Research

Numerical methods for fluids and plasma dynamics

Practical and theoretical works around the kinetic and Lattice Boltzmann methods

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.

Extension of the hydrostatic reconstruction to moving steady solutions

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.

Composition schemes for the guiding-center model

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.

Thermodynamically compatible discretization of a compressible two-fluid model with two entropy inequalities

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.

Hard congestion limit of the dissipative Aw-Rascle system

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.

Implicit-explicit solver for a two-fluid single-temperature model

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.

Two-Dimensional Linear Implicit Relaxed Scheme for Hyperbolic Conservation Laws

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.

Around machine learning and numerical schemes

Enrichment of Discontinuous Galerkin bases with Physics-Informed Neural Networks

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.

An optimal control framework for adaptive neural ordinary differential equations

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.

Deep optimal control to design hybrid numerical schemes

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.

Modelling solar coronal magnetic fields with physics-informed neural networks

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.

Reduced modeling and ML

Hamiltonian reduced model for wave equations

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.

Hyperbolic reduced model for Vlasov-Poisson equation with Fokker-Planck collision

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.

Other applications

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.

Micromagnetic simulations of the size dependence of the Curie temperature in ferromagnetic nanowires and nanolayers

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 $T_C$ that are in line with the experimental values for cobalt, iron and nickel and for finite-sized objects such as nanowires (1D) and nanolayers (2D), we study the variances of the Curie temperature with respect to the smallest size of the system. Moreover, optimization and parallelization of the python code solving the Landau-Lifshitz-Gilbert equation in the finite-temperature regime led to a 100-time acceleration of the code. We were therefore able to study the effect of the size of the material on the magnetization, in particular the value of the Curie temperature.

Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples

Considering a ferromagnetic material of ellipsoidal shape, the associated magnetic moment then has two asymptotically stable opposite equilibria, of the form $\pm \overline{m}$ . In order to use these materials for memory storage purposes, it is necessary to know how to control the magnetic moment. In 23, we use as a control variable a spatially uniform external magnetic field and consider the question of flipping the magnetic moment, i.e., changing it from the $+\overline{m}$ configuration to the $–\overline{m}$ one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic field used. We therefore include a constraint on the $L^{\infty }$ norm of the controls, assumed to be less than a threshold value $U$ . We show that, generically with respect to the dimensions of the ellipsoid, there is a minimal value of $U$ for this problem to have a solution. We then characterize it precisely. Finally, we investigate some particular configurations associated to geometries enjoying symmetry properties and show that in this case the magnetic moment can be controlled in minimal time without imposing a threshold condition on $U$ . This type of phenomenon (existence of a minimum time only if the control is powerful enough and non-controllability otherwise) seems new and leads to interesting extensions for more complex systems.

Reduced modeling and optimal control of epidemiological individual-based models with contact heterogeneity

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.

Numerical analysis of a shape optimization algorithm

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 $L^{\infty }–L^1$ optimal control problems. In this paper we consider the optimisation of the Dirichlet energy, of Dirichlet eigenvalues and of certain non-energetic problems. Our proofs rely on new diagonalisation procedure for shape Hessian matrices in optimal control problems, which leads to local stability estimates.

Optimal scenario for road evacuation in an urban environment

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.

Spontaneous rotations in epithelia as an interplay between cell polarity and RhoA activity at boundaries

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.