TONUS 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, tokamak design also requires mathematical modelling 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 1 and SCHNAPS 2 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 modelling 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 interaction 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 dimensionality of the problem. Such models are implemented in GYSELA and Selalib.

On the boundary of the plasma, the collisions can no more be neglected. Fluid models, such as MagnetoHydroDynamics (MHD) become again relevant. 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 modelling 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:

The Strasbourg team 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) 32 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 allow solving the equations in the 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 22.

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 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.

But 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 an as small as possible set of kinetic velocities. 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 33, 20, 29, 28.

In tokamaks, the reduced model generally involves many time scales. Among these time scales, many of them, associated to the fastest waves, are not relevant. 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 Lax) number.

Precise resolution of the electromagnetic fields is essential for proper plasma simulation. 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 19 (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 and this requires implementing implicit time solvers in order to efficiently capture the stationary state, the solution of the magnetic induction equation or the Poisson equation.

The search for alternative energy sources is a major issue for the future. Among others, controlled thermonu-clear 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 of engineering.

It is clear now that future computers will be made of a collection of thousands of interconnected multicore processors. Globally it appears as a classical distributed memory MIMD machine. But at a lower level, each of the multicore processors is itself made of a shared memory MIMD unit (a few classical CPU cores) and a SIMD unit (a GPU). When designing new algorithms, it is important to adapt them to this kind of architecture. Our philosophy will be to program our algorithms in such a way that they can be run efficiently on this kind of computers. Practically, we will use the MPI library for managing the coarse grain parallelism, while the OpenCL library will efficiently operate the fine grain parallelism.

We have invested for several years until now into scientific computing on GPUs, using the open standard OpenCL (Open Computing Language). We were recently awarded a prize in the international AMD OpenCL innovation challenge thanks to an OpenCL two-dimensional Vlasov-Maxwell solver that fully runs on a GPU. OpenCL is a very interesting tool because it is an open standard now available on almost all brands of multicore processors and GPUs. The same parallel program can run on a GPU or a multicore processor without modification.

Because of the envisaged applications of CLAC, which may be either academic or commercial, it is necessary to conceive a modular framework. The heart of the library is made of generic parallel algorithms for solving conservation laws. The parallelism can be both fine-grained (oriented towards GPUs and multicore processors) and coarse-grained (oriented towards GPU clusters). The separate modules allow managing the meshes and some specific applications. In this way, it is possible to isolate parts that should be protected for trade secret reasons.

Participants: L. Bois, E. Franck, L. Navoret, V. Vigon (IRMA)

In this work, which was the object of a M2 internship, we have considered the Vlasov-Poisson equation for plasma physics. To reduce the CPU cost, it is usual to consider a model on the first moments in the velocity space (Euler equations). To obtain the final model it is important to add a closure (writing the last moment using the others). The classical closures are obtained by asymptotic analysis and are valid only on some regimes. In this work 13 we have proposed a closure based on a convolutional neural network call V-net and reference simulations obtained with a kinetic code. We have obtained a closure accuracy uniformly compared to the regime.

During two M1 internships, we have also tested the applications of Convolutional Neural Networks (CNNs) to the resolution of PDEs. The goal of the two M1 internships was to test the accuracy of convolutional networks like V-net to capture the solution of PDE problems for different applications. Regarding the M2 internship, its goal was to provide new closures for the Vlasov-Poisson equations.

In the first M1 internship, we proposed to construct a reduced model for tsunami propagation using data obtained solving the Shallow water equation in 1D. In the end, a first model, based on CNN, was able to predict the size and arrival time of the tsunami wave. In a second time, the CNN managed to predict the full wave which arrived on the beach. This work showed the interesting ability of CNNs to capture some physical effects and design cheaper qualitative models.

The second M1 internship deals with an inverse problem for the radiative transfer equations, which model the light propagation in a medium. In a first work, we consider a medium density constant and low with localized high density areas. Using simulations of the PDE, a first CNN model has been trained to localize the high density area using the results of light propagation only on the boundary. With a second neural network model, we have extended this method to the reconstruction of smooth densities. These two examples give a good idea of the CNN's ability to capture simplified physical solutions of the PDE.

We have been working on building new numerical methods, as well as a simulation framework, for fluid and/or plasma dynamics. The development of numerical methods have been focused on the multi-scale aspect of the equations, and more specifically on the low-Mach and low-beta regimes. Namely, we constructed so-called asymptotic-preserving schemes, which behave well in every scale of the simulation.

Participants: L. Mendoza

The tofu (Tomography for Fusion) library (see section 5.1.4) aims at providing the fusion and plasma community with a tool for designing tomography diagnostics, computing synthetic signal (direct problem) as well as tomographic inversions (inverse problem). In order to compute the radiated power from the plasma on a certain volume received by a particle it is essential to accurately sample that volume. This procedure can be time- and memory-consuming. Most of the coding time was dedicated to this issue this year. The code was parallelized using OpenMP. Updates were made to use the latest and open source tools of CI/CD to ensure long-term maintenance and reliability. Finally, with the increasing number of users from different regions of the plasma community, we integrated more tokamak configurations to the code (DEMO, Asdex Upgrade, NSX, etc.)

Participants: F. Bouchut (CNRS & Gustave Eiffel University), C. Courtès, E. Franck, V. Michel-Dansac, L. Navoret

Previously, we have proposed implicit relaxation methods for fluid models that allow us to reverse a simple system. However, previous methods 21 were not very effective in the multi-scale regimes of interest. We have proposed in 2 a semi-implicit scheme based on a dynamic splitting and a relaxation allows to compute efficiently the low Mach limit of the Euler equations. During this year we have studied the MHD extension for plasmas simulations. First, we have written the relaxation model and proven the stability of the system.Second, we have proposed a spatial scheme for the relaxation which allows us to treat with a good accuracy the low Mach et low Beta flows. Currently we will try to write the splitting for the relaxation problem and the original equations.

Participants: V. Michel-Dansac, A. Thomann (JGU Mainz, Germany)

When simulating the solutions to multi-scale problems, special care must be provided to the treatment of the different scales. This work aims at developing and applying IMEX (IMplicit-EXplicit) schemes to such problems, like the MHD system. The goal is twofold: developing generic schemes for a toy multi-scale problem, under physical constraints of stability and precision, and applying these schemes to complex systems. Following 25, 31, we have developed in 18 a new paradigm to build stable and accurate IMEX schemes for a toy scalar equation. The application to the Euler system, which represents flows in fluid dynamics, is underway and shows promising results.

Participants : Ph. Helluy, M. Houillon, P. Gerhard, L. Mendoza, V. Michel-Dansac, B. Bramas (CAMUS project-team)

The Discontinuous Galerkin method is a general method for solving conservation laws. This year, the end of the thesis of Marie Houillon culminated with the realization of a challenging numerical simulation on the Jean Zay supercomputer, using several hundreds of GPUs. The objective is to simulate the electromagnetic waves generated by a BlueTooth antenna around a human body: https://

Participants : Ph. Helluy, L. Navoret, E. Franck, M. Boileau, R. Hélie, L. Quibel, H. Baty (ObAS, Strasbourg), C. Klingenberg (JMU Würzburg, Germany)

This year, we continued to explore the wide possibilities of kinetic schemes, with applications to such areas as plasma physics, multiphase flows and multi-scale problems. We first applied kinetic schemes to plasma physics 6 and to multiphase flows 7. The kinetic scheme is extremely simple and allows us to implement very efficient software. Indeed, with a Lattice Boltzmann version of the scheme, we were able to reach 80% of the maximal bandwidth of high-end GPU for computing MHD flows. This allowed us to propose simulations on extremely fine meshes. In 7, in the context of multiphase flows, we propose the simulation of the explosion of a hot-water pressurized vessel. The flow is complex, with shock waves, phase transitions and high variations of physical quantities. The kinetic scheme was able to capture the phenomena with good robustness and accuracy.

While developing numerical tools mainly for plasma physics, it appears that these tools can also be used for other applications. We list below two of these applications.

Participants: L. Almeida (Sorbonne University), M. Duprez (MIMESIS project-team), R. Hélie, Y. Privat and N. Vauchelet (Paris 13 University)

We are interested in the analysis and simulation of solutions to optimal control problems motivated by population dynamics issues. In order to control the spread of mosquito-borne arboviruses, the population replacement technique consists in releasing into the environment mosquitoes infected with the Wolbachia bacterium, which greatly reduces the transmission of the virus to the humans. Spatial releases are then sought in such a way that the infected mosquito population invades the uninfected mosquito population. In order to determine the best temporal strategy for releases, we first considered time-dependent dynamic models 12, 11. Having made good progress on this issue, we now seek to determine time-space strategies on concrete territories.

We have introduced an appropriate model on the proportion of infected mosquitoes in time and space, and then an optimal control problem to determine the best spatial strategy to achieve these releases. This problem has been theoretically analysed, including the optimality of natural candidates and we have carried out first numerical simulations in one dimension of space to illustrate the relevance of our approach. This first work having been done 14, we are now seeking to design powerful numerical algorithms, allowing us to address this problem in dimensions 2 and 3 of space.

In parallel with this work, in the same spirit, we are seeking to determine how to optimally allocate resources in a habitat, when these are in limited quantities, in order to optimize the survival of the population, see 17, 16, 15

Participants: S. Lo Vecchio (IGBMC, Strasbourg), R. Thiagarajan (IGBMC, Strasbourg), D. Caballero (IGBMC, Strasbourg), V. Vigon (IGBMC, Strasbourg), L. Navoret, R. Voituriez (Sorbonne University), D. Riveline (IGBMC, Strasbourg)

In collaboration with physicists, we are interested in the individual movements of cells and how migration can result from asymmetries in the cell environment. The experiment consists in cells moving over triangular adhesion zones arranged in a line. Focal contacts of cells on these adhesion zones were detected and analysed. These data allowed us to validate in 5 a geometric model that predicts the direction of migration from the geometry of the accessible adhesive zones to the cells.

E. Franck and L. Navoret gave online talks for the “fête de la science”.