinuous Galerkin (DG) approach 40. It is possible to make it of arbitrary order. It requires limiters when it is applied to nonlinear PDEs occurring for instance in fluid mechanics. But the reduced models that we intend to write are essentially linear. The nonlinearity is concentrated in a few coupling source terms.

In addition, this method, when written in a special set of variables, called the entropy variables, has nice properties concerning the entropy dissipation of the model. It opens the door to constructing numerical schemes with good conservation properties and no entropy dissipation, as already used for other systems of PDEs 43, 34, 41, 39.

In tokamaks, the reduced model generally involves a lot of different time scales. Most of them, associated to the fastest waves, are not relevant to the tokamak simulation. In order to filter them out, it is necessary to adopt implicit solvers in time. When the reduced model is based on a kinetic interpretation, it is possible to construct implicit schemes that do not impose solving costly linear systems. In addition the resulting solver is stable even at a very high CFL (Courant–Friedrichs–Lewy) number.

Accurate resolution of the electromagnetic fields is essential for proper plasma simulations. Thus it is important to use efficient solvers for the Maxwell systems and its asymptotics: Poisson equation and magnetostatics.

The proper coupling of the electromagnetic solver with the Vlasov solver is also crucial for ensuring conservation properties and stability of the simulation.

Finally, plasma physics implies very different time scales. It is thus very important to develop implicit Maxwell solvers and Asymptotic Preserving (AP) schemes in order to obtain good behavior on long time scales.

The coupling of the Maxwell equations to the Vlasov solver requires some precautions. The most important one is to control the charge conservation errors, which are related to the divergence conditions on the electric and magnetic fields. We will generally use divergence correction tools for hyperbolic systems presented for instance in 32 (and the references therein).

As already pointed out, in a tokamak, the plasma presents several different space and time scales. It is not possible in practice to solve the initial Vlasov-Maxwell model. It is first necessary to establish asymptotic models by letting some parameters (such as the Larmor frequency or the speed of light) tend to infinity. This is the case for the electromagnetic solver. To address this issue, time-implicit solvers have to be implemented, in order to provide good approximations of stationary states, or of the solution to the magnetic induction equation or to the Poisson equation.

The search for alternative energy sources is a major issue for the future. Among others, controlled thermonuclear fusion in a hot hydrogen plasma is a promising possibility. The principle is to confine the plasma in a toroidal chamber, called a tokamak, and to attain the necessary temperatures to sustain nuclear fusion reactions.The International Thermonuclear Experimental Reactor (ITER) is a tokamak being constructed in Cadarache, France. This was the result of a joint decision by an international consortium including the European Union, Canada, USA, Japan, Russia, South Korea, India and China. ITER is a huge project. As of today, the budget is estimated at 22 billion euros. The first plasma shot is planned for November 2025. Many technical and conceptual difficulties have to be overcome before the actual exploitation of fusion energy. Consequently, much research has been carried out around magnetically confined fusion. Among these studies, it is important to carry out computer simulations of the burning plasma. Thus, mathematicians and computer scientists are also needed in the design of ITER. The reliability and the precision of numerical simulations allow a better understanding of the physical phenomena and thus would lead to better designs. TONUS's main involvement is in such research. The required temperatures to attain fusion are very high, of the order of a hundred million degrees. Thus it is imperative to prevent the plasma from touching the tokamak inner walls. This confinement is obtained thanks to intense magnetic fields. The magnetic field is created by poloidal coils, which generate the toroidal component of the field. The toroidal plasma current also induces a poloidal component of the magnetic field that twists the magnetic field lines. The twisting is very important for the stability of the plasma. The idea goes back to research by Tamm and Sakharov, two Russian physicists, in the 50's. Other devices are essential for the proper operation of the tokamak: a divertor for collecting the escaping particles, microwave heating for reaching higher temperatures, a fuel injector for sustaining the fusion reactions, toroidal coils for controlling instabilities, etc.

The software and numerical methods that we develop can also be applied to other fields of physics or engineering.

Members of the TONUS team have organized the FVCA10 conference (Tenth International Conference on Finite Volumes for Complex Applications), held in Strasbourg, from October 30 to November 3, 2023. There were around 110 participants, mostly from across Europe, and also from Asia and North America. This conference led to the publication of two books of proceedings: 13, 14.

The work described in 3 is devoted to the simulation of magnetohydrodynamic (MHD) flows with a Lattice Boltzmann Method (LBM) that handles shock waves. The scheme is implemented on GPU using PyOpenCL and makes it possible to use very fine grids. The method is validated on complex resistive instabilities.

In 26 we develop a simple and efficient methodology for compressing data on the fly during LBM simulations. This approach makes it possible to handle simulations that would not enter the GPU memory. It is based on wavelet compression and presents a reasonable overhead.

In 29 we propose a rigorous entropy stability analysis of the LBM scheme. We also expose a new algorithm for performing an equivalent equation analysis of the kinetic scheme with over-relaxation.

In 30 we propose a simple general approach for applying boundary conditions in the LBM. The method is based on a correction that ensures entropy dissipation, while maintening scheme accuracy.

In 17, we focused on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we paid close attention to steady solutions with nonzero velocity. The goal of this work was to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution, by proposing an extension of the well-known hydrostatic reconstruction 33. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we proved that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we proposed a linear well-balanced high-order procedure.

In 31, we presented composition methods for solving non-linear transport equations, like the guiding-center model. By composing direct and adjoint transport steps on different well-chosen sub time steps, we were able to construct high-order in time numerical methods. The adjoint steps being implicit, they are replaced by explicit approximated adjoint steps using a given number of fixed point iterations. This does not modify the order of accuracy. Several numerical tests, using high-order semi-Lagrangian spatial discretization, assess the performance of the method. Despite the large number of substeps per iteration, the method can be interesting since it reduces the number of copies of the unknown.

In 8, we have introduced a new thermodynamically compatible finite volume scheme for the symmetric hyperbolic and thermodynamically compressible (SHTC) two-fluid model of Romenski et al. that is endowed with two entropy inequalities. The new method is able to discretize the two entropy inequalities directly and obtains total energy conservation as a mere consequence of the thermodynamically compatible discretization of all other equations. The thermodynamically compatible numerical flux is based on the seminal ideas of Abgrall, which was subsequently generalized and applied to SHTC systems. Compared to previous thermodynamically compatible finite volume schemes the new approach forwarded in this paper does not require the computation of any path integral but is totally general and can be applied to arbitrary overdetermined hyperbolic and thermodynamically compatible systems that admit an extra conservation law. The proposed HTC FV schemes satisfy two discrete entropy inequalities and are provably nonlinearly stable in the energy norm. We furthermore consider arbitrary high order accurate discontinuous Galerkin (DG) schemes applied to the vanishing viscosity limit of the overdetermined system. In this case, no particular care is taken to satisfy the additional conservation law exactly at the discrete level, apart from the mere resolution of all flow features. The numerical results clearly show that the schemes proposed in this project achieve their designed order of accuracy and that in the stiff relaxation limit the numerical solution tends to the asymptotically reduced Baer–Nunziato (BN) limit system with lift forces. The influence of the lift forces has been studied separately, showing that their presence in the BN limit is necessary to achieve a good agreement with the underlying SHTC system. This work was also the object of the short communication 9.

In 22, we analyse the famous Aw-Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called "hard congestion limit", and show the convergence of a subsequence of solutions towards a weak solution of an hybrid freecongested system. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contact between the grains.

In 7, we have derived an analyzed a new implicit-explicit finite volume (RS-IMEX FV) scheme for a single-temperature SHTC model. We note that the two-fluid model allows two velocities and pressures. Further, it includes two dissipative mechanisms: phase pressure and velocity relaxations. In the proposed scheme these processes are treated differently. The relative velocity relaxation term is linear and is resolved as a part of the implicit sub-system, whereas the pressure relaxation is strongly nonlinear and therefore is treated separately by the Newton method. Our RS-IMEX FV method is constructed in such a way that acoustic-type waves are linearized around a suitably chosen reference state (RS) and approximated implicitly in time and by means of central finite differences in space. The remaining advective-type waves are approximated explicitly in time and by means of the Rusanov FV method. The RS-IMEX FV scheme is suitable for all Mach number flows, but in particular it is asymptotic preserving in the low Mach number flow regimes. Many multi-phase flows, such as granular or sediment transport flows, can be modeled within the single-temperature approximation. In turn, many of these flows are weakly compressible and therefore impose severe time step restrictions if solved with a time-explicit numerical scheme. Therefore, the proposed RS-IMEX FV scheme is suitable to model various environmental flows. The proposed method was tested on a number of test cases for low and moderately high Mach number flows demonstrating the capability of the scheme to properly capture both regimes. The theoretical second order accuracy of the scheme was confirmed on a stationary vortex test case. We compared the second order scheme against its first order variant which showed that the second order scheme yields more accurate approximations of discontinuities. Finally, the theoretically proven asymptotic preserving property was verified numerically.

In 10, we present a two-dimensional extension to the linear implicit all-speed finite volume scheme for hyperbolic conservation laws based on Jin-Xin relaxation recently forwarded in 44. It is based on stiffly accurate SDIRK (Singly-Diagonal Implicit Runge-Kutta) methods in time and a convex combination of Rusanov and centered fluxes in space making it asymptotically consistent in the low Mach number regime and allows an accurate capturing of material waves under large time steps. The scheme is numerically tested on the Euler equations and a non-linear model for elasticity in the compressible and low Mach number regime.

In 28, we proposed to enrich Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we computed using a Physics-Informed Neural Network (PINN). We proved convergence results and error estimates, showing that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, parametric PINNs were used. Numerical experiments on 1D and 2D hyperbolic systems show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

In 16, we proposed an iterative adaptive algorithm where we progressively refined the time discretization (i.e. increasing the number of layers). Provided that certain tolerances are met across the iterations, we prove that the strategy converges to the underlying continuous problem. One salient advantage of such a shallow-to-deep approach is that it helps to benefit in practice from the higher approximation properties of deep networks by mitigating over-parametrization issues.

In 19, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontiuous Galerkin (DG) methods. To this end, the problem is reformulated as an optimal control problem for which the control is the viscosity function and the cost function involves comparison with a reference solution after several compositions of the scheme. The learning process is strongly based on a time propagation method which uses the ML framework to compute the gradient of the full scheme with respect to the parameters of the network at many time steps.

In 4, we present a novel numerical approach aiming at computing equilibria and dynamics structures of magnetized plasmas in coronal environments. A technique based on the use of neural networks that integrates the partial differential equations of the model, and called physics-informed neural networks (PINNs), is introduced. The functionality of PINNs is explored via calculation of different magnetohydrodynamic (MHD) equilibrium configurations, and also the derivation of exact two-dimensional steady-state magnetic reconnection solutions. The advantages and drawbacks of PINNs compared to traditional numerical codes are discussed in order to propose future improvements. Interestingly, PINNs are a meshfree method in which the obtained solution and associated different order derivatives are quasi-instantaneously generated at any point of the spatial domain. We believe that our results can help to pave the way for future developments of time dependent MHD codes based on PINNs.

In the paper 24 we propose a non-linear reduction method for models coming from the spatial discretization of partial differential equations: it is based on convolutional auto-encoders and Hamiltonian neural networks. Their training is coupled in order to simultaneously learn the encoder-decoder operators and the reduced dynamics. Several test cases on non-linear wave dynamics show that the method has better reduction properties than standard linear Hamiltonian reduction methods.

The paper 27 proposes a reduced model to simulate the one-dimensional Vlasov-Poisson equation. The model provides the space-time dynamics of a few macroscopic quantities constructed following the Reduced Order Method (ROM) in the velocity variable: the compression is thus applied to the semi-discretization of the Vlasov equation. To gain efficiency, a Discrete Empirical Interpolation Method (DEIM) is applied to the compressed non-linear Fokker-Planck operator. Furthermore, we propose a correction to the reduced collision operator which ensures that the reduced moments satisfy an Euler-type system. Numerical simulations of the reduced model show that the model can capture the plasma dynamics in different collisional regimes and initial conditions at a low cost.

While the main focus of the numerical tools we develop is plasma physics, they can also be used for other applications. We list below four such applications.

In 25, we solved the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. We obtained Curie temperatures

Considering a ferromagnetic material of ellipsoidal shape, the associated magnetic moment then has two asymptotically stable opposite equilibria, of the form

Modeling epidemics using classical population-based models suffers from shortcomings that so-called individual-based models are able to overcome. They are able to take into account heterogeneity features, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose in 5 a numerical approach to determine optimal health policies for a stochastic individual-based model taking into account heterogeneity in the population. More precisely, we introduce a deterministic reduced population-based model involving a neural network, designed to faithfully mimic the local dynamics of the more complex individual-based model. Then the optimal control is determined by sequentially training the network until an optimal strategy for the population-based model succeeds in also containing the epidemic when simulated on the individual-based model. After describing the practical implementation of the method, several numerical tests are proposed to demonstrate its ability to determine controls for models with contact heterogeneity.

In 20, 21, we prove a new result in shape optimization: the convergence of the fixed-point (also called thresholding) algorithm in three optimal control problems under large volume constraints. This algorithm was introduced by Céa, Gioan and Michel, and is of constant use in the simulation of

How to free a road from vehicle traffic as efficiently as possible and in a given time, in order to allow, for example, emergency vehicles to pass? In 18, we are interested in this question which we reformulate as an optimal control problem. We consider a macroscopic road traffic model on networks, semi-discretized in space and decide to give ourselves the possibility to control the flow at junctions. Our target is to smooth the traffic along a given path within a fixed time. A sparsity constraint is imposed on the controls, in order to ensure that the optimal strategies are feasible in practice. We perform an analysis of the resulting optimal control problem, proving the existence of an optimal control and deriving optimality conditions, which we rewrite as a single functional equation. We then use this formulation to derive a new mixed algorithm interpreting it as a mix between two methods: a descent method combined with a fixed point method allowing global perturbations. We verify with numerical experiments the efficiency of this method on examples of graphs, first simple, then more complex. We highlight the efficiency of our approach by comparing it to standard methods. We propose an open source code implementing this approach in the Julia language.

Directed flows of cells in vivo are essential in morphogenesis. They shape living matter in phenomena involving cell mechanics and regulations of the acto-myosin cytoskeleton. However the onset of coherent motion during collective cell migration is still poorly understood. In 6 we show that coherence is set by spontaneous alignments of cell polarity by designing cellular rings of controlled dimensions. A tug-of-war between opposite polarities dictates the onset of coherence, as assessed by tracking live cellular shapes and motions in various experimental conditions. In addition, we identify an internally driven constraint by cellular acto-myosin cables at boundaries as essential to ensure coherence. Moreover, active force is generated as evaluated by the high RhoA activity. Its contribution is required to trigger coherence as shown by our numerical simulations based on a novel Vicsek-type model including free active boundaries. Altogether, spontaneous coherent motion results from basic interplay between cell orientations and active cables at boundaries.

Collaboration with start-up Renaissance Fusion on the Stellacage project, co-funded by INRIA (Project Leader: Mario Sigalotti).

We collaborate with EDF through the CIFRE thesis of Gauthier Lazare in the optimization of nuclear plant simulation software. The optimization is realized through algorithmic improvements but also with software acceleration based on machine learning.