Publications HAL

2023

Journal articles

ref_biblio: Anna Chiara Lai, Paola Loreti, Michel Mehrenberger. Observability of a string-beams network with many beams. ESAIM: Control, Optimisation and Calculus of Variations, 2023, 29, pp.61. ⟨10.1051/cocv/2023054⟩. ⟨hal-04175250⟩
resume: We prove the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [C. Baiocchi, V. Komornik and P. Loreti, Acta Math. Hung. 97 (2002) 55–95.]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [K. Ammari, M. Jellouli and M. Mehrenberger, Networks Heterogeneous Media 4 (2009) 2009.] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present.
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Preprints, Working Papers, ...

ref_biblio: Anh-Tuan Vu, Michel Mehrenberger. Semi-Lagrangian Vlasov-Poisson solvers with a strong external uniform magnetic field. 2023. ⟨hal-04016348v2⟩
resume: In this article, we numerically solve the long-time Vlasov-Poisson system with a strong external magnetic field. For that, we consider a backward semi-Lagrangian method as follows: we first propose an approximation of the characteristics based on first and second order explicit numerical schemes; then, a 4-D interpolation is performed to update a numerical unknown. We show that when the magnitude of the external magnetic field becomes large while the time step is independent of the fast oscillation in time, this scheme is able to provide a consistent semi-Lagrangian discretization of the guiding center model. In order to avoid 4-D interpolation, we apply a splitting scheme suited for strong magnetic field which is inspired by J. Ameres [1] but uses the semi-Lagrangian solver instead of a Fourier spectral discretization solver. Finally, we present some numerical simulations to validate the capabilities and limits of the methods under the Kelvin-Helmholtz instability test case.
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2021

Journal articles

ref_biblio: C Yang, Michel Mehrenberger. Highly accurate monotonicity-preserving Semi-Lagrangian scheme for Vlasov-Poisson Simulations. Journal of Computational Physics, In press, ⟨10.1016/j.jcp.2021.110632⟩. ⟨hal-03126595v2⟩
resume: In this paper, we study a high accurate monotonicity-preserving (MP) Semi-Lagrangian scheme for Vlasov-Poisson simulations. The classical Semi-Lagrangian scheme is known to be high accurate and free from CFL condition, but it does not satisfy local maximum principle. To remedy this drawback, using the conservative form of the Semi-Lagrangian scheme, we recast existing MP schemes for the numerical flux in a common framework, and then substitute the local minimum/maximum by some "better" guess, in order to avoid as much as possible loss of accuracy and clipping near extrema, while keeping the monotonicity on monotone portions. With the limiter, on the one hand, the scheme keeps the good properties of the unlimited scheme: it is conservative, free from CFL condition and high accurate. On the other hand, for locally monotonic data, the monotonicity of the solution is preserved. Numerical tests are made on free transport equation and Vlasov-Poisson system to illustrate the robustness of our method.
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2020

Journal articles

ref_biblio: Michel Mehrenberger, Laurent Navoret, Nhung Pham. Recurrence phenomenon for Vlasov-Poisson simulations on regular finite element mesh. Communications in Computational Physics, 2020, ⟨10.4208/cicp.OA-2019-0022⟩. ⟨hal-01942708⟩
resume: In this paper, we focus on one difficulty arising in the numerical simulation of the Vlasov-Poisson system: when using a regular grid-based solver with periodic boundary conditions, perturbations present at the initial time artificially reappear at a later time. For regular finite-element mesh in velocity, we show that this recurrence time is actually linked to the spectral accuracy of the velocity quadrature when computing the charge density. In particular, choosing trigonometric quadrature weights optimally defers the occurence of the recurrence phenomenon. Numerical results using the Semi- Lagrangian Discontinuous Galerkin and the Finite Element / Semi-Lagrangian method confirm the analysis.
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ref_biblio: Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic and Related Models , 2020, 13 (1), pp.129-168. ⟨10.3934/krm.2020005⟩. ⟨hal-02070138v3⟩
resume: The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the 1D×1D and 2D×2D Vlasov-Poisson system illustrate the effectiveness of this approach.
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2019

Journal articles

ref_biblio: David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, Laurent Navoret. High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation. Computers and Fluids, 2019, 190, pp.485--502. ⟨10.1016/j.compfluid.2019.06.007⟩. ⟨hal-01706614⟩
resume: We construct a high order discontinuous Galerkin method for solving general hyperbolic systems of conservation laws. The method is CFL-less, matrix-free, has the complexity of an explicit scheme and can be of arbitrary order in space and time. The construction is based on: (a) the representation of the system of conservation laws by a kinetic vectorial representation with a stiff relaxation term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport solver; and (c) a stiffly accurate composition method for time integration. The method is validated on several one-dimensional test cases. It is then applied on two-dimensional and three-dimensional test cases: flow past a cylinder, magnetohydrodynamics and multifluid sedimentation.
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ref_biblio: Ksander Ejjaaouani, Olivier Aumage, Julien Bigot, Michel Mehrenberger, Hitoshi Murai, et al.. InKS, a Programming Model to Decouple Algorithm from Optimization in HPC Codes. Journal of Supercomputing, 2019, ⟨10.1007/s11227-019-02950-2⟩. ⟨hal-02281963⟩
resume: Existing programming models tend to tightly interleave algorithm and optimization in HPC simulation codes. This requires scientists to become experts in both the simulated domain and the optimization process and makes the code difficult to maintain or port to new architectures. In this paper, we propose the InKS programming model that decouples these concerns with two distinct languages: InKS pia to express the simulation algorithm and InKS pso for optimizations. We define InKS pia and evaluate the feasibility of defining InKS pso with three test-languages: InKS o/C++ , InKS o/loop , InKS o/XMP. We evaluate the approach on synthetic benchmarks (NAS and heat equation) as well as on a more complex example (6D Vlasov-Poisson solver). Our evaluation demonstrates the soundness of the approach as it improves the separation of algorithmic and optimization concerns at no performance cost. We also identify a set of guidelines for the later full definition of the InKS pso language.
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ref_biblio: Shuangxi Zhang, Michel Mehrenberger, Christophe Steiner. Computing the double-gyroaverage term incorporating short-scale perturbation and steep equilibrium profile by the interpolation algorithm. plasma, 2019, 2 (2), pp.91-126. ⟨10.3390/plasma2020009⟩. ⟨hal-02009446v2⟩
resume: In gyrokinetic model and simulations, when the double-gyroaverage term incorporates the combining effect contributed by the finite Larmor radius, short-scales of the perturbation and steep gradient of the equilibrium profile, the low-order approximation of this term could generate unignorable error. This paper implements an interpolation algorithm to compute the double-gyroaverage term without low-order approximation to avoid this error. For a steep equilibrium density, the obvious difference between the density on the gyrocenter coordinate frame and the one on the particle coordinate frame should be accounted for in the quasi-neutrality equation. An Euler-Maclaurin-based quadrature integrating algorithm is developed to compute the quadrature integral for the distribution of the magnetic moment. The application of the interpolation algorithm to computing the double-gyroaverage term and to solving the quasi-neutrality equation is benchmarked by comparing the numerical results with the known analytical solutions. At last, to make the advantage of the interpolation solver clearer, the numerical comparison between the interpolation solver and a classical second order solver is carried out in a constant theta-pinch magnetic field configuration using SELALIB code. When the equilibrium profile is not steep and the perturbation only has the nonzero mode number along the parallel spatial dimension, the results computed by the two solvers match each other well. When the gradient of the equilibrium profile is steep, the interpolation solver provides a bigger driving effect for the ion-temperature-gradient modes which possess large polar mode numbers.
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ref_biblio: Roberto Ferretti, Michel Mehrenberger. STABILITY OF SEMI-LAGRANGIAN SCHEMES OF ARBITRARY ODD DEGREE UNDER CONSTANT AND VARIABLE ADVECTION SPEED. Mathematics of Computation, In press, ⟨10.1090/mcom/3494⟩. ⟨hal-02302625⟩
resume: The equivalence between semi-Lagrangian and Lagrange-Galerkin schemes has been proved in [9, 10] for the case of centered Lagrange interpolation of odd degree p ≤ 13. We generalize this result to an arbitrary odd degree, for both the case of constant-and variable-coefficient equations. In addition, we prove that the same holds for spline interpolations.
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2018

Journal articles

ref_biblio: Mehdi Badsi, Michel Mehrenberger, Laurent Navoret. Numerical stability of plasma sheath. ESAIM: Proceedings and Surveys, 2018, ⟨10.1051/proc/201864017⟩. ⟨hal-01676656⟩
resume: We are interested in developing a numerical method for capturing stationary sheaths, that a plasma forms in contact with a metallic wall. This work is based on a bi-species (ion/electron) Vlasov-Ampère model proposed in [3]. The main question addressed in this work is to know if classical numerical schemes can preserve stationary solutions with boundary conditions, since these solutions are not a priori conserved at the discrete level. In the context of high-order semi-Lagrangian method, due to their large stencil, interpolation near the boundary of the domain requires also a specific treatment.
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ref_biblio: Nicolas Bouzat, Camilla Bressan, Virginie Grandgirard, Guillaume Latu, Michel Mehrenberger. Targeting realistic geometry in Tokamak code Gysela *. ESAIM: Proceedings and Surveys, 2018, ⟨10.1051/proc/201863179⟩. ⟨hal-01653022⟩
resume: In magnetically confined plasmas used in Tokamak, turbulence is responsible for specific transport that limits the performance of this kind of reactors. Gyrokinetic simulations are able to capture ion and electron turbulence that give rise to heat losses, but require also state-of-the-art HPC techniques to handle computation costs. Such simulations are a major tool to establish good operating regime in Tokamak such as ITER, which is currently being built. Some of the key issues to address more realistic gyrokinetic simulations are: efficient and robust numerical schemes, accurate geometric description, good parallelization algorithms. The framework of this work is the Semi-Lagrangian setting for solving the gyrokinetic Vlasov equation and the Gysela code. In this paper, a new variant for the interpolation method is proposed that can handle the mesh singularity in the poloidal plane at r = 0 (polar system is used for the moment in Gysela). A non-uniform meshing of the poloidal plane is proposed instead of uniform one in order to save memory and computations. The interpolation method, the gyroaverage operator, and the Poisson solver are revised in order to cope with non-uniform meshes. A mapping that establish a bijection from polar coordinates to more realistic plasma shape is used to improve realism. Convergence studies are provided to establish the validity and robustness of our new approach.
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ref_biblio: Yann A Barsamian, Joackim Bernier, Sever Adrian Hirstoaga, Michel Mehrenberger. Verification of 2D × 2D and two-species Vlasov-Poisson solvers. ESAIM: Proceedings and Surveys, 2018, 63, pp.78-108. ⟨10.1051/proc/201863078⟩. ⟨hal-01668744⟩
resume: In [18], 1D × 1D two-species Vlasov-Poisson simulations are performed by the semi-Lagrangian method. Thanks to a classical first order dispersion analysis, we are able to check the validity of their simulations; the extension to second order is performed and shown to be relevant for explaining further details. In order to validate multi-dimensional effects, we propose a 2D × 2D single species test problem that has true 2D effects coming from the sole second order dispersion analysis. Finally, we perform, in the same code, full 2D × 2D non linear two-species simulations with mass ratio √ 0.01, and consider the mixing of semi-Lagrangian and Particle-in-Cell methods.
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ref_biblio: Guillaume Latu, Michel Mehrenberger, Yaman Güçlü, Maurizio Ottaviani, Eric Sonnendrücker. Field-aligned interpolation for semi-Lagrangian gyrokinetic simulations. Journal of Scientific Computing, 2018, 74 (3), pp.1601-1650. ⟨10.1007/s10915-017-0509-5⟩. ⟨hal-01315889v3⟩
resume: This work is devoted to the study of field-aligned interpolation in semi-Lagrangian codes. 1 In the context of numerical simulations of magnetic fusion devices, this approach is motivated by the observation that gradients of the solution along the magnetic field lines are typically much smaller than along a perpendicular direction. In toroidal geometry, field-aligned interpolation consists of a 1D interpolation along the field line, combined with 2D interpolations on the poloidal planes (at the intersections with the field line). A theoretical justification of the method is provided in the simplified context of constant advection on a 2D periodic domain: unconditional stability is proven, and error estimates are given which highlight the advantages of field-aligned interpolation. The same methodology is successfully applied to the solution of the gyrokinetic Vlasov equation, for which we present the ion temperature gradient (ITG) instability as a classical test-case: first we solve this in cylindrical geometry (screw-pinch), and next in toroidal geometry (circular Tokamak). In the first case, the algorithm is implemented in Selalib (semi-Lagrangian library), and the numerical simulations provide linear growth rates that are in accordance with the linear dispersion analysis. In the second case, the algorithm is implemented in the Gysela code, and the numerical simulations are benchmarked with those employing the standard (not aligned) scheme. Numerical experiments show that field-aligned interpolation leads to considerable memory savings for the same level of accuracy; substantial savings are also expected in reactor-scale simulations.
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Conference papers

ref_biblio: Ksander Ejjaaouani, Olivier Aumage, Julien Bigot, Michel Mehrenberger, Hitoshi Murai, et al.. InKS, a Programming Model to Decouple Performance from Algorithm in HPC Codes. Repara 2018 - 4th International Workshop on Reengineering for Parallelism in Heterogeneous Parallel Platforms, Aug 2018, Turin, Italy. pp.1-12, ⟨10.1007/978-3-030-10549-5_59⟩. ⟨hal-01890132⟩
resume: Existing programming models tend to tightly interleave algorithm and optimization in HPC simulation codes. This requires scientists to become experts in both the simulated domain and the optimization process and makes the code difficult to maintain and port to new architectures. This paper proposes the InKS programming model that decouples these two concerns with distinct languages for each. The simulation algorithm is expressed in the InKS pia language with no concern for machine-specific optimizations. Optimizations are expressed using both a family of dedicated optimizations DSLs (InKS O) and plain C++. InKS O relies on the InKS pia source to assist developers with common optimizations while C++ is used for less common ones. Our evaluation demonstrates the soundness of the approach by using it on synthetic benchmarks and the Vlasov-Poisson equation. It shows that InKS offers separation of concerns at no performance cost.
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ref_biblio: Yann A Barsamian, Arthur Charguéraud, Sever Adrian Hirstoaga, Michel Mehrenberger. Efficient Strict-Binning Particle-in-Cell Algorithm for Multi-Core SIMD Processors. Euro-Par 2018 - 24th International European Conference on Parallel and Distributed Computing, Aug 2018, Turin, Italy. ⟨10.1007/978-3-319-96983-1_53⟩. ⟨hal-01890318⟩
resume: Particle-in-Cell (PIC) codes are widely used for plasma simulations. On recent multi-core hardware, performance of these codes is often limited by memory bandwidth. We describe a multi-core PIC algorithm that achieves close-to-minimal number of memory transfers with the main memory, while at the same time exploiting SIMD instructions for numerical computations and exhibiting a high degree of OpenMP-level parallelism. Our algorithm keeps particles sorted by cell at every time step, and represents particles from a same cell using a linked list of fixed-capacity arrays, called chunks. Chunks support either sequential or atomic insertions, the latter being used to handle fast-moving particles. To validate our code, called Pic-Vert, we consider a 3d electrostatic Landau-damping simulation as well as a 2d3v transverse instability of magnetized electron holes. Performance results on a 24-core Intel Sky-lake hardware confirm the effectiveness of our algorithm, in particular its high throughput and its ability to cope with fast moving particles.
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2017

Journal articles

ref_biblio: Fernando Casas, Nicolas Crouseilles, Erwan Faou, Michel Mehrenberger. High-order Hamiltonian splitting for Vlasov-Poisson equations. Numerische Mathematik, 2017, 135 (3), pp.769-801. ⟨10.1007/s00211-016-0816-z⟩. ⟨hal-01206164⟩
resume: We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nyström type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.
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Conference papers

ref_biblio: David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, Laurent Navoret. Palindromic Discontinuous Galerkin Method. Finite volumes for complex applications VIII—hyperbolic, elliptic and parabolic problems, Jun 2017, Lille, France. pp.6, ⟨10.1007/978-3-319-57394-6_19⟩. ⟨hal-01653049⟩
resume: We present a high-order scheme for approximating kinetic equations with stiff relaxation. The construction is based on a high-order, implicit, upwind Discontinuous Galerkin formulation of the transport equations. In practice, because of the triangular structure of the implicit system, the computations are explicit. High order in time is achieved thanks to a palindromic composition method. The whole method is asymptotic-preserving with respect to the stiff relaxation and remains stable even with large CFL numbers.
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Poster communications

ref_biblio: Laurent Navoret, Michel Mehrenberger. About recurrence time for a semi-Lagrangian discontinuous Galerkin Vlasov solver. Collisionless Boltzmann (Vlasov) Equation and Modeling of Self-Gravitating Systems and Plasmas, Oct 2017, Marseille, France. ⟨hal-01653023⟩
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Preprints, Working Papers, ...

ref_biblio: Olivier Aumage, Julien Bigot, Ksander Ejjaaouani, Michel Mehrenberger. InKS, a programming model to decouple performance from semantics in simulation codes. 2017. ⟨cea-01493075⟩
resume: Existing programming models lead to a tight inter-leaving of semantics and computer optimization concerns in high-performance simulation codes. With the increasing complexity and heterogeneity of super-computers this requires scientists to become experts in both the simulated domain and the optimization process and makes the code difficult to maintain and port to new architectures. This report proposes InKS, a programming model that aims to improve the situation by decoupling semantics and optimizations in code so as to ease the collaboration between domain scientists and expert of high-performance optimizations. We define the InKS language that enables developers to describe the semantic of a simulation code with no concern for performance. We describe the implementation of a compiler able to automatically execute this InKS code without making any explicit execution choice. We also describe a method to manually specify these choices to reach high-performance. Our preliminary evaluation on a 3D heat equation solver demonstrates the feasibility of the automatic approach as well as the ability to specify complex optimizations while not altering the semantic part. It shows promising performance where two distinct specifications of optimization choices in InKS offer similar performance as existing hand-tailored versions of the solver.
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2016

Journal articles

ref_biblio: Fabien Rozar, Christophe Steiner, Guillaume Latu, Michel Mehrenberger, Virginie Grandgirard, et al.. Optimization of the gyroaverage operator based on hermite interpolation. ESAIM: Proceedings and Surveys, 2016, ⟨10.1051/proc/201653012⟩. ⟨hal-01261427⟩
resume: Gyrokinetic modeling is appropriate for describing Tokamak plasma turbulence, and the gyroaverage operator is a cornerstone of this approach. In a gyrokinetic code, the gyroaveraging scheme needs to be accurate enough to avoid spoiling the data but also requires a low computation cost because it is applied often on the main unknown, the 5D guiding-center distribution function, and on the 3D electric potentials. In the present paper, we improve a gyroaverage scheme based on Hermite interpolation used in the Gysela code. This initial implementation represents a too large fraction of the total execution time. The gyroaverage operator has been reformulated and is now expressed as a matrix-vector product and a cache-friendly algorithm has been setup. Different techniques have been investigated to quicken the computations by more than a factor two. Description of the algorithms is given, together with an analysis of the achieved performance.
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ref_biblio: Virginie Grandgirard, Jérémie Abiteboul, Julien Bigot, Thomas Cartier-Michaud, Nicolas Crouseilles, et al.. A 5D gyrokinetic full-f global semi-lagrangian code for flux-driven ion turbulence simulations. Computer Physics Communications, 2016, 207, pp.35-68. ⟨10.1016/j.cpc.2016.05.007⟩. ⟨cea-01153011v3⟩
resume: This paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic Gysela code is one of the few international 5D gyrokinetic codes able to perform global, full-f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the Gysela code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties.
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ref_biblio: Adnane Hamiaz, Michel Mehrenberger, Aurore Back, Pierre Navaro. Guiding center simulations on curvilinear grids. ESAIM: Proceedings, 2016, 53, pp.99-119. ⟨10.1051/proc/201653007⟩. ⟨hal-00908500v3⟩
resume: Semi-Lagrangian guiding center simulations are performed on sinusoidal perturbations of cartesian grids, thanks to the use of a B-spline finite element solver for the Poisson equation and the classical backward semi-Lagrangian method (BSL) for the advection. We are able to reproduce the standard Kelvin-Helmholtz instability test on such grids. When the perturbation leads to a strong distorted mesh, we observe that the solution differs if one takes standard numerical parameters that are used in the cartesian reference case. We can recover good results together with correct mass conservation, by diminishing the time step.
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ref_biblio: Adnane Hamiaz, Michel Mehrenberger, Hocine Sellama, Eric Sonnendrücker. The semi-Lagrangian method on curvilinear grids. Communications in Applied and Industrial Mathematics, 2016, ⟨10.1515/caim-2016-0024⟩. ⟨hal-01213366⟩
resume: We study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.
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Preprints, Working Papers, ...

ref_biblio: David Coulette, Emmanuel Franck, Philippe Helluy, Michel Mehrenberger, Laurent Navoret. Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation. 2016. ⟨hal-01422922⟩
resume: We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several ingredients: (i) a high order implicit upwind Discontinuous Galerkin approximation of the kinetic equations with easy-to-solve triangular linear systems; (ii) a second order asymptotic-preserving time integration based on symmetry arguments; (iii) a palindromic composition of the second order method for achieving higher orders in time. The method is then tested at orders 2, 4 and 6. It is asymptotic-preserving with respect to the stiff relaxation and accepts high CFL numbers.
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2015

Journal articles

ref_biblio: Christophe Steiner, Michel Mehrenberger, Nicolas Crouseilles, Virginie Grandgirard, Guillaume Latu, et al.. Gyroaverage operator for a polar mesh. The European Physical Journal D : Atomic, molecular, optical and plasma physics, 2015, 69 (1), pp.221. ⟨10.1140/epjd/e2014-50211-7⟩. ⟨hal-01090681⟩
resume: In this work, we are concerned with numerical approximation of the gyroaverage operators arising in plasma physics to take into account the effects of the finite Larmor radius corrections. The work initiated in [5] is extended here to polar geometries. A direct method is proposed in the space configuration which consists in integrating on the gyrocircles using interpolation operator (Hermite or cubic splines). Numerical comparisons with a standard method based on a Padé approximation are performed: (i) with analytical solutions, (ii) considering the 4D drift-kinetic model with one Larmor radius and (iii) on the classical linear DIII-D benchmark case [6]. In particular, we show that in the context of a drift-kinetic simulation, the proposed method has similar computational cost as the standard method and its precision is independent of the radius. PACS. PACS-key discribing text of that key – PACS-key discribing text of that key
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ref_biblio: Michel Mehrenberger, Laura S. Mendoza, Charles Prouveur, Eric Sonnendrücker. Solving the guiding-center model on a regular hexagonal mesh. ESAIM: Proceedings and Surveys, 2015, 53, pp.149-176. ⟨10.1051/proc/201653010⟩. ⟨hal-01117196v3⟩
resume: This paper introduces a Semi-Lagrangian solver for the Vlasov-Poisson equations on a uniform hexagonal mesh. The latter is composed of equilateral triangles, thus it doesn't contain any singularities, unlike polar meshes. We focus on the guiding-center model, for which we need to develop a Poisson solver for the hexagonal mesh in addition to the Vlasov solver. For the interpolation step of the Semi-Lagrangian scheme, a comparison is made between the use of Box-splines and of Hermite finite elements. The code will be adapted to more complex models and geometries in the future.
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Conference papers

ref_biblio: Vilmos Komornik, Paola Loreti, Michel Mehrenberger. Observability of a Ring Shaped Membrane via Fourier Series. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.322-330, ⟨10.1007/978-3-319-55795-3_30⟩. ⟨hal-01626896⟩
resume: We study the inverse Ingham type inequality for a wave equation in a ring. This leads to a conjecture on the zeros of Bessel cross product functions. We motivate the validity of the conjecture through numerical results. We do a complete analysis in the particular case of radial initial data, where an improved time of observability is available.
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Preprints, Working Papers, ...

ref_biblio: Christophe Steiner, Michel Mehrenberger, Nicolas Crouseilles, Philippe Helluy. Quasi-neutrality equation in a polar mesh. 2015. ⟨hal-01248179⟩
resume: In this work, we are concerned with the numerical resolution of the quasi-neutrality equation arising in plasma physics. A classic method is based on a Padé approximation. Two other methods are proposed in this paper: a Padé approximation of higher order and a direct method in the space configuration which consists in integrating on the gyrocircles using interpolation operator. Numerical comparisons are performed with analytical solutions and considering the 4D drift-kinetic model with one Larmor radius. This is a preliminary study; further study in GYSELA is envisioned.
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2014

Journal articles

ref_biblio: Bedros Afeyan, Fernando Casas, Nicolas Crouseilles, Adila Dodhy, Erwan Faou, et al.. Simulations of Kinetic Electrostatic Electron Nonlinear (KEEN) Waves with Variable Velocity Resolution Grids and High-Order Time-Splitting. The European Physical Journal D : Atomic, molecular, optical and plasma physics, 2014, 68 (10), DOI: 10.1140/epjd/e2014-50212-6. ⟨10.1140/epjd/e2014-50212-6⟩. ⟨hal-00977344v2⟩
resume: KEEN waves are nonlinear, non-stationary, self-organized asymptotic states in Vlasov plasmas outside the scope or purview of linear theory constructs such as electron plasma waves or ion acoustic waves. Nonlinear stationary mode theories such as those leading to BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force used to drive them. Smaller amplitude drives create highly localized structures attempting to coalesce into KEEN waves. These cases have much more chaotic and intricate time histories than strongly driven ones. The narrow range in which one must maintain adequate velocity resolution in the weakly driven cases challenges xed grid numerical schemes. What is missing there is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational e ort. This is needed for low amplitude runs which take a long time to set up KEEN waves, if they are able to do so at all. The new code's performance is compared to uniform grid simulations and the advantages quanti ed. The birth pains associated with KEEN waves which are weakly driven is captured in these simulations. These techniques allow the e cient simulation of KEEN waves in multiple dimensions which will be tackled next as well as generalizations to Vlasov-Maxwell codes which are essential to understanding the impact of KEEN waves in practice.
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ref_biblio: Nicolas Crouseilles, Pierre Glanc, Sever Adrian Hirstoaga, Eric Madaule, Michel Mehrenberger, et al.. A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence. The European Physical Journal D : Atomic, molecular, optical and plasma physics, 2014, 68 (9), DOI: 10.1140/epjd/e2014-50180-9. ⟨10.1140/epjd/e2014-50180-9⟩. ⟨hal-00977342v2⟩
resume: While developing a new semi-Lagrangian solver, the gap between a linear Landau run in 1Dx1Dand a 5D gyrokinetic simulation in toroidal geometry is quite huge. Intermediate test cases are welcomefor testing the code. A new fully two-dimensional conservative semi-Lagrangian (CSL) method is presentedhere and is validated on 2D polar geometries. We consider here as building block, a 2D guiding-centertype equation on an annulus and apply it on two test cases. First, we revisit a 2D test case previouslydone with a PIC approach [18] and detail the boundary conditions. Second, we consider a 4D drift-kineticslab simulation (see [10]). In both cases, the new method appears to be a good alternative to deal withthis type of models since it improves the lack of mass conservation of the standard semi-Lagrangian (BSL)method.
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ref_biblio: Mohamed Jellouli, Michel Mehrenberger. Optimal decay rates for the stabilization of a string network. Comptes Rendus. Mathématique, 2014, ⟨10.1016/j.crma.2014.03.023⟩. ⟨hal-00977345⟩
resume: We study the decay of the energy for a degenerate network of strings, and obtain optimal decay rates when the lengths are all equal. We also de ne a classical space semi-discretization and compare the results with the exact method.
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ref_biblio: Guillaume Latu, Virginie Grandgirard, Jérémie Abiteboul, Nicolas Crouseilles, Guilhem Dif-Pradalier, et al.. Improving conservation properties in a 5D gyrokinetic semi-Lagrangian code. The European Physical Journal D : Atomic, molecular, optical and plasma physics, 2014, 68 (11), pp.345. ⟨10.1140/epjd/e2014-50209-1⟩. ⟨hal-00966162⟩
resume: In gyrokinetic turbulent simulations, the knowledge of some stationary states can help reducing numerical artifacts. Considering long-term simulations, the qualities of the Vlasov solver and of the radial boundary conditions have an impact on the conservation properties. In order to improve mass and energy conservation mainly, the following methods are investigated: fix the radial boundary conditions on a stationary state, use a 4D advection operator that avoids a directional splitting, interpolate with a delta-f approach. The combination of these techniques in the semi-Lagrangian code GYSELA leads to a net improvement of the conservation properties in 5D simulations.
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Reports

ref_biblio: Guillaume Latu, Michel Mehrenberger, M Ottaviani, Eric Sonnendrücker. Aligned interpolation and application to drift kinetic semi-Lagrangian simulations with oblique magnetic field in cylindrical geometry. [Research Report] IRMA. 2014. ⟨hal-01098373⟩
resume: We introduce field aligned interpolation for Semi-Lagrangian schemes, adapting a method developed by Hariri-Ottaviani [7] to the semi-Lagrangian context. This approach is vali-dated on the constant oblique advection equation and on a 4D drift kinetic model with oblique magnetic field in cylindrical geometry. The strength of this method is that one can reduce the number of points in the longitudinal direction. More precisely, we observe that we gain a factor |n| |n+mι| (where ι is the inverse safety factor), with respect to the classical approach, for the typical function sin(mθ + nϕ).
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2013

Journal articles

ref_biblio: Luca Marradi, Bedros Afeyan, Michel Mehrenberger, Nicolas Crouseilles, Christophe Steiner, et al.. Vlasov on GPU (VOG Project). ESAIM: Proceedings, 2013, 43, p. 37-58. ⟨10.1051/proc/201343003⟩. ⟨hal-00908498⟩
resume: This work concerns the numerical simulation of the Vlasov-Poisson set of equations using semi- Lagrangian methods on Graphical Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Second, to overcome the limitation of single precision inherent in GPU, a {\delta}f type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order reconstruction methods and a large number of grid points in phase space). We show results for rather academic test cases on Landau damping and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.
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Poster communications

ref_biblio: Michel Mehrenberger, Nicolas Crouseilles, Eric Sonnendrücker, Bedros Afeyan. High-Order Numerical Methods for KEEN Wave Vlasov-Poisson Simulations. PPPS, Jun 2013, San Francisco, United States. ⟨10.1109/PLASMA.2013.6634958⟩. ⟨hal-01298979⟩
resume: KEEN waves [1] provided a challenging test case for Vlasov Poisson numerical solvers since they involve highly non stationary, multiple-harmonic self-organized kinetic states. They require high resolution in the phase space region around the phase velocity of the drive wave. Different interpolation strategies are discussed and compared to classical cubic splines.
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Reports

ref_biblio: Eric Madaule, Sever Adrian Hirstoaga, Michel Mehrenberger, Jérôme Pétri. Semi-Lagrangian simulations of the diocotron instability. [Research Report] 2013. ⟨hal-00841504⟩
resume: We consider a guiding center simulation on an annulus. We propose here to revisit this test case by using a classical semi-Lagrangian approach. First, we obtain the conservation of the electric energy and mass for some adapted boundary conditions. Then we recall the dispersion relation and discussions on diff erent boundary conditions are detailed. Finally, the semi-Lagrangian code is validated in the linear phase against analytical growth rates given by the dispersion relation. Also we have validated numerically the conservation of electric energy and mass. Numerical issues/diffi culties due to the change of geometry can be tackled in such a test case which thus may be viewed as a fi rst intermediate step between a classical guiding center simulation in a 2D cartesian mesh and a slab 4D drift kinetic simulation.
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Preprints, Working Papers, ...

ref_biblio: Christophe Steiner, Michel Mehrenberger, Daniel Bouche. A semi-Lagrangian discontinuous Galerkin convergence. 2013. ⟨hal-00852411⟩
resume: We show a superconvergence property for the Semi-Lagrangian Discontinuous Galerkin scheme of arbitrary degree in the case of constant linear advection equation with periodic boundary conditions.
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2012

Journal articles

ref_biblio: Nicolas Crouseilles, Pierre Glanc, Michel Mehrenberger, Christophe Steiner. Finite Volume Schemes for Vlasov. ESAIM: Proceedings, 2012, CEMRACS'11: Multiscale Coupling of Complex Models in Scientific Computing, 38, pp.275-297. ⟨10.1051/proc/201238015⟩. ⟨hal-00653038⟩
resume: We present finite volumes schemes for the numerical approximation of the one-dimensional Vlasov-Poisson equation (FOV CEMRACS 2011 project). Stability analysis is performed for the linear advection and links with semi-Lagrangian schemes are made. Finally, numerical results enable to compare the diff erent methods using classical plasma test cases.
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Habilitation à diriger des recherches

ref_biblio: Michel Mehrenberger. Inégalités d'Ingham et schémas semi-lagrangiens pour l'équation de Vlasov. Equations aux dérivées partielles [math.AP]. Université de Strasbourg, 2012. ⟨tel-00735678⟩
resume: Dans une première partie, on rassemble plusieurs résultats en théorie du contrôle autour des inégalités d'Ingham, généralisations de l'égalité de Parseval, qui inter- viennent pour montrer l'observabilité, la contrôlabilité ou la stabilisation frontière ou interne de l'équation des ondes ou d'équations similaires dans certains cas parti- culiers. On s'intéresse dans un premier temps à l'optimalité de ce type d'inégalités en généralisant un résultat précédent au cas vectoriel. On développe ensuite un théo- rème de type Ingham adapté pour traiter le cas d'une géométrie cartésienne. Enfin, on donne des résultats d'observabilité dans le cas d'approximations numériques. Dans une seconde partie, on présente les méthodes semi-Lagrangiennes qui sont composées essentiellement de deux ingrédients : calcul des caractéristiques le long desquelles la fonction de distribution est constante et étape d'interpolation. On ana- lyse des schémas d'ordre élevé en temps pour le système de Vlasov-Poisson 1D×1D, basés sur le splitting directionnel, qui est une succession d'étapes de transport li- néaire. On étudie alors les méthodes semi-Lagrangiennes dans ce cas particulier et on fait le lien entre différentes formulations. On obtient également un théorème de convergence pour le système de Vlasov-Poisson dans ce cadre, qui reste valable pour des petits déplacements. On développe ensuite ce type de méthodes dans un cadre plus général, en se basant sur le splitting uni-dimensionnel conservatif, avec une variante de type Galerkin discontinu. Dans une dernière partie, on étudie l'opérateur de gyromoyenne qui intervient en physique des plasmas pour prendre en compte des corrections de rayon de Larmor fini. Enfin, on discute de la problématique de la divergence discrète nulle qui donne une compatibilité entre le calcul du champ et la méthode numérique de transport.
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Poster communications

ref_biblio: Michel Mehrenberger, Nicolas Crouseilles, Virginie Grandgirard, Sever Adrian Hirstoaga, Eric Madaule, et al.. Berk-Breizman and diocotron instability testcases. AMVV Algorithm and Model Verification and Validation, Nov 2012, East-Lansing, United States. ⟨hal-01298988⟩
resume: Berk-Breizman and diocotron testcases are proposed.
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ref_biblio: Michel Mehrenberger, Morgane Bergot, Virginie Grandgirard, Guillaume Latu, Hocine Sellama, et al.. Conservative Semi-Lagrangian solvers on mapped meshes. ICOPS International Conference on Plasma Science, Jul 2012, Edinburgh, United Kingdom. 2012, ⟨10.1109/PLASMA.2012.6383972⟩. ⟨hal-00759823⟩
resume: We are interested in the numerical solution of the collisionless kinetic or gyrokinetic equations of Vlasov type needed for example for many problems in plasma physics. Different numerical methods are classically used, the most used is the Particle In Cell method, but Eulerian and Semi- Lagrangian (SL) methods that use a grid of phase space are also very interesting for some applications. Rather than using a uniform mesh of phase space which is mostly done, the structure of the solution, as a large variation of the gradients on different parts of phase space or a strong anisotropy of the solution, can sometimes be such that it is more interesting to use a more complex mesh. This is the case in particular for gyrokinetic simulations for magnetic fusion applications. We develop here a generalization of the Semi-Lagrangian method on mapped meshes. Classical Backward Semi-Lagrangian methods (BSL), Conservative Semi-Lagrangian methods based on one-dimensional splitting or Forward Semi- Lagrangian methods (FSL) have to be revisited in this case of mapped meshes. A first use of the classical advective BSL method on a mapped mesh has been described in 1. We consider here the problematic of conserving exactly some equilibrium of the distribution function, by using an adapted mapped mesh, which fits on the isolines of the Hamiltonian. This could be useful in particular for Tokamak simulations where instabilities around some equilibrium are investigated. We also consider the problem of mass conservation. In the cartesian framework, the FSL method automatically conserves the mass, as the advective and conservative form are shown to be equivalent. This does not remain true in the general curvilinear case. Numerical results are given on some gyrokinetic simulations performed with the GYSELA code and show the benefit of using a mass conservative scheme like the conservative version of the FSL scheme.
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Reports

ref_biblio: Guillaume Latu, Virginie Grandgirard, Jérémie Abiteboul, Morgane Bergot, Nicolas Crouseilles, et al.. Accuracy of unperturbed motion of particles in a gyrokinetic semi-Lagrangian code. [Research Report] RR-8054, INRIA. 2012, pp.17. ⟨hal-00727118⟩
resume: Inaccurate description of the equilibrium can yield to spurious effects in gyrokinetic turbulence simulations. Also, the Vlasov solver and time integration schemes impact the conservation of physical quantities, especially in long-term simulations. Equilibrium and Vlasov solver have to be tuned in order to preserve constant states (equilibrium) and to provide good conservation property along time (mass to begin with). Several illustrative simple test cases are given to show typical spurious effects that one can observes for poor settings. We explain why Forward Semi-Lagrangian scheme bring us some benefits. Some toroidal and cylindrical GYSELA runs are shown that use FSL.
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Preprints, Working Papers, ...

ref_biblio: Nicolas Crouseilles, Michel Mehrenberger, Francesco Vecil. A Discontinuous Galerkin semi-Lagrangian solver for the guiding-center problem. 2012. ⟨hal-00717155⟩
resume: In this paper, we test an innovative numerical scheme for the simulation of the guiding-center model, of interest in the domain of plasma physics, namely for fusion devices. We propose a 1D Discontinuous Galerkin (DG) discretization, whose basis are the Lagrange polynomials interpolating the Gauss points inside each cell, coupled to a conservative semi-Lagrangian (SL) strategy. Then, we pass to the 2D setting by means of a second-order Strangsplitting strategy. In order to solve the 2D Poisson equation on the DG discretization, we adapt the spectral strategy used for equally-spaced meshes to our Gauss-point-based basis. The 1D solver is validated on a standard benchmark for the nonlinear advection; then, the 2D solver is tested against the swirling deformation ow test case; nally, we pass to the simulation of the guiding-center model, and compare our numerical results to those given by the Backward Semi-Lagrangian method.
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2011

Journal articles

ref_biblio: Nicolas Crouseilles, Michel Mehrenberger, Francesco Vecil. Discontinuous Galerkin semi-Lagrangian method for Vlasov-Poisson. ESAIM: Proceedings, 2011, CEMRACS 2010, pp.21. ⟨10.1051/proc/2011022⟩. ⟨hal-00544677⟩
resume: We present a discontinuous Galerkin scheme for the numerical approximation of the one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.
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ref_biblio: Frédérique Charles, Bruno Després, Michel Mehrenberger. Enhanced convergence estimates for semi-lagrangian schemes Application to the Vlasov-Poisson equation. SIAM Journal on Numerical Analysis, 2011, ⟨10.1137/110851511⟩. ⟨inria-00629081⟩
resume: We prove enhanced error estimates for high order semi-lagrangian discretizations of the Vlasov-Poisson equation. It provides new insights into optimal numerical strategies for the numerical solution of this problem. The new error estimate is based on advanced error estimates for semi-lagrangian schemes, also equal to shifted Strang's schemes, for the discretization of the advection equation.
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ref_biblio: Kaïs Ammari, Michel Mehrenberger. Study of the nodal feedback stabilization of a string-beams network. Journal of Applied Mathematics and Computing, 2011, ⟨10.1007/s12190-010-0412-9⟩. ⟨hal-01298967⟩
resume: We consider a stabilization problem for a string-beams network. We prove an exponential decay result. The method used is based on a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. Moreover, we give a numerical illustration based on the methodology introduced in Ammari and Tucsnak (ESAIM Control Optim. Calc. Var. 6, 361–386, 2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system
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Preprints, Working Papers, ...

ref_biblio: Jean-Philippe Braeunig, Nicolas Crouseilles, Virginie Grandgirard, Guillaume Latu, Michel Mehrenberger, et al.. Some numerical aspects of the conservative PSM scheme in a 4D drift-kinetic code.. 2011. ⟨inria-00435203v2⟩
resume: The purpose of this work is simulation of magnetised plasmas in the ITER project framework. In this context, kinetic Vlasov-Poisson like models are used to simulate core turbulence in the tokamak in a toroidal geometry. This leads to heavy simulations because a 6D dimensional problem has to be solved, even if reduced to a 5D in so called gyrokinetic models. Accurate schemes, parallel algorithms need to be designed to bear these simulations. This paper describes the numerical studies to improve robustness of the conservative PSM scheme in the context of its development in the GYSELA code. In this paper, we only consider the 4D drift-kinetic model which is the backbone of the 5D gyrokinetic models and relevant to build a robust and accurate numerical method.
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ref_biblio: Nicolas Crouseilles, Erwan Faou, Michel Mehrenberger. High order Runge-Kutta-Nyström splitting methods for the Vlasov-Poisson equation. 2011. ⟨inria-00633934⟩
resume: In this work, we derive the order conditions for fourth order time splitting schemes in the case of the $1D$ Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system : this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to the fourth order. Numerical results are performed and show the benefit of using high order splitting schemes in that context.
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2010

Journal articles

ref_biblio: Frédéric Alauzet, Michel Mehrenberger. P1-conservative solution interpolation on unstructured triangular meshes. International Journal for Numerical Methods in Engineering, 2010, pp.48. ⟨10.1002/nme.2951⟩. ⟨inria-00354509⟩
resume: This document presents an interpolation operator on unstructured triangular meshes that verifies the properties of mass conservation, P1-exactness (order 2) and maximum principle. This operator is important for the resolution of the conservation laws in CFD by means of mesh adaptation methods as the conservation properties is not verified throughout the computation. Indeed, the mass preservation can be crucial for the simulation accuracy. The conservation properties is achieved by local mesh intersection and quadrature formulae. Derivatives reconstruction are used to obtain an order 2 method. Algorithmically, our goal is to design a method which is robust and efficient. The robustness is mandatory to apply the operator to highly anisotropic meshes. The efficiency will permit the extension of the method to dimension three. Several numerical examples are presented to illustrate the efficiency of the approach.
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ref_biblio: Nicolas Crouseilles, Michel Mehrenberger, Hocine Sellama. Numerical solution of the gyroaverage operator for the finite gyroradius guiding-center model.. Communications in Computational Physics, 2010, 8 (3), pp.484-510. ⟨10.4208/cicp.010909.220110a⟩. ⟨inria-00507301⟩
resume: In this work, we are concerned with numerical approximation of the gyroav-erage operators arising in plasma physics to take into account the effects of the finite Larmor radius corrections. Several methods are proposed in the space configuration and compared to the reference spectral method. We then investigate the influence of the different approximations considering the coupling with some guiding-center models available in the literature.
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ref_biblio: Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker. Conservative semi-Lagrangian schemes for Vlasov equations. Journal of Computational Physics, 2010, pp.1927-1953. ⟨10.1016/j.jcp.2009.11.007⟩. ⟨hal-00363643⟩
resume: Conservative methods for the numerical solution of the Vlasov equation are developed in the context of the one-dimensional splitting. In the case of constant advection, these methods and the traditional semi-Lagrangian ones are proven to be equivalent, but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity. In the non constant advection case, they present an alternative to the traditional semi-Lagrangian schemes which can suffer from bad mass conservation, in this time splitting setting.
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ref_biblio: Alia Barhoumi, Vilmos Komornik, Michel Mehrenberger. A vectorial Ingham-Beurling type theorem. Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Mathematica, 2010, www.cs.elte.hu/~annalesm/vol53trt.pdf. ⟨10.1023/a:1020806811956⟩. ⟨hal-01298976⟩
resume: Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend these results to vector coefficient sums
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Poster communications

ref_biblio: Nicolas Crouseilles, Michel Mehrenberger, Hocine Sellama. Résolution numérique de l'opérateur de gyromoyenne. CANUM, May 2010, Carcans-Maubuisson, France. ⟨hal-01298977⟩
resume: L'opérateur de gyromoyenne est défini par J(f)(x, y) = 1 2π 2π 0 f (x + ρ cos(θ), y + ρ sin(θ))dθ. Dans un champ magnétique uniforme, les particules décrivent une trajectoire hélicoïdale et la projection sur le plan perpendiculaire est un cercle. L'opérateur de gyromoyenne traduit alors, dans la théorie gyrocinétique, l'idée de moyenner la fonction de distribution des particules autour d'un cercle d'un rayon caractéristique (le rayon de Larmor ρ) représentant le mouvement de gyration très rapide des particules autour des lignes de champs. On s'intéresse icì à la résolution numérique de cet opérateur en présentant et comparant différentes méthodes numériques. On suppose f 2π p ´ eriodique en x et en y. On définit une grille cartésienne de taille N x × N y .
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Reports

ref_biblio: Jerome Guterl, Jean-Philippe Braeunig, Nicolas Crouseilles, Virginie Grandgirard, Guillaume Latu, et al.. Test of some numerical limiters for the conservative PSM scheme for 4D Drift-Kinetic simulations.. [Research Report] RR-7467, INRIA. 2010, pp.66. ⟨inria-00540948⟩
resume: The purpose of this work is simulation of magnetised plasmas in the ITER project framework. In this context, Vlasov-Poisson like models are used to simulate core turbulence in the tokamak in a toroidal geometry. This leads to heavy simulation because a 6D dimensional problem has to be solved, 3D in space and 3D in velocity. The model is reduced to a 5D gyrokinetic model, taking advantage of the particular motion of particles due to the presence of a strong magnetic field. However, accurate schemes, parallel algorithms need to be designed to bear these simulations. This paper describes a Hermite formulation of the conservative PSM scheme which is very generic and allows to implement different semi-Lagrangian schemes. We also test and propose numerical limiters which should improve the robustness of the simulations by diminishing spurious oscillations. We only consider here the 4D drift-kinetic model which is the backbone of the 5D gyrokinetic models and relevant to build a robust and accurate numerical method.
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2009

Journal articles

ref_biblio: Michel Mehrenberger. An Ingham type proof for the boundary observability of a N-d wave equation. Comptes Rendus. Mathématique, 2009, ⟨10.1016/j.crma.2008.11.002⟩. ⟨hal-00311730⟩
resume: The boundary observability of the wave equation has been studied by many authors. A method of choice is to use the multiplier method. Recently, a first Fourier based proof is given in the case where the domain is a square, thanks to a new Hautus type test. We give here a new self-contained proof with an Ingham type approach in the more general case where the domain is a product of intervals; this leads to explicit time and constants. However, we do not reach the optimal time which can be obtained for this problem by the multiplier method.
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ref_biblio: Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, ⟨10.3934/nhm.2009.4.19⟩. ⟨hal-01298964⟩
resume: We consider a stabilization problem for a coupled string-beam system. We prove some decay results of the energy of the system. The method used is based on the methodology introduced in Ammari and Tucsnak [2] where the exponential and weak stability for the closed loop problem is reduced to a boundedness property of the transfer function of the associated open loop system. Morever, we prove, for the same model but with two control functions, independently of the length of the beam that the energy decay with a polynomial rate for all regular initial data. The method used, in this case, is based on a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
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ref_biblio: Kaïs Ammari, Michel Mehrenberger. Stabilization of coupled systems. Acta Mathematica Hungarica, 2009, ⟨10.1007/s10474-009-8011-7⟩. ⟨hal-01298966⟩
resume: We characterize the stabilization for some coupled infinite dimensional systems. The proof of the main result uses the methodology introduced in Ammari and Tucsnak [2], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system and a result in [11].
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Conference papers

ref_biblio: Jean-Philippe Braeunig, Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker. Guiding-center simulations on curvilinear meshes. Numerical Models for Controlled Fusion (NMCF09), 2009, Porquerolles, France. pp.271-282, ⟨10.3934/dcdss.2012.5.271⟩. ⟨hal-01298727⟩
resume: The purpose of this work is to design simulation tools for magne-tised plasmas in the ITER project framework. The specific issue we consider is the simulation of turbulent transport in the core of a Tokamak plasma, for which a 5D gyrokinetic model is generally used, where the fast gyromotion of the particles in the strong magnetic field is averaged in order to remove the associated fast timescale and to reduce the dimension of 6D phase space involved in the full Vlasov model. Very accurate schemes and efficient parallel algorithms are required to cope with these still very costly simulations. The presence of a strong magnetic field constrains the time scales of the particle motion along and accross the magnetic field line, the latter being at least an order of magnitude slower. This also has an impact on the spatial variations of the observables. Therefore, the efficiency of the algorithm can be improved considerably by aligning the mesh with the magnetic field lines. For this reason, we study the behavior of semi-Lagrangian solvers in curvilinear coordinates. Before tackling the full gyrokinetic model in a future work, we consider here the reduced 2D Guiding-Center model. We introduce our numerical algorithm and provide some numerical results showing its good properties.
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2008

Journal articles

ref_biblio: Paola Loreti, Michel Mehrenberger. An Ingham type proof for a two-grid observability theorem. ESAIM: Control, Optimisation and Calculus of Variations, 2008, ⟨10.1051/cocv:2007062⟩. ⟨hal-01298963⟩
resume: Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $t>2\sqrt{2}$; this time, if the observation is made in $(-T/2,T/2)$, is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I 338 (2004) 413–418]. Our proof follows an Ingham type approach.
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ref_biblio: Paola Loreti, Michel Mehrenberger. Observabilite uniforme de l'equation des ondes 1D . ESAIM: Proceedings, 2008, ⟨10.1051/proc:082505⟩. ⟨hal-01298974⟩
resume: This proceedings is a complement of [7]=hal-01298963. In [7], new Ingham type theorems have been developped in order to establish the uniform observability of the wave equation in 1D. We gather here in a self-contained and compact form several results obtained by the mean of discrete multipliers. In a second part, we give an example of Ingham type inequality where the position of the interval is a key point in the determination of the optimal time. This example enforces the idea that such phenomenon, already mentionned in [7], can occur.
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ref_biblio: Nicolas Besse, Michel Mehrenberger. Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system. Mathematics of Computation, 2008, 77 (261), pp.93-123. ⟨10.1090/S0025-5718-07-01912-6⟩. ⟨hal-00594785⟩
resume: Abstract: In this paper we present some classes of high-order semi-Lagran- gian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function $ f(t,x,v)$ and the electric field $ E(t,x)$ converge in the $ L^2$ norm with a rate of $\displaystyle \mathcal{O}\left(\Delta t^2 +h^{m+1}+ \frac{h^{m+1}}{\Delta t}\right),$ where $ m$ is the degree of the polynomial reconstruction, and $ \Delta t$ and $ h$ are respectively the time and the phase-space discretization parameters
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2007

Journal articles

ref_biblio: Martin Campos Pinto, Michel Mehrenberger. Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system. Numerische Mathematik, 2007, 108 (3), pp.407-444. ⟨10.1007/s00211-007-0120-z⟩. ⟨hal-00320364⟩
resume: An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two- dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L infinity metric. The numerical solutions are proved to converge in L infinity towards the exact ones as ε and Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to ${W^{1,\infty} \cap W^{2,1}}$ . The rate of convergence is ${\mathcal{O}({\Delta}t^2 + \varepsilon/{\Delta}t)}$ , which should be compared to the results of Besse who recently established in (SIAM J Numer Anal 42(1):350–382, 2004) similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in ${{\mathcal C}^2}$ . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.
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2006

Journal articles

ref_biblio: Michel Mehrenberger, Eric Violard, Olivier Hoenen, Martin Campos Pinto, Eric Sonnendrücker. A Parallel Adaptive Vlasov Solver Based on Hierarchical Finite Element Interpolation. Nuclear Instruments and Methods in Physics Research, 2006, Proceedings of the 8th International Computational Accelerator Physics Conference - ICAP 2004, 558 (1), pp.188-191. ⟨10.1016/j.nima.2005.11.225⟩. ⟨inria-00111165⟩
resume: We present a parallel adaptive scheme for the Vlasov equation. Our method is based on a way of reducing dependencies between data, thanks to a hierarchical finite element interpolation approach. A specific data distribution pattern yields an efficient implementation. Numerical results are exhibited for a classical beam simulation in the 1D phase space.
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ref_biblio: Michel Mehrenberger, Eric Violard. A Hermite type adaptive semi-Lagrangian scheme. International Journal of Applied Mathematics and Computer Science, 2006, ⟨10.2478/v10006-007-0027-y⟩. ⟨inria-00110865⟩
resume: We study a new Hermite type interpolating operator in a semi-Lagrangian scheme for solving the Vlasov equation in the 2D phase space. Numerical results on uniform and adaptive grid are shown and compared with biquadratic Lagrange interpolation in the case of a rotating Gaussian.
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Conference papers

ref_biblio: Michaël Gutnic, Michel Mehrenberger, Eric Sonnendrücker, Olivier Hoenen, Guillaume Latu, et al.. Adaptive 2-D Vlasov Simulation of Particle Beams. ICAP 2006, Oct 2006, Chamonix, France. ⟨hal-01300207⟩
resume: This paper presents our progress for the solution of the 4D Vlasov equation on a grid of the phase space, using two adaptive methods. We briefly recall the principle of the two methods and then particularly focus on computer science features - as data structures or parallelization - for the efficient implementation of the methods. Some relevant numerical results are presented.
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2005

Journal articles

ref_biblio: Michel Mehrenberger. Critical length for a Beurling type theorem. Bollettino dell'Unione Matematica Italiana, 2005, 8, http://www.bdim.eu/item?id=BUMI_2005_8_8B_1_251_0. ⟨hal-00139352⟩
resume: In a recent paper [3] C. Baiocchi, V. Komornik and P. Loreti obtained a generalisation of Parseval's identity by means of divided differences. We give here a proof of the optimality of that theorem.
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Book sections

ref_biblio: Martin Campos Pinto, Michel Mehrenberger. Adaptive numerical resolution of the Vlasov equation.. Numerical Methods for Hyperbolic and Kinetic Problems, 7, EMS, pp 43--58, 2005, IRMA Lectures in Mathematics and Theoretical Physics, ⟨10.4171/012-1/3⟩. ⟨hal-00137877⟩
resume: A fully adaptive scheme (based on hierarchical continuous finite element decomposition) is derived from a semi-Lagrangian method for solving a periodic Vlasov-Poisson system. The first numerical results establish the validity of such a scheme.
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Poster communications

ref_biblio: Michel Mehrenberger. High order semi Lagrangian schemes. Atelier du GdR CHANT sur les méthodes numériques pour les équations cinétiques, hyperboliques et de Hamilton-Jacobi, Nov 2005, Strasbourg, France. , 2005. ⟨hal-01298992⟩
resume: We present some results concerning high order semi-Lagrangian schemes in the 1Dx1D phase space. In the uniform case, we consider the stability of several reconstructions : B-splines, Lagrange interpolation and interpolets. In the adaptive case, we give first numerical results concerning a Hermite reconstruction.
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Reports

ref_biblio: Martin Campos Pinto, Michel Mehrenberger. Convergence of an Adaptive Scheme for the one dimensional Vlasov-Poisson system. [Research Report] RR-5519, INRIA. 2005, pp.49. ⟨inria-00070487⟩
resume: An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic one-dimensional Vlasov-Poisson system is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on the analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter which represents the local interpolation error at each time step.The numerical solutions are proved to converge in sup-norm towards the exact ones as the toloerance parameter and the time step tend to zero provided the initial data is Lipschitz and has a finite total curvature. t)$, Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.
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2004

Journal articles

ref_biblio: Michel Mehrenberger. Observability of coupled systems. Acta Mathematica Hungarica, 2004, 4 (103), pp.321-348. ⟨10.1023/B:AMHU.0000028832.47891.09⟩. ⟨hal-00139351⟩
resume: By applying the theory of semigroups, we generalize an earlier result of Komornik and Loreti [5] on the observability of compactly perturbed systems. As an application, we answer a question of the same authors concerning the observability of weakly coupled linear distributed systems.
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Conference papers

ref_biblio: Olivier Hoenen, Michel Mehrenberger, Eric Violard. Parallelization of an Adaptive Vlasov Solver. EuroPMV/MPI, 2004, Budapest, Hungary. ⟨10.1007/978-3-540-30218-6_59⟩. ⟨hal-01298972⟩
resume: This paper presents an efficient parallel implementation of a Vlasov solver. Our implementation is based on an adaptive numerical scheme of resolution. The underlying numerical method uses a dyadic mesh which is particularly well suited to manage data locality. We have developed an adapted data distribution pattern based on a division of the computational domain into regions and integrated a load balancing mechanism which periodically redefines regions to follow the evolution of the adaptive mesh. Experimental results show the good efficiency of our code and confirm the adequacy of our implementation choices. This work is a part of the CALVI project.
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Theses

ref_biblio: Michel Mehrenberger. Inégalites d'observabilité et résolution adaptative de l'équation de Vlasov par éléments finis hiérarchiques. Informatique [cs]. Université Louis Pasteur - Strasbourg I, 2004. Français. ⟨NNT : 2004STR13124⟩. ⟨tel-00008254⟩
resume: La premiere partie est consacree a l'etude d'inegalites d'observabilite, qui interviennent en theorie du controle. On donne ainsi un theoreme abstrait qui permet de deduire l'observabilite d'un systeme par perturbation compacte, avec une condition affaiblie sur l'operateur perturbe. Ce theoreme est ensuite applique a l'observabilite de certains systemes faiblement couples. On demontre aussi l'optimalite d'un theoreme recent concernant une generalisation de l'identite de Parseval aux differences divisees d'exponentielles. La deuxieme partie de ce travail est consacree a la resolution numerique de l'equation de Vlasov en utilisant des schemas de type semi-lagrangien. On demontre dans un premier temps la convergence de schemas d'ordre eleve arbitraire, en completant des resultats precedents. On developpe ensuite une nouvelle methode numerique basee sur une interpolation par elements finis hierarchiques biquadratiques, qui permet ici une parallelisation efficace. Dans le cadre d'une reconstruction affine par maille, on definit une strategie de raffinement et des quantites qui controlent l'erreur produite a chaque pas de temps pour construire finalement un algorithme adaptatif dont on montre la convergence.
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