MUFFIN - sorbonne-universite.fr

AAPG2019 MUFFIN Funding instrument : PRC

Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros

scientific evaluation committee : 5.6. Modeles numeriques, simulation, applications

MUFFIN (MUltiscale and treFFtz for numerIcal transport)

Summary table of persons involved in the project:

Partner Name First nameCurrentposition

Role & responsibilities in theproject (4 lines max)

Involvement(person.month)throughout theproject's total

duration

Sorbonne Univ.

Després Bruno PR

- PI and coordinator of the whole project- PI of the LJLL Team- Researcher on TDG

20 months, over4 years, that is 41.66 %.

Sorbonne Univ.

Tournier Pierre-Henri Ing.-CNRS - Developer FBL and TDG 10 months

Sorbonne Univ.

Campos-Pinto

Martin CR-CNRS - Researcher/developer FBL 0 month

Sorbonne Univ.

Charles Frédérique MCF - Researcher/developer FBL 12 months

Sorbonne Univ.

Hirstoaga Sever CR Inria - Researcher TDG 12 months

Sabatier Univ.

Filbet Francis PR- PI of the Sabatier team,- Researcher LRM

24 months

Sabatier Univ.

Vignal Marie-Hélène

MCF - Researcher/developer FKS 12 months

Sabatier Univ.

Loubere Raphael DR-CNRS- Researcher/developer FKS/AP

12 months

Sabatier Univ.

Narski Jacek MCF - Researcher/developer FKS 12 months

Nantes Univ.

Berthon Christophe PR- PI of the Nantes Team- researcher WB

15 months

Nantes Univ.

Crestetto Anais CR - researcher WB 12 months

Nantes Univ.

Badsi Mehdi CR- Researcher/developer WB/Sheath

15 months

Modifications with respect to the pre-project.

a) With respect to the pre-project, Sever Hirstoaga is a new member, incorporated in the LJLL team. Sever is anexpert on transport methods in magnetized plasma. The reason of the modification is that Sever recentlymoved in fall 2018 from Strasbourg to Inria-Paris, so it is an opportunity. Also, Martin Campos Pinto will moveto a Max-Planck institute in Munich. He will continue to participate in the scientific development of the project,but as a member of a foreign institute (with his own funding): therefore his involvement is not accounted for inthe budget, so the 0 month in the last column.

b) The requested funding has increased from 370 k€ (which was announced in the pre-proposition) to 408.78k€, essentially to the application of administrative costs (+ 8%).

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I. Proposal’s context, positioning and objective(s)

Context and positioning: In many applications, one is faced with the problem of solving numerically a

set of transport equations, also called kinetic equations in this document. The huge difficulty attached

to this task is that it may be computationally exhausting, mainly because of the high dimension1 and

of the multi-scale nature of the model (with strong gradients/filamentation, or with boundary

layers/jump/variations in the coefficients). Therefore one is forced to admit that the numerical

solution of transport/kinetic equations is a bottleneck for advanced modelling and simulation in

applied multi-physics sciences. Two exemplary recent references are [49] in the context of

supercomputing and [27] in the context of computational magnetized plasma2.

a. Objectives and research hypothesis

Objectives: The objective of the proposed research is to explore and optimize original computational

and numerical scenarios for multi-scale and high dimensional transport codes, with priority

applications in plasma physics. It is at the frontier of computing and numerical analysis and intends to

reduce the computational burden in the context of intensive calculation.

To tackle this problem our approach will consist of developing and extending a series of numerical

methods in a collaborative environment.

The general idea is to exploit the local/global information (this typically is a multi-scale information

about the local equilibria) present in the physics or in the partial differential equations and to

introduce this information in the design of advanced approximation strategies. An important

objective of our project is also to develop a proper testing stage to assess the numerical efficiency of

the proposed methods on challenging test cases, and compare them when that is feasible. Indeed,

many numerical issues (implementation per se, CPU requirements, memory requirements, linear

algebra issues for implicit schemes, compatibility with modern architectures) can be revealed only by

implementing the methods and testing on challenging physical configurations (issued from

magnetized and non magnetized plasma physics in our case). Within this project, the physical

configurations will be taken mainly from the modeling of plasma with or without magnetic field. Our

interest in these methods is because they introduce a priori information in the discretization

procedure to save computational resources. Preliminary publications detailed afterward assess that

1� A transport equation is typically written for particles, which have a position (x) and a velocity (v). Therefore the numerical

transport equation may be written in total dimension d=4 for 2x-2v problem, and in total dimension d=6 for a 3x-3v

problem. A scenario with 100 cells per direction yields 100^d cells at minimum, so out of reach with standard methods in

dimension d=6.

2� A transport equation with magnetic field but no collision is typically as Vlasov equation. A transport equation with

collision becomes more a Boltzmann like equation.

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the scientific foundations of the methods/algorithms are now solid enough so that numerical analysis

and implementation can be advanced in parallel.

The scientific barriers addressed in the project concern the development of the new transport

algorithms per se (stability, accuracy, degree of parallelism). The technical barriers will be more the

implementation, analysis, optimization and performance measurements of advanced transport

codes. Numerical magnetized transport near walls or the numerical coupling of transport and fluid

models will be considered as specific challenging applicative problems which have their own

modeling challenges.

The outputs will consist in advanced transport codes optimized on local and national computational

servers, for the solutions of various fundamental problems in transport for plasmas, in various

scientific publications in the best journals of the disciplines (applied mathematics, numerical physics,

scientific computing), and in the organization of small Workshops with invitation of international

experts at the end of which a publication in a LNSCE is planed.

b. Position of the project as it relates to the state of the artThe different approximations methods that we plan to develop are now presented with respect to the

state of the art, both in terms of mathematical foundations and practical implementation.

FBL: The Forward-Backward Lagrangian (FBL) method for transport equations has been developed at

LJLL [10, 11] to improve the accuracy of density reconstructions in particle codes: it relies on the

backward Lagrangian representation of the solution like a standard backward semi-Lagrangian (BSL)

method, but it is uses a collection of markers pushed forward in time to evaluate the (backward)

characteristics. Using centered finite difference formulas one can indeed evaluate the Jacobian and

Hessian matrices of the flow using only these markers, and then approximate the backward flow by

local Taylor expansions. Compared to existing smooth particle methods with either fixed or

transformed shapes (such as FBL [42] or LTP/QTP methods [9]), the proposed reconstruction achieves

higher locality and accuracy. For linear transport equations the FBL method is proven second order

convergent [11] in any dimension. Moreover, it preserves the positivity of the solution. The method

has been applied to simulate 2D test cases of passive transport problems, and also non-linear Vlasov-

Poisson and incompressible Euler equation in vorticity-velocity form.

FKS: Fast Kinetic Schemes (FKS) [43, 21] recently invented (at the Univ. Paul Sabatier) use redundancy

in the velocity space to discretize with less degrees of freedom. On the contrary to classical semi-

Lagrangian methods, one does not reconstruct the distribution function for each time step. This

allows to tremendously reduce the computational cost and to perform efficient numerical simulations

up to the 3Dx3Dx1D (space x velocity v time t) case. The resulting scheme shares analogies with semi-

Lagrangian methods and Monte-Carlo methods. Numerical simulation on 3Dx3D BGK equation can be

solved on laptop, while Boltzmann equation simulations can be performed on small clusters: so the

gain in CPU time and memory consumption is dramatic with this family of algorithms. For the

moment FKS is restricted to 1st order approximation. One of our goals cover the extension of FKS to

high order, to improve upon the efficiency for Boltzmann simulations by allowing different mesh

resolutions in velocity and exploring the asymptotic preserving (AP) property of numerical methods.

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TDG: The Trefftz Discontinuous Galerkin method (TDG) is similar to a Discontinuous Galerkin method

[44], except that a local sub-scale information about the equilibrium (close to Maxwellians for

plasmas) is incorporated in the local approximation space. Very recently during a PhD at LJLL, it has

been applied to the transport of particles with relaxation for a linear Boltzmann equation [39, 40, 41]

(the particles were neutrons or photons) with internal boundary layers.

For a magnetized plasma (now the particles are charges, ions or electrons), the construction and

implementation the new basis functions adapted to a Vlasov equation is a fully original task with

respect to the literature: a huge difference with respect to the classical approach [44] is that the basis

functions will contain exponential terms which increase a lot the capabilities of a purely polynomial

space of approximation. For charges particles, the coefficients of the exponential terms correspond to

the physics (decomposition between parallel and perpendicular temperatures).

The mathematical structure of TDG can be described as a moment model with respect to the velocity

variables: it yields Friedrichs systems of large size (equal to the number of moments); the recent

theory [39, 40, 41] shows that the basis functions contain the coefficients of the underlying local

physical models; for neutrons, the underlying physics is typically characterized with an absorption

coefficient and a scattering coefficient which are function of the space variable; for magnetized

transport, the underlying physics (and so the basis functions) will contain the coefficients for the

parallel and perpendicular temperature plus the anisotropic magnetic field. Such systems are

endowed with strong stability properties in quadratic norm. After discretization with standard

methods, one gets a linear system, which need to be inverted at each time. The implicit version of the

scheme is stable in quadratic norm unconditionally with respect to the time step while the explicit

version needs a CFL condition for stability. The recent numerical experiments [39, 40, 41] show that

the linear implicit system display good condition properties consequence of the strong mathematical

properties. Therefore standard linear solvers are prone to be very efficient. The recent study [4] in an

engineering context (model of wave propagation from the Total company) confirms this global

tendency.

The application of TDG methods to transport equations in strongly magnetized plasma is our purpose

in the context of this project. Essentially, a Lorentz force is like a filter: the kernel of the corresponding

operator determines a reduced set of functions which can be used as basis functions for an original

Trefftz Discontinuous Galerkin applied to transport. One can make comparisons with a similar

selection principle performed in gyro-kinetic models. The expertise of two members of the project,

namely A. Crestetto (one author of [44], on DG methods for Vlasov equation) and P.H. Tournier

(expert on linear solver [7, 33]), will be important assets to compare the TDG method with respect to

state of the art of DG and linear solvers.

LRM: Low-Rank Methods (LRM) have been recently introduced in the frame of kinetic equations by

Einkemmer and Lubich in [25, 26] for the Vlasov-Poisson system. This approach allows to reduce the

storage requirements and to save computational effort, which are crucial for the numerical simulation

of kinetic equation. The main idea is to approximate numerical solutions of high-dimensional

evolutionary partial differential equations by dynamical low-rank approximation allowing

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compression of the solution via high-order singular value decomposition. In [25, 26], LRM has been

applied in the Eulerian framework using spectral and semi-Lagrangian methods. Very good numerical

accuracy for Landau damping has been reported and also for the study of instabilities (two stream

instability) in plasma physics. In [35] a low-rank numerical algorithm for the Vlasov–Poisson equation

has been proposed. It is done using the common technique of splitting in time and then applying a

spline based semi-Lagrangian scheme to the resulting advection equations. Each step of such an

algorithm can then be written as a linear combination of low-rank approximations.

Our aim is to apply the LRM in a different frame, based on the Particle-In-Cell method, which can be

interpreted as a static low-rank approximation of the distribution function. Indeed, the solution

f(t,x,v) is classically approximated by a sum of Dirac measures or fixed regularized approximations

whereas we will now approximate it by a set of functions which will evolve according to a dynamical

low rank model. By using such a dynamical approach, our aim is to reduce the inherent statistical

noise of the Particle-In-Cell algorithms.

WB-Sheath: This part of the project addresses the mathematical and numerical modeling of a

particular type of boundary layers that arise in plasma transport problems. In terms of mathematical

modeling, it is much more ambitious than the previous tasks.

In the presence of absorbing (e.g. metallic) walls, isolated plasmas develop steep, non-neutral

boundary layers called plasma sheaths: close to the walls and due to the small electron/ion mass

ratio, a strong electric field builds up and self-consistently interacts with the flow of charged particles

in order to preserve the global neutrality of the plasma. Although this phenomenon is well-known in

plasma physics [14, 46], at the numerical level it results in a nonlinear boundary condition for which

many open questions remain, in particular for the choice of appropriate models to capture a plasma

sheath, be it with or without a magnetic field [45, 38, 12, 34].

Previous works of ours [1, 2, 3] on this topic have shown that electrostatic sheaths are now better

understood both at the mathematical level and the numerical level. For univariate problems, semi-

Lagrangian schemes (for the transient regime) and gradient algorithm (for the stationary state) have

been proposed. However efficient, these studies as well as other numerical studies on this subject

have proved that due to non-linearities in the models and to the multi-scale nature of the physics,

solutions may develop strong localized gradients or even discontinuities, making the computational

effort for the simulation exhausting.

This part of our project will thus be devoted to the mathematical construction of reference sheaths

solutions in magnetized plasmas, and to the development of accurate and efficient numerical

methods for the simulation of these phenomena. Because of the small electron/ion mass ratio this

problem can be related to existing mathematical works on Asymptotic Preserving (AP) methods for

Vlasov-Poisson models [5], however the presence of a nonlinear boundary condition on the electric

field is an original feature of our problem. In particular, a key issue will consist of designing Well-

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Balanced (WB) schemes that preserve steady states in the presence for such nonlinear boundary

layers.

Computational tools/Computer environment/HPC:

Here we describe the computational environment of our studies by detailing, firstly the expertise at

LJLL for linear algebra and HPC, and secondly the HPC expertise at Sabatier for the FKS code.

HPC expertise at LJLL: Pierre-Henri Tournier is a member of the joint INRIA-LJLL research group

Alpines, focusing on scientific computing and in particular on parallel linear solvers for sparse and

dense linear systems [33, 7, 47, 48]. The team has experience in developing optimized libraries and

implementing scalable state-of-the-art linear solvers for distributed environments using hybrid

MPI/Open MP parallelism, in particular using domain decomposition methods. These developments

benefit from GENCI grants (960,000 hours in 2017 through project A0020607330, 1,050,000 hours in

2018 through project A0040607330) giving access to the Occigen supercomputer operated at CINES,

equipped with 2106 Haswell nodes ([email protected] GHz, 50544 cores total) and 1260 Broadwell

nodes ([email protected], 35280 cores total). Code development at LJLL also benefits from a local

server equipped with 32 Intel(R) Xeon(TM) 64 bits EvyBridge [email protected] processors, with a total

of 320 cores and 2048Gb of shared memory with a Numalink 7 interconnect.

These existing developments will directly benefit the work package TDG regarding the parallel

assembly and solution of the linear systems on distributed environments. Moreover, the HPC

expertise at LJLL will allow the development of efficient parallel code for the work package FBL in

order to assess the efficiency of FBL on higher dimensional problems. The method should be

embarrassingly parallel and should scale to thousands of cores with no significant implementation

effort.

FKS under HPC: FKS is a C++ parallel code working in 1Dx1D up to 3Dx3D on Cartesian grids for BGK or

Boltzmann operators. It is developed by R. Loubere and collaborators. The FKS code is much more

advanced than the other parts of the project so it will serve as the HPC workhorse of the project for

the firsts years.

The available versions are:

Version 1) a massively CPU parallel version, MPI for large distributed memory architectures;

Version 2) a light parallel version, OpenMP for laptops, some clusters;

Version 3) a GPU/CUDA mix OpenMP parallel version, for graphic card clusters available and/or laptop

that have more than one GPU card. As reported in [43], the code has shown an embarrassingly

parallel behavior (for Boltzmann collisional operator) due to the fact that the collision operator

represent more than 90% of the computation needs. The code has already run on the following

architectures: Hybrid//MPI/Open MP on two facilities:

1) Méso-centre CALMIP (Centre de CAlcul en Midi-Pyrénées), Projet P1542, 30000+30000 hours en

2015-2016. Machine EOS: this supercomputer is equipped with 612 computational nodes, each of

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mailto:[email protected]






them containing two Intel@Ivybridge 2.8GHz 10 core CPUs and 64 GB of RAM. Each CPU is equipped

with 25MB of cache memory. The code was executed on 90 computational nodes, ie on 1800

computational cores in parallel.

2) GENCI ( http://www.genci.fr CT-6 Informatique, algorithmique et mathématiques). Projet Grant

2016-AP010610045), 1 million hours in 2017. Machines TGCC: thin nodes of the supercomputer

Curie equipped with 5040 B510 Bullx nodes(called thin nodes), each containing two Intel@Sandy

Bridge 2.7GHz 8 core CPUs (20MBof cache memory) and 64GB of RAM. The conclusion is that the

code has been granted a strong scaling closed to ideal (up to 1024 computational nodes, i.e. 2048

processors and 16384 computational cores) for 3x-3v problems.

OpenMP solely, the code has run on:

- Server at IMT (Toulouse) equipped with 4 Intel(R) Xeon(TM) E5-4650 processors running at 2.7 GHz

giving a total of 32 physical cores and 64 logical) with 512GB of RAM under Debian Wheezy;

- Several laptops (Mac book air, HP, ...) using at best the computational power available. The

conclusion is that the code shows a close-to-linear scaling with the number of threads up to 64, i.e.

close to ideal.

On GPU machines. The version has been tested on one and two GPU cards so far, on the following

machines:

- Computational server equipped with dual Intel(R) Xeon(TM) E5-2650 processor running at 2.0GHz

(16 physical and 32 virtual cores) with 128GB and 2 Nvidia GTX 780 units (3GB of memory, 2304

CUDA cores at 900MHz each) running under Debian Wheezy.

- Laptops equipped with two GPU cards such as a HP Elitebook, HP Zbook for instance.

c. Methodology and risk managementThe project is organized in Work-Packages (WP) which are either individual objectives for each of the

methods described above, or global objectives:

- WP-FBL (on Forward-Backward Lagrangian methods),

- WP-FKS (on Fast-kinetic Schemes),

- WP-TDG (on Trefftz Discontinuous Galerkin methods),

- WP-LRM (on Low Rank Methods),

- WP-WB/Sheaths (on Well-Balanced methods for plasma sheaths),

- WP-Global (on global collaboration, comparisons and exchange of techniques during the project).

These Work-Packages are described below in details. We plan to advance them in parallel for the first

stages of the project. To answer the instructions in the document, a scale of risk assessment 0 ≤ R ≤ 5

is proposed, with 0 = no risk/certain success and 5 = high risk = improbable success.

1. WP-FBL (M. Campos Pinto, F. Charles and P.-H. Tournier).

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http://www.genci.fr/




We plan to develop and extend the FBL method along two main directions:

- WP FBL-1 (M. Campos Pinto, F. Charles): For pure transport problems we will extend the FBL method

by adding adaptive discretizations of the density. An interesting feature of the FBL method compared

to Backward Semi-Lagrangian schemes lies in fact that it couples in a weak fashion the approximation

of the transport flow and that of the density. Here we intend to take advantage of that feature to

propose efficient adaptive strategies for FBL approximations.

To assess the numerical efficiency of the method, a 3D version of a current code developed at LJLL

will be written and the results will be compared to challenging benchmark problems (e.g. [37] or [16])

where simple volumes in 2D and 3D are stretched along very nonlinear flows and mapped back to

their original position. The challenge is to resolve the thinner and thinner stretched shapes, which is

eventually measured in terms of volume loss at the final time. Nonlinear test-cases will also be

performed, including standard Vlasov-Poisson test problems showing phase-space filamentation, and

the performances of the method will be compared with those of the LRM Work-Package. The HPC

expertise of Tournier at LJLL will be used for this task.

Risk assessment: R = 2.

- WP FBL-2 (M. Campos Pinto, F. Charles, P.-H. Tournier and F. Filbet):

A second direction will be to extend the FBL method to linear and non-linear Fokker-Planck

equations, in order to model diffusive phenomena in the velocity space. A standard approach consists

of rewriting the diffusive model as a non-linear transport equation. This idea has already been used in

1D particle methods for the linear heat equation in [18] and [36]. Here we will adapt this idea in the

framework of FBL approximations. Due to the accuracy and flexibility of the basis method for non-

linear transport problems, we are convinced that the resulting method will be a promising one.

To assess the numerical accuracy of our method, we will compare it with the famous Chang-Cooper

scheme [13], and a meaningful test will be to reproduce the convergence trends to equilibrium

recently observed in [24, 6]. We note that these approaches are restricted to the linear Fokker-Planck

equation or/and are not easily extended to dimensions higher than 1D x 1V.


2. WP FKS (Loubere, Vignal, Narski, Després) For the last few years the large-scale goal of the team

involved in this task was to simulate with the best possible numerical methods the Kinetic models

(BGK or Boltzmann equations) and the continuous models (Navier-Stokes/Euler equations), and to

simulate the transition zone between those two family of models.

Within this work-package, our aim is to continue the development ultra-efficient fast kinetic schemes.

Some ideas may arise from our works about the design an accurate and efficient scheme dedicated to

transition zones between a kinetic model and a continuous (or fluid) model.

- WP FKS-1 (Loubere, Vignal, Narski): Improving the efficiency of the kinetic scheme by reducing

velocity mesh size.

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Main idea : Reduce the velocity mesh size and compensate this reduction with an accurate

reconstruction procedure (sine, polynomials, Beziers…) because the variations of Maxwellian M

(equilibrium) and distribution function “f” must be smooth enough to retrieve the accuracy at lower

cost. The choice of reconstruction procedure should be studied further, the type of functions

(trigonometric, polynomials, Beziers), with or without limiting (slope limiting, WENO…). Also some

inspiration will come from the parallelism with models coupling of different natures, because it is the

same issue at the numerical level for numerical methods. It is now standard to address such issues

with AP (asymptotic preserving) methods [22, 23] for which the theory is well developed for fluid

macroscopic equations; the extension of these ideas towards fast kinetic schemes will be

investigated.

Justification : Transport operator is inexpensive, therefore a fine velocity mesh is painless but

maintains the accuracy. Contrarily the Collisional operator is (very) expensive, hence a coarse velocity

mesh is preferable. The passage fine←→coarse mesh is performed by high accurate reconstructions

of M and/or f.

Test cases : being an improvement of the existing FKS simulation code, the validation test cases are

taken from the already published literature [21, 43]. The methodology of testing will consider 2Dx2D

BGK and Boltzmann on Sod test case, then possibly 3Dx3D BGK and Boltzmann later. We will check

the CPU time of the adaptive-FKS against the original FKS code and we expect: a drastic CPU time

reduction for an equivalent accuracy.

There is no "industrial" test case to run at this moment, however with the adaptive-FKS scheme for

Boltzmann 3Dx3D simulation, we will show that larger meshes can be simulated on small clusters,

hence for the same cost, a better resolution is expected. Runs on massively parallel machine (GENCI)

and on GPU cards like in [43] will be performed to assess the gain in CPU times.

Diagnostic of success: improvement of CPU time is difficult to anticipate (factor 3, 10, 100,...). A

justified strategy of choosing the reconstruction functions (WENO, CWENO, sine, other) is needed

and this part demands some work.

Risk assessment: weak risk R=1. Almost no risk in terms of development, because FKS exists in

2Dx2D, so concerning the proof of concept there is no risk in the development. The 3Dx3D

reconstructions, polynomials or spline, are already in the literature but technical details may generate

troubles, i.e. large stencils for parallel machinery, global neighborhood for spline or trigonometric

functions, which will nonetheless be solved by dedication.

- WP FKS-2 (Loubere, Narski, Després):

Improving the accuracy of the kinetic scheme, by increasing representation of functions of the

Maxwellian “M” and the kinetic function “f” by polynomials.

Main idea: use high accurate polynomial reconstructions to improve upon the general FKS algorithm

for BGK equation first, then Boltzmann equations.

Justification: In [20] some accuracy evidences in 1D for BGK and Boltzmann equations with piecewise

linear polynomials have been gathered as a proof of concept. The goal is to extend this approach to

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higher polynomial degrees in multi-dimensions (2Dx2D for BGK). Then, when the procedure is

genuinely validated, we may extend the procedure for BGK 3Dx3D and implement a version for the

Boltzmann collisional operator. We have already observed in 1D on BGK model, that the 1st order of

accuracy of FKS can be increased to reach a 2nd order with piecewise reconstructions of the

Maxwellian. There we assume that there exists a direct link between the Maxwellian reconstruction

and the distribution function representation. With such an assumption we can avoid the costly

reconstruction of the distribution functions, and it opens the path to use higher polynomial degrees

at relative low cost.

Test cases: we have observed that FKS supplemented with piecewise linear reconstruction (R-FKS) is

of the same accuracy that a semi-Lagrangian (SL) scheme with MUSCL like reconstruction on a

relative smooth flow. However when the flow presents many tiny variations, like an highly oscillating

kinetic problem, the numerical dissipation is drastically reduced with our R-FKS compares to any high

accurate SL scheme. Then we expect to gain even more with higher polynomial degrees (third-fourth

orders of accuracy).

Diagnostic of success: on smooth solution we expect to retrieve at least the second order of accuracy

for BGK and Boltzmann models, then when higher degrees of reconstructions are used, we expect to

see an improvement in accuracy at fixed mesh. On non-regular solutions, or solutions with steep

gradients, we expect to observe steeper and steeper gradient compared to SL schemes for at least a

comparable cost.

Risk assessment: R=2. The extension to 2nd order multi-dimensions (BGK and Boltzmann collisional

models) presents no real risk apart from code development and its associated troubles. The more

risky part of this work-package is the extension to higher polynomial degrees where the gain to

expect is not clear. At this moment we do not know if a perfect gain in accuracy will be observed; for

instance like the ones for Finite Volume schemes on smooth solutions. Consequently, the numerical

experimentation is of paramount importance for this matter.

3. WP TDG (Després, Tournier, Hirstoaga)

The objective is to construct a prototype code, which implements the TDG method for a general

transport equation in magnetized plasma. The constraints will be that the prototype must be

relatively simple to write and test. But at the same time, the model problem must be relevant for

magnetized transport with a given magnetic field function of the space variable, and so the geometry

of the problem will be full 3D: this is quite heavy for such prototype.

To satisfy this constraint and keep the prototype in reasonable size, the solution will be to use a 3D

cartesian finite volume mesh with periodic boundary conditions, so as to concentrate on the main

difficulty in the velocity/kinetic space. The basis functions (per cell) will be describe by kinetic

functions with a reduced number of degrees of freedom, dependent in priority of the parallel

velocity, with a dependance with respect to the modulus of the transverse velocity. Another

ingredient will be to choose such function in compatibility with local Maxwellian (it can be done with

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Hermite polynomials and Hermite functions which are orthogonal with respect to the Maxwellian

weight).

Using a standard DG or Finite space integration of these functions, it ends up with the calculation of

the fluxes. Preliminary investigations show that it can be done with integration in half-spaces (in

velocity space) of products of such Hermite polynomials: primarily with numerical integration with

weights based on Hermite quadratures (for the transverse velocities) and a Laguerre quadrature (for

the velocity variable normal to the interface). A variant will be a second order flux with the mid rule

formula (in this case, only Hermite quadratures should be involved). Mathematically, special care

must be given to the fact the bulk magnetic field depends on the space variable, so are the basis

functions. On this basis, a numerical prototype will be detailed, implemented and tested. The steps

will be:

- WP TDG 1 (Després, Tournier, Hirstoaga)

We split our program in three steps:

write an efficient routine for the numerical coupling of 2 basis functions;

implement a 3D (in space) code with a variable bulk magnetic field;

assemble the matrix for the implicit solver and invert it with in-house linear solvers on local high performances facilities. Assessment of the computing performance will be made at LJLL on the basis of the expertise of Pierre-Henri Tournier.


- WP TDG 2 (Després, Tournier, Hirstoaga)

Assessment of the numerical performance and accuracy of the computed solutions will be made bycomparison with:

the solutions constructed in with the Berstein modes techniques, it yields exact solutions which are developed by Alexandre Rege (PHD at LJLL since October 2018);

the established literature, for example papers by Crouseilles-Lemou [17] and therein, which are based on completely different techniques but seem so far restricted to 2D configurations (contrary to the one investigated in this study, but with coupling with a self consistent electricfield). The numerical analysis will be performed in collaboration between LJLL and LaboratoryJean Leray (Nantes, expertise of A. Crestetto). The implementation will be performed at LJLL. The optimization of the code and the inversion of the linear systems will be performed at LJLL, and with the post-doc which is planned.


4. WP LRM (Filbet)

The basic objective of LRM is to constrain the dynamics of the transport kinetic equation to a

manifold of low-rank functions by a tangent space projection which is then split into its summands

over a time step, adapting the projector-splitting approach to time integration, so it yields a sequence

of advection equations in a lower-dimensional space. Then appropriate numerical techniques can be

applied easily to obtain a numerical solution. In the frame of kinetic equations, this approach is used

to obtain equations separately in x and in v, which reduces a 2d-dimensional problem to a sequence

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of d-dimensional problems. This is the key point to get rid of the curse of dimensionality of kinetic

equations. This approach is independent to the numerical method applied to the discretization of the

resulting PDE model. The first step of the present project would consist in applying LRM to the frame

of forward particle methods: the low rank approximation is applied to the shape function, which

evolves at each time step.

- WP LRM-1 (Filbet) : Classical numerical tests in plasma physics. Before to treat physical problems,

the first step is to validate low-rank algorithms for the Vlasov-Poisson system in a simple

configuration. Of course, there are a large variety of classical numerical tests available in the literature

(Landau damping, two-stream instability, plasma echo, bump on the tail). This new algorithm will be

compared to well known numerical methods (semi-Lagrangian, Particle-In-Cell) in term of accuracy,

computational effort and storage.

Risk assessment: R=2: LRM has to be applied to the Vlasov-Poisson system in the particle framework.

We have already materials on the topic with a strong experience on classical numerical tests.

- WP LRM-2 (Filbet) : Phase space filamentation. For most of the numerical tests performed in WP

LRM-1 we only consider small perturbations of space homogeneous plasmas (Landau damping, two

stream instability) hence filamentation (small oscillations) of the distribution function only occurs in

velocity space. However, for charged particle beams, the situation is more complicated since

filamentation occurs in phase space (physical and velocity space) and the mixing is more

sophisticated as halo formations for charged particle beams [28]. Again the Vlasov-Poisson system is

considered with an external electric or magnetic field applied to confine the particle beam. Due to

the nonlinearity, small scales are generated and need to be quantified.

Risk assessment: R=4. This is a crucial step to apply this approach for non-homogeneous plasma. At

this moment we do not know if a perfect gain in accuracy will be observed when comparing with

more standard methods.

5. WP WB/Sheath (Badsi, Crestetto, Berthon, Després, Campos-Pinto)

We will focus on those two main subprojects, which are the construction of reference solutions for

transport in magnetized plasma with sheath and the development of AP (asymptotic-preserving) and

WB (well-balanced) schemes for kinetic plasma sheaths models.

- WP WB/Sheath-1 (Badsi, Després, Campos-Pinto).

Our aim is to construct stationary sheath solutions in a three dimensional geometry with a non-zero

magnetic field B and to identify the role of the boundary condition in the establishment of the

sheath. A very fist step has been done in this direction, in the Phd of Badsi, showing that a scaling

hypothesis is necessary between the Debye length ε and the ion gyrofrequency ω to be able to

construct and simulate the model in full generality. This part of the project is the more theoretical

one: it is however crucial in terms of applications and potential applications, typically for Tokamak.

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Risk assessment = 4. This part of the project is challenging, since the mathematical foundations of

magnetized kinetic sheaths models are still under construction. The risk is high enough, we therefore

plan to decompose this project into several less risky tasks which may include:

The construction of reference magnetized sheath solution in simple geometry, using symmetries of the solutions;

Extension to more general geometry will be the ultimate goal of this part of the project.

- WP WB/Sheath-2 (Badsi, Berthon, Crestetto).

We will discretize the model selected in WP WB/Sheath-1, to use the techniques of asymptotic

preserving schemes to overcome the burden of computational cost and then to investigate the

stability of plasma sheath. Asymptotic preserving scheme are numerical schemes that are design in

order to be consistent and stable in different limit regimes. For plasma sheaths, the physical regime

we are interested in is small Debye length ε ≪ 1. The usual approach for asymptotic preserving

scheme lies on two steps:

A reformulation of the model so that the limit of the model do not degenerate in the limit,

the use of a discretization in time where stiff operators yields implicit solvers.

The choice of an adequate spatial discretization in order to reach good accuracy in space. In

this direction, well-balanced schemes are a natural choice. Following this principle, the first

step to develop a well-balanced scheme, is to start from any AP scheme and to reformulate

the problem in a perturbative regime where the initial condition is the stationary sheath

solution.

A specific treatment of the boundary conditions for both the electric field and the densities, notably

by an assessment of the accuracy of the scheme must be carefully addressed. It is too early to detail a

specific test case: for the development of such methods, we expect strong interactions with the

subtask FKS-2 also involved in the development of AP schemes.

Risk assessment = 3. This part presents a moderate risk since the discretization techniques (that will

be used at first) are standard [11, 44]. The novelty, with respect to the state of the art, steems from

the discretization of the boundary conditions together with discretization in time which must result in

the investigations of the stability of the electrostatic sheath [3].

6. WP Global: Common objectives (all participants)

In this part, we detail the global objective of the project and how it will be implemented among theparticipants.

- WP-G-1. Coordination will be centralized in LJLL by Després. A web page of the project will be editedand maintained, as for the other ANR projects ran in the past by the participants. It will serve as acommon media for collaboration, code exchange, announcements...

- WP-G-2. Common test cases are common to transport codes in different WP. We plan to worktogether on several WPs. In particular:

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when possible the results of the different methods will be compared on the same test-cases. E.g., standard Vlasov-Poisson test problems showing phase-space filamentation can be addressed in WP FBL-1 (LJLL-Paris) and in WP LRM-2 (Sabatier-Toulouse),

the coupling of kinetic equations with continuous models will be addressed in WP FKS-2 (Sabatier-Toulouse) and WP WB/Sheath-2 (Nantes),

the construction of references solutions for sheath in magnetized transport will be addressed in common by Badsi (Nantes) with Després and Campos-Pinto in WP WB/Sheath 2,

the design of high order polynomial extension of FKS concerns the Sabatier team and the LJLLteam, in WP FKS-2,

the development/numerical analsysi of TDG methods WP TDG-1 can be done in parallel/collaboration with the construction of WB numerical schemes WP WB/Sheath 2, since these techniques share many practical similarities. This will be the occasion of strong collaboration between the 3 poles (Paris, Toulouse and Nantes).

WP G-3 Good practice. As MUFFIN is devoted to an efficient practical implementation of multi-scaletransport algorithms, within the MUFFIN task force, we will share practical implementation tools,exchange techniques, share pieces of codes, discuss adaptation to the underlying modeling, andaccelerate the numerical developments and the comparison of the various algorithms.

WP-G-4 A first internal workshop is planned at T0+12, essentially for scientific organization. Themiddle term Workshop approximatively at T0+30 will be the main one for internal scientificcollaboration and presentation of the results.

WP-G-5, Dissemination of the results. At the end of the project, we plan to organize a final workshop(medium size, with international invitees) and the possible publication of a research book, typically int h e LNSCE=Lecture Notes in Computational Science and Engineering series, with a series ofbenchmarks.

The MUFFIN project does not focus on test problems proposed by industrials: as well, it does nothave industrial partners. However, since our the planned major numerical developments,implementation and test cases evaluation are for problems in dimensions 2x-2v and 3x-3v (WP FKS-1,WP FKS-2, WP TDG-1, WP FBL-1), we strongly believe that it will address the current challenges in thetransport simulation which are studied in Research Institutes worldwide. In this sense, the describedstudies will try to address numerical solutions for pre-industrial applications. On this basis, we areconfident that we will be able to foster strong scientific interactions with researchers in theseinstitutes.

Risk assessment: 0.

II. Organisation and implementation of the project

a. Scientific coordinator and its consortium / its team

The proposed scientific coordinator of the PRC is Bruno Després. He was research fellow at the CEA

(Commissariat a l’Energie Atomique) from 1992 to 2009 (a position similar to “Directeur de

Recherche CNRS”), specialist in Scientific Computing for applied physics. He is now Professor in

Applied Mathematics and numerical analysis at the LJLL/Sorbonne University and is still scientific

consultant at CEA. Among publications which assess his experience in the field of numerical modeling

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and computation, let us mention [19] on the basis of long time collaboration at CEA. B. Després ran

the CHROME ANR project 2012-2016.

F. Filbet is member of IUF. Filbet, Campos-Pinto and Charles were (or are) members of Eurofusion

projects with plasma physicists. F. Charles intends to ask for a CNRS delegation. M. Campos-Pinto is

CR CNRS. B. Després is IUF senior 2016-2021, and part of his extra-time for research will be dedicated

to MUFFIN. R. Loubère is DR CNRS. J. Narski and M. H. Vignal are assistant professors and members of

ANR Moonrise (end june 2019). C. Berthon is professor at Nantes, M. Badsi and A. Crestetto are

assistant professors. A. Crestetto is member of the MOHYCON ANR projet (JCJC project, piloted by M.

Bessemoulin).

The consortium is based on 3 pillars. The LJLL one (represented by B. Despres), the Toulouse/

Bordeaux one (represented by F. Filbet) and the Nantes one (represented by C. Berthon). The

LJLL/team (Campos-Pinto, Charles, Després, Hirstoaga, Tournier) is engaged in numerical methods for

applied physics: Campos Pinto and Charles are specialists of the numerical analysis and

implementation of FBL. Després is specialist of Trefftz methods. Tournier is research

engineer/specialist of HPC at LLL. Hirstoaga’s research is on numerical methods for magnetize

transport. The Bordeaux/Toulouse team is made of Filbet (specialist of numerical methods for Vlasov

equation), Loubère and Narski (the fathers of the recent FKS scheme) and Vignal an expert in

numerical analysis of plasma-oriented schemes. The Nantes team is made of Badsi (expert on the

mathematical modelling and numerics for Sheath), together with Berthon (expert on WB) and

Crestetto (kinetic equation). The consortium is well balanced between senior scientists (Campos-

Pinto, Despres, Loubère, Filbet, Berthon, Vignal), younger researchers (Charles, Narski, Crestetto,

Badsi, Tournier, Hirstoaga), and will be reenforced by post-docs. We plan to invite experts and

researchers at the international level to accelerate the scientific developments: a preliminary list is

Ilaria Perugia (specialist of Trefftz methods, Vienna university), Christian Lubich (Numerical analysis

group, Tubingen university), Giacomo Dimarco (univ. Ferrara, Italy, kinetic schemes), George-Henri

Cottet (Grenoble U., supercomputing and transport). In particular, the very strong collaboration

between Loubere and Dimarco are already taken into account in the budget for the Sabatier team (in

the form of higher mission costs). The Gantt chart below implements the planned schedule, which,

we believe, is standard for such collaborations.

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Implication of the partner’s scientific leader in on-going project(s)Name of theresearcher

Person.month Call, funding agency,grant allocated Project’s title

Name of thescientific coordinator

Start - End

Campos-Pinto 20% Eurofusion Magyk Sonnedrucker 2019-2021

Charles 5%

5%

Campus France

(2050 and 6000euros)

- PHC/Galilee

- PHC/Sakura

C. Negulescu

A. Moussa

End 2019

End 2019

Crestetto 14.4 months ANR MOHYCON M. Bessemoulin-Chatard

End 2020

Narski 10 months ANR MOONRISE F. Mehats End 2019

Vignal 20 months ANR MOONRISE F. Mehats End 2019

Tournier 5% - Eurofusion - Defi infinity

- Magyk

The scientific coordinator is not involved in any project.

b. Implemented and requested resources to reach the objectivesThe requested funds correspond to 48 months of post-doctoral fellowships, plus small equipment,missions and organization of the final workshop. In what follows, we describe for each partners theadditional resources dedicated to the project and the requested resources. The highest “general andadministrative costs” line is justified by the organization of the final Worskhop, typically at Cargese

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(CNRS center) with invitation of international researchers, and the Sabatier team will serve as the costcenter for this event ((see also end of previous section a preliminary list of international researchersthat we plan to invite).

Partner 1: LJLL-Sorbonne University. Total= 157 000 euros+8%=169 560 €

Staff expenses: 2 year of post-doctoral fellowship: 2x 55 000 euros, for working on the WP-TDG. The

justification is that the code will implemented almost from scratch, even if preliminary small loops

will be written before: the time to explain the approach and perform basic implementation will take

around one year, so another year is needed to test, optimize and perform physically sound test cases.

So a two years post-doctoral would really be a plus.

We mention that 2 additional PHDs at LJLL are founded and worked on topics related to the project.

One PhD has a university grant and works on transport models with magnetic fields (2018-2021): his

work will be used for constructing references for task WP-TDG-2. The other PHD has an international

grant (CNRS-Lebanon) and works of the numerical theory of TREFFTZ methods. They will participate

to develop the mathematical/numerical analysis of TDG for transport in magnetized plasmas, but a

priori not on the specific implementation developments.

Instruments-material costs: Laptops with advanced facilities for postdocs/particpants: 3 x 3 k€ = 9 k€.

General-administrative costs & other operating expenses: French conferences and internal meetings:

12 x 500 €= 6 k€.

International conferences (USA-like): 8 x 4 k€ = 32 k€.

Subtotal = 38 k€.

Partner 2: Sabatier. Total= 121.5 k€+8%=135.54 k€

Staff expenses

One year of post-doc for the development of numerical solver based on LRM for magnetized plasmas.

The first step is to validate this approach on classical numerical tests and to perform some numerical

analysis to justify its validity for well-suited initial data or linear cases. Cost= 50 k€.

Instruments and material costs

Small equipment is planned: 3 laptops 2.5 k€ each = 7.5k€

General and administrative costs & other operating expenses

The specificity of the collaborations between Loubere and Dimarco (Ferrara-Italy) will be covered by

4 trips during 4 years = 16 k€.

2 workshops will be organized, that is 2x10 k€ = 20k€.

Finally, the highest budget is for international conferences, that is 4k€ x 2 confs x 4 years = 32k€

Partner N: Nantes. Total = 96 k€ + 8% = 103.68 k€

Staff expenses: We request a one year post-doc funding to recruit a young researcher specialized in

scientific computing and numerical analysis throughout the second year of the project. The person

will be recruited mainly as a support for task WP- of this work package. A strong interaction between

permanent researchers and the recruited person is intended, be it in the development of asymptotic

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preserving schemes and their implementation. An important task of the recruited person will consist

in the implementation and the assessment of the numerical methods on physically based (namely

magnetized or unmagnetized sheath) test cases for which a huge literature is available. Cost: 60 k€

(charged salary).

Small equipment and others: We ask for two laptops computers (2 x 2 k€) for the project’s

participants (including the Post-Doc). We also require 2000 euros to buy books and possibly subscribe

to some scientific software licence that are not already granted by our university. Total: 6 k€.

Total=66 k€.

General and administrative costs & other operating expenses Missions: Travel expenses will cover an

annual project advancement meeting for the members of the team since they belong to different

laboratories (3x500 euros). We also ask for support as regards to the participation to national and

international conferences (3 x 3 k€/year) including:

a) Participation to a top international conference, where we aim to present our results and exchange

with specialists;

b) Participation to an annual workshop participation to Numkin (between Strasbourg and Garching

(Germany)) on numerical method for kinetic models is also considered;

c) Participation of the members to the CEMRACS 2020, a one month summer school that takes place

in Luminy, whose topic will be devoted to the numerical and mathematical modeling of system of

large number of particles.

Total: 30 k€.

Requested means by item of expenditure and by partner*

Partner LJLL Partner IMT Partner LJL

Staff expenses 2X55 k€ = 110 k€ 50 k€ 60k€

Instruments and material costs(including the scientific consumables)

9k€ 7.5 k€ 6 k€

General andadministrative c o s t s &otheroperatingexpenses

Travel costs 38 k€ 68 k€ 30 k€

Administrativem a n a g e m e n t &structure costs**

12 560 € 10 040 € 7 680 €

Sub-total 149 040 € 135 540 € 103 680 €

Requested funding 388 260 €

The calculation of the administrative management and structure cost has been done on a 8 % basis,which is the rule at Sorbonne University. It corresponds to 4% for university costs and 4% forlaboratory costs.

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I. Impact and benefits of the project

The expected impact of the project will be in 3 different fields.

- The first one is the community of numerical analysis and applied mathematics, which is ourprimarily field. Indeed we truly believe that some of the methods that will be investigated will beinnovative (or very innovative), so we expect good dissemination among specialized journals.

- The second field is applied science and numerical plasma physics for ITER oriented applications. Themembers of the team are already well implicated. Actually some of the project parts correspond toscientific questions addressed in this more physically oriented community. We are regularparticipants of the Numkin series of Workshops for example. It will be a natural applicativecommunity to present our results.

- The development of new HPC oriented algorithms and use of HPC techniques will also be a greatopportunity to have fruitful interactions with the computer oriented community.

We strongly believe our participation to these three fields will naturally guarantee the impact of ourresults in the scientific community.

I. References related to the project

[1] Badsi et al, A minimization formulation of a bi-kinetic sheath, KRM (2016).

[2] Badsi, Linear electron stability for a bi-kinetic sheath model, J. of Math. Anal. and App. (2017).

[3] Badsi, Mehrenberger and Navoret Numerical stability of a plasma sheath, ESAIM Proc. (2018).[4] Baruq, Calandra, Diaz and Shishenina, Space-Time TDG app. for Elasto-Acoustics, HAL (2018).

[5] Ben Abdallah, Weak sol. of the initial-boundary value prob. for the Vlasov-P. sys. M2AS (1994).[6] Bessemoulin-Chatard, Herda and Rey, Hypocoercivity and diffusion limit of a finite volume schemefor linear kinetic equations, prep. (2018).[7] Bonazzoli, Dolean, Graham, Spence and Tournier. Two-level preconditioners for the Helmholtzequation, HAL (2018).[8] Birdsall and Langdon, Plasma physics via computer simulation. McGraw-Hill (1985).[9] Campos-Pinto and Charles, Uniform Convergence of a Linearly Transformed Particle Method forthe Vlasov--Poisson System, SINUM (2016).[10] Campos Pinto,Charles, From particle methods to forward-backward Lag. schemes, (2018).

[11] Campos Pinto, Unstructured Forward-Backward Lag. Scheme for Transport Prob, FVCA8 (2017).

[12] Chalise,Khanal, A kinetic trajectory simulation model for mag. plasma sheath, Plas. Phys, (2012).[13] Chang and Cooper, A practical difference scheme for Fokker-Planck equations, JCP (1970).

[14] Chen, Introduction to Plasma Physics and controlled fusion. Springer (1984).[15] Chodura, Plasma-wall transition in an oblique magnetic field, AIP Publishing (1982).

[16] Cottet, Etanceli, Perignon and Picard, High order Semi-Lagrangian particles for transportequations: numerical analysis and implementation issues, M2AN (2014).[17] Crouseilles, Lemou, Mehats and Zhao, Uniformly accurate forward semi-Lagrangian methods forhighly oscillatory Vlasov-Poisson equations, MMS (2017).[18] Degond,Mustieles, A deterministic approximation of diffusion eq. using particles, SISC (1990).[19] Després. Numerical methods for Eulerian and Lagrangian conservation laws. (2017).[20] Dimarco, Hauck and Loubère, Class of low dissipative schemes for solving kinetic eq., JSC (2018).

[21] Dimarco, Loubère, Narski and Rey An efficient numerical method for solving the Boltzmann

equation in multidimensions, JCP (2018).

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https://hal.archives-ouvertes.fr/hal-01525424

https://hal.archives-ouvertes.fr/hal-01525424

https://hal.archives-ouvertes.fr/search/index/?q=*&authIdHal_s=pierre-henri-tournier

https://hal.archives-ouvertes.fr/search/index/?q=*&authFullName_s=Euan+Spence

https://hal.archives-ouvertes.fr/search/index/?q=*&authFullName_s=Ivan+Graham

https://hal.archives-ouvertes.fr/search/index/?q=*&authIdHal_s=victorita-dolean

https://hal.archives-ouvertes.fr/search/index/?q=*&authIdHal_s=marcella-bonazzoli




[22] Dimarco, Loubère, Michel-Dansac and Vignal, Second order Implicit-Explicit Total Variation

Diminishing schemes for the Euler system in the low Mach regime, JCP (2018).

[23] Dimarco, Loubère and Vignal, Study of a new Asymptotic Preserving scheme for the Euler system

in the low Mach number limit, SISC (2017).

[24] Dujardin, Hérau and Lafitte Coercivity, hypocoercivity, exponential time decay and simulations fordiscrete Fokker-Planck equations, submitted (2018).[25] Einkemmer and Lubich, Low-rank projector-splitting integrator for the Vlasov-P., prep. (2018).

[26] Einkemmer and Lubich, A quasi-conservative dynamical low-rank algo. for the Vlasov eq., (2018).[27] Erlacher et al, Progress rep. on the implementation of kinetic elec. in GYSELA, Numkin (2016).

[28] F. Filbet and E. Sonnendrücker, Modeling and Numerical Simulation of Space Charge Dominated

Beams in the Paraxial Approximation, M3AS (2006).

[29] Filbet and Xiong, A hybrid discontinuous Galerkin scheme for multi-scale kinetic eq., JCP (2018).

[30] Francis, Xiong,Sonnendrücker, On Vlasov-Maxwell system with strong mag. field, (SIAP) (2018).

[31] Filbet and Rodrigues, Asymptotically preserving particle-in-cell…, SINUM (2017).

[32] Grandgirard and Sarazin, Gyrokinetic simulations of magnetic fusion plasmas, (2013).

[33] Haddad, Sayah, Hecht and Tournier, Parallel computing investigations for the projection method

applied to the interface transport scheme of a 2-phase flow ..., Num Alg. (2019).[34] Heuraux et al. Plasma sheath properties in a magnetic field parallel to the wall, PoP (2016).

[35] Kormann, A semi-Lagrangian Vlasov solver in tensor train format, SISC (2015).

[36] Lions and Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives nonlinéaires, CRAS (2001).[37] Magni and Cottet, Accurate, non-oscillatory, remeshing schemes for particle methods. JCP (2012).[38] Manfredi and Devaux, Magnetized plasma wall transition. Consequences for wall sputtering anderosion, Institute of Physics publishing (2008).[39] Morel, Buet and Després, Trefftz Discontinuous Galerkin Method for Friedrichs Systems with

Linear Relaxation: Application to the P1 Model, CMAM (2017).

[40] Morel, Buet and Després, Trefftz Discontinuous Galerkin basis functions for a class of Friedrichs

systems coming from linear transport, HAL online (2019).

[41] Morel, Asymptotic-preserving and well-balanced schemes for transport models using Trefftz

discontinuous Galerkin method, PhD Thesis, Sorbonne University, (2018).

[42] Nair, Scroggs and Semazzi, A forward-trajectory global semi-Lagrangian trans. sch. JCP, (2003).[43] Narski, Dimarco and Loubère, Ultra efficient kinetic scheme Part III: High Perf. Comp. JCP (2015).

[44] Pham, Helluy,Crestetto, Space-only hyperbolic approximation of the Vlasov eq., ESAIM (2013).[45] Riemann, The bohm criterion and sheath formation, Physics of Plasmas (1991).[46] Stangeby, The plasma boundary of magnetic fusion devices. IOP publishing, (2000).

[47] Tournier et al, Microwave Tomographic Imaging of Cerebrovascular Accidents by Using High-

Performance Computing, Parallel Computing (2019).

[48] Tournier, Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers Proc.

ICHPC- Networking, Storage and Analysis (2016).[49] Umeda and Fukazawa, Performance comparison of Eulerian kinetic Vlasov code between flat-MPI

parallelism and hybrid parallelism on Fujitsu FX100 supercomputer, (2016).

[50] Valsaque,Manfredi, Numerical study of plasma wall transition in an oblique mag. field, (2001).

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