TONUS has started in January 2014. It is a team of the Inria Nancy-Grand Est center. 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 are 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.
25) 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. 24.

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) 30 for the simulation of core turbulence that has been developed at CEA Cadarache in collaboration with our team 27.

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 23.

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.

As previously indicated, an efficient method for solving the reduced models is the Discontinuous Galerkin (DG) approach. 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 31, 22, 29, 28.

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 21 (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 20 billion euros. The first plasma shot is planned for 2025 and the first deuterium-tritium operation for 2027. 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.

The Discontinuous Galerkin method is a general method for solving conservation laws. In 8, we described a parallel and quasi-explicit Discontinuous Galerkin (DG) kinetic scheme for solving systems of balance laws. The solver is unconditionally stable (i.e., the CFL number can be arbitrary) and has the complexity of an explicit scheme. It can be applied to any hyperbolic system of balance laws. In this work, we assessed the performance of the scheme in the particular case of the three-dimensional wave equation and of Maxwell's equations. We measured the benefit of the unconditional stability by performing experiments with very large CFL numbers. In addition, we investigated how to parallelize this method.

A version of this solver (presented in section 5.1.1), called KOUGLOFV, was implemented in the RUST language. It is a very reliable language that allows us to avoid most of the common memory bugs at compile time. It also provides nice tools for automatic and robust shared-memory parallelization. This parallelization was tested in the code, and good efficiency results were obtained. In 2022, KOUGLOFV was enhanced with distributed-memory parallelization (via MPI) and with the capability to handle large-scale simulations of electromagnetic waves within the human body, see figure 1. This led to the preprints 18 as well as the short report 19; the research was performed in the context of the grant detailed in section 7.1.

In 5, we considered a general framework to build a linear high-order well-balanced scheme for systems of balance laws. Such a framework is suited to almost any equation describing the motion of a fluid or a plasma. This article deals with a well-known issue of high-order well-balanced schemes. Indeed, such high-order schemes are based on a polynomial reconstruction, which must preserve the steady states under consideration in order to get the required well-balancedness property. A priori, to capture such a steady state, one needs to solve some strongly nonlinear equations. Here, a very easy, linear correction is designed under the generic framework of a system of hyperbolic balance laws, which describe most of the fluid or plasma systems. This correction can be applied to any scheme of order greater than or equal to 2, such as a MUSCL-type scheme, and ensures that this scheme exactly preserves the steady solutions. The main discrepancy with usual techniques lies in never having to invert the nonlinear function governing the steady solutions.

In 11, we developed of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. 26 gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its

In 12, we have been interested, with M. Sigalotti and R. Robin, in the optimal design of stellarators, devices for the production of controlled nuclear fusion reactions, alternative to tokamaks. The confinement of the plasma is entirely achieved by a helical magnetic field created by the complex arrangement of coils powered by high currents around a toric domain. These coils describe a surface called “coil winding surface” (CWS). We modeled the design of the CWS as a shape optimization problem, so that the cost function reflects both the optimal plasma properties, through a least squares functional, and also manufacturability, through geometric terms involving the lateral surface and curvature of the CWS.

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.

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 17 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 collaboration with L. Almeida, J. Bellver Arnau, M. Duprez, G. Nadin, I. Mazari and N. Vauchelet, we pursued a series of works dedicated to the analysis and simulation of solutions of an optimal control problem motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the sterile insect technique (SIT) consists in releasing mosquitoes infected with a bacterium called Wolbachia into the environment, which considerably reduces the transmission of the virus to humans. The goal is to effectively release the mosquitoes spatially so that the population of infected ones overwhelms the population of uninfected mosquitoes. Assuming very high fecundity rates, an asymptotic model on the proportion of infected mosquitoes is introduced, leading to an optimal control problem to determine the best spatial strategy to adopt.

We tackled the optimal strategy problem for SIT in 3 and 4, more specifically studying the robustness of optimal strategy for a general family of criteria and introduced an adapted optimization algorithm for computing optimal control strategies.

In addition, 10 is dedicated to the study of some qualitative properties of optimal control problems involving weighted eigenvalues and diffusion reaction systems used to describe the spatio-temporal dynamics of mosquito vectors of diseases such as dengue. This work, although fundamental and not dedicated to any particular application, have notably led to efficient optimization algorithms and analysis for the mentioned applied problems.

In 13, we were interested in the reconstruction of the shape of a parallelepipedic room from measurements made by microphones during a certain time. A difficulty of this work was the fact that only the low frequency part of the signal is measured. We introduced a model and an efficient reconstruction algorithm for such shapes. We now seek to generalize our approach to any shape of room.

In 6, algorithms specific to a particular application, epidemiology or quantum chemistry, were obtained using a precise analysis of the model and the optimality conditions.

Finally, 9 contains fine analysis results of the "geometric quantity" for the wave equation with an internal control, corresponding to the minimum time taken on average by a geodesic to intersect the control set.

We are involved in a common project with the company AxesSim in Strasbourg. The objective is to help to the development of a commercial software for the numerical simulation of electromagnetic phenomena. The applications are directed towards antenna design and electromagnetic compatibility. This project was partly supported by DGA through "RAPID" funds. A software engineer position (P. Gerhard) has started in 2021 on this subject. The objective is to implement a matrix-free CFL-less Discontinuous Galerkin (DG) scheme for solving Maxwell equations. In this way it is possible to get rid of the very small time steps imposed by small cells in automatically generated meshes. Preliminary results were very promising: stability and second order accuracy are observed for simulations with CFL numbers greater than 10. This project is supported by the Labex IRMIA++ (engineer position of P. Gerhard) and the French 2021 "Plan de relance": it allowed to host an AxesSim engineer in the Tonus team for one year for working on the joint DG solver.