AAPG2019 MUFFIN Funding instrument : PRC Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros scientific evaluation committee : 5.6. Modles numriques, simulation, applications MUFFIN (MUlscale and treFFtz for numerIcal transport) Summary table of persons involved in the project: Partner Name First name Current posion Role & responsibilies in the project (4 lines max) Involvement (person.month) throughout the project's total duraon Sorbonne Univ. Després Bruno PR - PI and coordinator of the whole project - PI of the LJLL Team - Researcher on TDG 20 months, over 4 years, that is 41.66 %. Sorbonne Univ. Tournier Pierre-Henri Ing.-CNRS - Developer FBL and TDG 10 months Sorbonne Univ. Campos- Pinto Marn 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 Sabaer Univ. Filbet Francis PR - PI of the Sabaer team, - Researcher LRM 24 months Sabaer Univ. Vignal Marie- Hélène MCF - Researcher/developer FKS 12 months Sabaer Univ. Loubere Raphael DR-CNRS - Researcher/developer FKS/AP 12 months Sabaer 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. Cresteo Anais CR - researcher WB 12 months Nantes Univ. Badsi Mehdi CR - Researcher/developer WB/Sheath 15 months Modificaons 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 an expert on transport methods in magnezed plasma. The reason of the modificaon is that Sever recently moved in fall 2018 from Strasbourg to Inria-Paris, so it is an opportunity. Also, Marn Campos Pinto will move to a Max-Planck instute in Munich. He will connue to parcipate in the scienfic development of the project, but as a member of a foreign instute (with his own funding): therefore his involvement is not accounted for in the budget, so the 0 month in the last column. b) The requested funding has increased from 370 k€ (which was announced in the pre-proposion) to 408.78 k€, essenally to the applicaon of administrave costs (+ 8%). 1
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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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.
Risk assessment: R = 2.
- 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.
Risk assessment: R = 3.
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
(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).
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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros
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
[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
[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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AAPG2019 MUFFIN Funding instrument : PRC
Coordinated by: Bruno DESPRES Duration 4 years Requested Funding : 408 780 euros