Asymptotic-Preserving Schemes for anisotropic diffusion problems related to magnetized plasma simulation

Transcription

1 Asymptotic-Preserving Schemes for anisotropic diffusion problems related to magnetized plasma simulation Fabrice Deluzet 1 joint work with C. Besse 2, S. Brull 3, A. Crestetto 1, P. Degond 1, A. Lozinski 4, A. Mouton 2, J. Narski 1, C. Negulescu 1, C. Yang 5 1 Université de Toulouse, UPS, INSA, UT1, UTM & CNRS Institut de Mathématiques de Toulouse UMR 5219, F Toulouse 2 Université de Lille, Lab. Paul Painlevé UMR 8524, Villeneuve d Ascq 3 Université de Bordeaux 1, MATMECA, IMB-MAB, Talence 4 Université de Franche-Comté, Lab. de Mathématiques, UMR 6623, Besançon 5 Université de Lyon, Institut Camille Jordan, Villeurbanne Fabrice.Deluzet@math.univ-toulouse.fr GDR Chant «Transport dans les milieux micro-structurés» Les Ramayes des 7-Laux, 7-11 January 2013

2 Outline Introduction 1 Introduction 2 3 4

3 Talk goals Introduction «Asymptotic-Preserving» numerical methods Numerical method efficient for singular perturbation problems. Introduced by S. Jin a for diffusion limits of kinetic models. a. S. Jin, Efficient Asymptotic-Preserving (AP) Schemes for Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., 21(1999),

4 Talk goals Introduction «Asymptotic-Preserving Framework» numerical : Numerical methods resolution of a multiscale problem (P ε ) Numerical method efficient for singular perturbation problems. f Introduced by S. Jin a ε is the solution of the singularly perturbated problem (P ε ) for diffusion limits of kinetic models. the sequence f ε converges towards f 0, solution of the limit a. S. Jin, Efficient Asymptotic-Preserving problem (P 0 )(AP) Schemes for Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., (P 0 ) 21(1999), is different in nature from the system of equations (P ε ) Numerical difficulties (P ε ) numerically useless for small ε-values, (P 0 ) not accurate for ε = O(1), Coupling strategy difficult to implement : different unknowns, different numerical methods, moving interface,... Consistancy with (P ε ) for ε = O(1), Consistancy with (P 0 ) for ε 1, Inconditional stability with respect to ε.

5 Talk goals Introduction «Asymptotic-Preserving» numerical methods Numerical method efficient for singular perturbation problems. Introduced by S. Jin a for diffusion limits of kinetic models. a. S. Jin, Efficient Asymptotic-Preserving (AP) Schemes for Some Multiscale Kinetic Equations, SIAM J. Sci. Comput., 21(1999), Special focus on anisotropic diffusion problems Model problem 2 φ (P ε ) x ε 2 φ z 2 = f(x,z), in Ω = Ω x Ω z, φ(x, ) = 0, on Ω x, z φ(,z) = 0, on Ω z. Physical background motivating the investigation of these class of singular perturbation problems : magnetized plasma simulation.

6 Sponsors Introduction Instability of the ionospheric plasma IODISSEE «IOnospheric DIsturbanceS and SatEllite-to-Earth communications», ANR (National Research Agency) SYSCOM project call ( ) http ://iodissee.math.cnrs.fr/ Long term collaboration with the French Nuclear Agency (CEA CESTA) in the frame of the STRIATIONS, DYNAMO, ACADIA, PICCADI, SINEVOCADI, COLLICADIA, HYPARCADIA research contracts.

7 plasma The European Atomic Energy Community (Euratom CEA/Cadarache) BOOST «Building the future Of numerical methods for iter», ANR (National Research Agency) BLANC project call ( ) http ://perso.math.univ-toulouse.fr/boost RTRA STAE «Réseau Thématique de Recherche avancée Sciences et Technologie pour l Aéronautique et l Espace» Toulouse University in the frame of the MOSITER «MOdelling and Simulation for ITER» project CNRS «Centre National de la Recherche Scientifique» INRIA «Action d Envergure Nationale» Fr-FCM «Fédération de Recherche Fusion par Confinement Magnétique»

8 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

9 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

10 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

11 The earth environment Height (km) Density of neutrals (n.cm 3 ) n Thermosphere Mesosphere Stratosphere Troposhere T Temperature ( o K) Magnetosphere Ionosphere D region ne E region Day F region Night Density of e (n e.cm 3 ) Figure 1 Atmosphere stratification. The atmosphere is a stratified and ionosed medium, The ionosphere is characterized by altitudes ranging from 90 to 1500 kilometers, a density for neutral particles larger thant that of the plasma, a maximum density value 10 6 cm 3, in the F-region (altitudes 300 km).

12 Properties of the ionosphere Ionosphere long wave space wave sky wave TV, phones short wave sky wave UHF VHF AM radio ground wave ground wave FM radio, TV Figure 2 Radio waves transmission through the ionosphere. The ionosphere : reflects low frequency waves (ω ω p ), transmits high frequency waves (ω ω p ), submitted to numerous instabilities (solar eruptions, striations,...).

13 Introduction Ionospheric plasma instability : Striations Northern Hemisphere m Plasma bubble evolution in the F-region (altitudes 300 km). 0k 10 km 15 wind Neutral wind Ionosphere Troposphere Sea level Cosmic Ray Time Southern Hemisphere The plasma bubble is stretched along the magnetic field lines In a plane orthogonal to the magnetic field the plasma is submitted to the E B instability. Figure 3 Striations instability formation. F. Deluzet AP-Schemes for anisotropic diffusion problems

14 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

15 Notations Introduction n e, n i u e, u i u n P e, P i ν e, ν i m e, m i j = e(n i u i n e u e ) current density ρ c = e(n i n e ) charge density electronic, ionic densities electronic, ionic velocities velocity of neutral particles electronic, ionic pressure electron to neutral, ion to neutral collision frequency electronic, ionic, mass

16 The Euler-Maxwell system Euler equations L ue (n e ) = 0, { Lorentz }} { m e L ue (n e u ek ) = xk P e en e (E k +(u e B) k )+n e F ek, L ui (n i ) = 0, m i L ui (n i u ik ) = xk P i +qn i (E k +(u i B) k )+n i F ik, Maxwell s system 1 c 2 te B = µ 0 j, t B + E = 0, E = ρ c /ε 0, B = 0, ρ c = q(n i n e ), j = q(n i u i n e u e ). Where, the transport operator is defined as : L u (ρ) = tρ+ (ρu), and the friction forces read : F e = ν em e(u e u n), F i = ν im i(u i u n).

17 Scaling relations Introduction Physical quantity Order of Dimensionless grandeur parameter Typical value Time t t = t/ t 10 3 s Length x x = x/ x 10 5 m Velocity ū = x/ t u e,i,n = u e,i,n/ū 10 2 ms 1 Density n n e,i = n e,i/ n m 3 Magnetic field B B = B/ B 10 5 T Electric field Ē = ū B E = E/Ē 10 3 Vm 1 Collision freq. e-n ν e ν e = ν e/ ν e 10 2 s 1 Collision freq. i-n ν i = me m i ν e ν i = ν i/ ν i 10 2 s 1

18 Dimensionless parameters Writing the equations with dimensionless variables introduces β induced magnetic field intensity relative to that of the earth magnetic field, κ number of collisions during a cyclotron period, ε electron to ion mass ratio, α typical velocity to speed of light ratio, τ ratio of the time between two collisions to the typical time scale, η thermal energy scale.

19 The dimensionless Euler-Maxwell Euler Equations t n e + (n e u e ) = 0, Lorentz friction {}}{{}}{ τε(l ue (n e u ek )) = η xk P e κ 1 n e (E k +(u e B) k ) ν e n e (u ek u nk ), t n i + (n i u i ) = 0, τ (L ui (n i u ik )) = η xk P i +κ 1 n i (E k +(u i B) k ) ν i n i (u ik u nk ), Maxwell system α t E B = βj, t B + E = 0, κα β E = ρ c, B = 0, ρ c = n i n e, κj = n i u i n e u e. Typical values n i,e = m 3 ε = 10 4, τ = 10 1, η = 10 1, κ = 10 4 α = 10 12, β =

20 The dimensionless Euler-Maxwell Euler Equations t n e + (n e u e ) = 0, Lorentz friction {}}{{}}{ τε(l ue (n e u ek )) = η xk P e κ 1 n e (E k +(u e B) k ) ν e n e (u ek u nk ), t n i + (n i u i ) = 0, τ (L ui (n i u ik )) = η xk P i +κ 1 n i (E k +(u i B) k ) ν i n i (u ik u nk ), Maxwell system α t E B = βj, t B + E = 0, κα β E = ρ c, B = 0, ρ c = n i n e, κj = n i u i n e u e. Typical values n i,e = m 3 ε = 10 4, τ = 10 1, η = 10 1, κ = 10 4 α = 10 12, β =

21 The dimensionless Euler-Maxwell Euler Equations t n e + (n e u e ) = 0, Lorentz friction {}}{{}}{ 0 = η xk P e κ 1 n e (E k +(u e B) k ) ν e n e (u ek u nk ), t n i + (n i u i ) = 0, τ (L ui (n i u ik )) = η xk P i +κ 1 n i (E k +(u i B) k ) ν i n i (u ik u nk ), Maxwell system α t E B = βj, t B + E = 0, κα β E = ρ c, B = 0, ρ c = n i n e, κj = n i u i n e u e. Typical values n i,e = m 3 ε = 10 4, τ = 10 1, η = 10 1, κ = 10 4 α = 10 12, β =

22 The dimensionless Euler-Maxwell Euler Equations t n e + (n e u e ) = 0, Lorentz friction {}}{{}}{ 0 = η xk P e κ 1 n e (E k +(u e B) k ) ν e n e (u ek u nk ), t n i + (n i u i ) = 0, τ (L ui (n i u ik )) = η xk P i +κ 1 n i (E k +(u i B) k ) ν i n i (u ik u nk ), Maxwell system B = βj, Typical values n i,e = m 3 t B + E = 0, ε = 10 4, τ = 10 1, 0 = ρ c, quasi-neutrality η = 10 1, κ = 10 4 B = 0, α = 10 12, β = ρ c = n i n e, κj = n i u i n e u e.

23 Euler-Maxwell ε 0, α 0 MHD hierarchy Hall-MHD Dynamo hierarchy κ 0 τ 0 Finite conductivity-mhd Massless Hall-MHD τ 0 β 0 Massless MHD Dynamo β 0 κ 0 Striation Figure 4 Model hierarchies Dynamo hierarchy : regular plasma density, MHD hierarchy : large plasma density.

24 Model hierarchy publications Introduction C. Besse, J. Claudel, P. Degond, F. Deluzet, G. Gallice, C. Tessieras, A model hierarchy for ionospheric plasma modeling, Mathematical Models and MHD Methods hierarchy in Applied Sciences 14 (2004), pp Instability of the ionospheric κ 0 plasma : modeling and analysis, SIAM Appl. Math. 65 (2005), pp Finite conductivity-mhd Ionospheric plasmas : model derivation, stability analysis and numerical simulations, in Numerical Methods for Hyperbolic β 0 Hall-MHD ε 0, α 0 τ 0 τ 0 β 0 Massless MHD Euler-Maxwell Massless Hall-MHD Dynamo κ 0 and Kinetic Problems, European Striation Mathematical Society, Zürich, Figure 4 Model hierarchies Dynamo hierarchy : regular plasma density, MHD hierarchy : large plasma density. Dynamo hierarchy

25 Introduction Simulations thanks to the «Striation» model Upper ionosphere Lower ionosphere β=constante γ=constante α β γ Figure 6 Curvilinear coordinates for a bi-polar magnetic field. Magnetic field tube Computational domain Figure 5 Computational domain in the earth environment Figure 7 Discretized magnetic field tube

26 Simulations thanks to the «Striation» model Figure 8 Plasma density and velocity after the instability development

27 Numerical simulation publication Simulations thanks to the «Striation» model C. Besse, J. Claudel, P. Degond, F. Deluzet, G. Gallice, C. Tessieras, Numerical simulations of the ionospheric striation model in a non-uniform magnetic field, Computer Physics Communications, 176 (2007), pp Figure 8 Plasma density and velocity after the instability development Figure 9 Density measured during the checkmate experiment.

28 Limitations of the «Striation» model Euler-Maxwell ε 0, α 0 MHD hierarchy Hall-MHD Dynamo hierarchy κ 0 τ 0 Finite conductivity-mhd Massless Hall-MHD τ 0 β 0 Massless MHD Dynamo β 0 κ 0 Striation Figure 10 Model Hierarchy.

29 Limitations of the «Striation» model κ 0 β 0 Euler-Maxwell ε 0, α 0 Main assumptions MHD hierarchy of the Striation model Hall-MHD Quasi-neutral inertial-less plasma ; Magnetic field reduced to the earth magnetic field Finite ; conductivity-mhd electron-ion collisions negligible ; Compared to the transverse (Perdersen and Hall) Massless mobilitymhd the aligned mobility is infinite ( 1/κ). τ 0 Massless Hall-MHD τ 0 β 0 Dynamo κ 0 Dynamo hierarchy Striation Figure 10 Model Hierarchy.

30 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

31 The «3D-Dynamo» model quasi-neutral plasma : n = n e = n i, B = B z e z coincides with the earth magnetic field, on the typical time scales the magnetic field is assumed constant B/ t = 0, t n+ (nu i ) = 0, u e = M e ( E +κν e u n ), u i = M i (E +κν i u n ), E = φ, j = 0, κj = n(u i u e ), µ P e µ H e 0 M e = µ H e µ P e 0, 0 0 µ e µ P i µ H i 0 M i = µ H i µ P i 0, 0 0 µ i with µ P e,i = κν e,i (κν e,i ) 2 + B 2, B µh e,i = (κν e,i ) 2 + B 2, µ e,i = 1. κν e,i

32 The «3D-Dynamo» model Electrostatic potential equation (3D-Dynamo model) (nm φ) = J n, with (a) Perdersen to aligned mobility ratio µ P /µ (estimated by IRI and MSISE models). (b) Hall to aligned mobility ratio µ H /µ (estimated by IRI and MSISE models). Figure 11 Transverse to aligned mobility ratios (decimal log-scale : color scale associated to the power of ten). M = M e +M i µ P µ H 0 = µ H µ P 0, 0 0 µ and µ P = µ P e +µ P i, µ H = µ H e µ H i, µ = µ e +µ i.

33 Electric field computation j = 0 with j = σe = σ φ, σ = nm is the conductivity tensor σ σ σ 0, σ σ, 0 0 σ reconnecting to the neutral atmosphere j n = 0 ( x φ = 0) Toy problem 2 φ x φ ε z 2 = f, φ(,z) = 0, on Ω z z φ(x, ) = 0, on Ω x. in Ω = Ω x Ω z

34 Electric field computation j = 0 with j = σe = σ φ, σ = nm is the conductivity tensor σ σ σ 0, σ σ, 0 0 σ earth B x3 B x2 x1 B B ρ B reconnecting to the neutral atmosphere j n = 0 ( x φ = 0) Toy problem 2 φ x φ ε z 2 = f, φ(,z) = 0, on Ω z z φ(x, ) = 0, on Ω x. in Ω = Ω x Ω z

35 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

36 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

37 ITER : «Fusion is an energy source of the future» Recipe to overcome the Coulomb barrier Heat to thousands of million of degrees a dense plasma. Confine this plasma to maximize collisions and minimize energy loss in the plasma core thanks to an intense magnetic field ( times higher than that of the Earth). (a) Device schematic representation. Figure 12 ITER (International Thermonuclear Experimental Reactor) : magnetically confined fusion.

38 ITER : «Fusion is an energy source of the future» Recipe to overcome the Coulomb barrier Heat to thousands of million of degrees a dense plasma. Confine this plasma to maximize collisions and minimize energy loss in the plasma core thanks to an intense magnetic field ( times higher than that of the Earth). (a) Device schematic representation. (b) Plasma heating sources. Figure 12 ITER (International Thermonuclear Experimental Reactor) : magnetically confined fusion.

39 ITER : «Fusion is an energy source of the future» Recipe to overcome the Coulomb barrier Heat to thousands of million of degrees a dense plasma. Confine this plasma to maximize collisions and minimize energy loss in the plasma core thanks to an intense magnetic field ( times higher than that of the Earth). (a) Device schematic representation. (b) Plasma heating sources. (c) Magnetic field lines. Figure 12 ITER (International Thermonuclear Experimental Reactor) : magnetically confined fusion.

40 Context of magnetically confined plasmas : hot dense plasma, almost collisionless, large magnetic field (time varying quantity), quasi-neutrality (almost everywhere). Goals Develop a numerical method with no constraints on the discretization parameters ( x, t) related to the space of time scales of particle gyro motion, no constraints on ( x, t) related to acoustic waves propagation, no constraints on ( x, t) related to the Debye length or plasma frequency, no use of aligned coordinates with respect to the magnetic field.

41 Context of magnetically confined plasmas : hot dense plasma, almost collisionless, large magnetic field (time varying quantity), quasi-neutrality (almost everywhere). Goals Develop a numerical method with no constraints on the discretization parameters ( x, t) related to the space of time scales of particle gyro motion, no constraints on ( x, t) related to acoustic waves propagation, no constraints on ( x, t) related to the Debye length or plasma frequency, no use of aligned coordinates with respect to the magnetic field. Figure 13 Particle gyromotion

42 Context of magnetically confined plasmas : hot dense plasma, almost collisionless, large magnetic field (time varying quantity), quasi-neutrality (almost everywhere). Goals Develop a numerical method with no constraints on the discretization parameters ( x, t) related to the space of time scales of particle gyro motion, no constraints on ( x, t) related to acoustic waves propagation, no constraints on ( x, t) related to the Debye length or plasma frequency, no use of aligned coordinates with respect to the magnetic field.

43 Context of magnetically confined plasmas : hot dense plasma, almost collisionless, large magnetic field (time varying quantity), quasi-neutrality (almost everywhere). Goals Develop a numerical method with no constraints on the discretization parameters ( x, t) related to the space of time scales of particle gyro motion, no constraints on ( x, t) related to acoustic waves propagation, no constraints on ( x, t) related to the Debye length or plasma frequency, no use of aligned coordinates with respect to the magnetic field.

44 Context of magnetically confined plasmas : hot dense plasma, almost collisionless, large magnetic field (time varying quantity), quasi-neutrality (almost everywhere). Goals Develop a numerical method with no constraints on the discretization parameters ( x, t) related to the space of time scales of particle gyro motion, no constraints on ( x, t) related to acoustic waves propagation, no constraints on ( x, t) related to the Debye length or plasma frequency, no use of aligned coordinates with respect to the magnetic field.

45 Outline Introduction 1 Introduction 2 The earth ionosphere A hierarchy of models The Dynamo model Context and goals The Euler-Lorentz model in the drift limit 3 4

46 The bi-fluid Euler-Lorentz system The E-L system (for species α) n + q = 0, t (1a) ( q m α t + (q q ) ) + p = q α (ne +q B), (1b) n W + ((W +p)u )) = q α E q, (1c) t Electrostatic field computation W = 1 2 m q 2 α n p, p = nk BT. (1d) φ = q ε 0 (n n e ), E = φ. (2a) (2b)

47 The dimensionless bi-fluid Euler-Lorentz system Phenomena characteristics x, t, ū and T are the typical values of the length, time, velocity and temperature of the experiment. Dimensionless parameters electron to ion mass ration ε = me m i, dimensionless Debye length λ = ( ) 1 λd x = ε0k B T 2 1 x n q 2 number of Larmor rotations per characteristic time tω c = t qb the Mach number M = ū2 c, c 2 s 2 s Scaling relations = kb T m i. typical velocity ū = x t, equal electric and magnetic force magnitude : Ē = ū B, Mach number scaled to the dimensionless ions cyclotron period reciprocal : 1 M 2 = tω c = 1 τ, m i,

48 Dimensionless E-L system n + q = 0, t q t + (q q ) + 1 n τ p = 1 (ne +q B), τ W + ((W +p) q = E q, t n)) W = τ 1 q 2 2 n + 3p 2, p = nt. Drift regime τ 0 Momentum equations degenerate into p = (ne +q B) giving q = B/ B 2 ( P +ne) but no information for q. Reformulation of the problem Compute the pressure in order to guarantee p = (ne +q B) = O(τ), Need to solve an anisotropic ( diffusion equation p [1 ] ) t τ (b b)+(id b b) p Periodic boundary conditions on Γ. = f,

49 Euler-Lorentz in the drift limit [Degond] a, [Negulescu] b Overview of AP methods for magnetized plasmas. [DDSV09] c Iso-thermal E-L model with external (E,B) ( 1D, ions). [BDD12] d Iso-thermal E-L with external (E,B) (2D, ions). [BDDM11] e Quasi-neutral iso-thermal bi-fluid E-L with external B (2D). Ill-posed problem for the electric potential in the drift limit. a. P. Degond, Asymptotic-Preserving Schemes for Fluid Models of Plasmas, to appear in the collection Panoramas et Syntheses of the SMF. http :// degond/ b. C. Negulescu, Asymptotic-Preserving schemes. Modeling, simulation and mathematical analysis of magnetically confind plasmas, submitted to Riv. Mat. Univ. Parma. http :// cnegules/ c. P. Degond, F. Deluzet, A. Sangam, M-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, JCP 228 (2009), pp d. S. Brull, P. Degond, F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasma simulations,cicp, 11 (2012), pp e. S. Brull, P. Degond, F. Deluzet, A. Mouton, Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model, KRM 4 (2011), pp

50 Outline Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 1 Introduction 2 3 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 4

51 Outline Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 1 Introduction 2 3 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 4

52 Problematic statement Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Dynamo problem { φ = 0 on Ωx (nm φ) ( = J n, ) in Ω, nφ = 0 on Ω y Ω z µ P µ H 0, with M = µ H µ P 0, ε = µ P /µ = µ H /µ µ Model problem { (P ε x (A ) x φ ε ) 1 /ε z (A z z φ ε ) = f in Ω, ( ) φ = 0 on Ω x, z φ ε = 0 on Ω z. What can we expect from standard numerical methods? For ε 1 consistency with the degenerated problem { ( P 0 z (A ) z z φ 0 ) = 0 in Ω z φ 0 = 0 on Ω z. Blow-up of the condition number with ε 0.

53 Limit regime Problematic statement Problematic statement Derivation of Asymptotic-Preserving methods Main achievements φ 0 = lim ε 0 φ ε is { the solution of Dynamo problem (P 0 x ) (Ā xφ 0 ) = f in Ω, φ = 0 on{ Ω x. φ = 0 on Ωx (nm φ) ( = J n, ) in Ω,, Indeed nφ = 0 on Ω y Ω z µ P: µ H 0 with M = φ ε µ H verifies µ P 0, ε µ P /µ = µ H /µ x (A x φ ε ) = f in Ω, pour ε > 0, µ with : f = 1 Lz /L z 0 f(x,z)dz. Eq. ( ) in the limit ε 0 gives Model problem z φ 0 = 0 { (P ε x (A ) x φ ε ) 1 /ε z (A z z φ ε ) = f in Ω, ( ) φ = 0 on Ω x, z φ ε = 0 on Ω z. What can we expect from standard numerical methods? For ε 1 consistency with the degenerated problem { ( P 0 z (A ) z z φ 0 ) = 0 in Ω z φ 0 = 0 on Ω z. Blow-up of the condition number with ε 0.

54 Limit regime : Singular limit Problematic statement Problematic statement Derivation of Asymptotic-Preserving methods Main achievements φ 0 = lim ε 0 φ ε is { the solution of Dynamo problem (P 0 x ) (Ā xφ 0 ) = f in Ω, φ = 0 on{ Ω x. φ = 0 on Ωx (nm φ) ( = J n, ) in Ω,, Indeed nφ = 0 on Ω y Ω z µ P: µ H 0 with M = φ ε µ H verifies µ P 0, ε µ P /µ = µ H /µ x (A x φ ε ) = f in Ω, pour ε > 0, µ with : f = 1 Lz /L z 0 f(x,z)dz. Eq. ( ) in the limit ε 0 gives Model problem z φ 0 = 0 { (P ε x (A ) x φ ε ) 1 /ε z (A z z φ ε ) = f in Ω, ( ) φ = 0 on Ω x, z φ ε = 0 on Ω z. What can we expect from standard numerical methods? For ε 1 consistency with the degenerated problem { ( P 0 z (A ) z z φ 0 ) = 0 in Ω z φ 0 = 0 on Ω z. Blow-up of the condition number with ε 0.

55 Limit regime : Singular limit Problematic statement Problematic statement Derivation of Asymptotic-Preserving methods Main achievements φ 0 = lim ε 0 φ ε is { the solution of Dynamo problem (P 0 x ) (Ā xφ 0 ) = f in Ω, φ = 0 on{ Ω x. φ = 0 on Ωx (nm φ) ( = J n, ) in Ω,, Indeed nφ = 0 on Ω y Ω z µ P: µ H 0 with M = φ ε µ H verifies µ P 0, ε µ P /µ = µ H /µ x (A x φ ε ) = f in Ω, pour ε > 0, µ with : f = 1 Lz /L z 0 f(x,z)dz. Eq. ( ) in the limit ε 0 gives Model problem z φ 0 = 0 { (P ε x (A ) x φ ε ) 1 /ε z (A z z φ ε ) = f in Ω, ( ) Relation φ = with 0 onthe Ω x model, z hierarchy φ ε = 0 on Ω z. What can we expect (P ε ) from refersstandard to the Dynamo numerical model, methods? For ε 1 consistency (P 0 ) refers with to the the Striation degenerated model. problem { ( P 0 z (A ) z z φ 0 ) = 0 in Ω z φ 0 = 0 on Ω z. Blow-up of the condition number with ε 0.

56 Outline Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 1 Introduction 2 3 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 4

57 Asyptotic preserving schemes Problematic statement Derivation of Asymptotic-Preserving methods Main achievements General concepts AP-schemes benefits? Consistency with the original problem P ε for ε = O(1) Consistency with the limit problem P 0 when ε 0 Uniform stability / asymptotic parameter ε AP-Scheme derivation Identification of the limit problem Reformulation of the equations (singular limit) Implicit discretization

58 Asyptotic preserving schemes Problematic statement Derivation of Asymptotic-Preserving methods Main achievements General concepts AP-schemes benefits? Consistency with the original problem P ε for ε = O(1) Consistency with the limit problem P 0 when ε 0 Uniform stability / asymptotic parameter ε AP-Scheme derivation Identification of the limit problem Reformulation of the equations (singular limit) Implicit discretization P ε,h h 0 P ε ε 0 ε 0 P 0,h h 0 P 0 Consistency properties of.

59 Asyptotic preserving schemes Problematic statement Derivation of Asymptotic-Preserving methods Main achievements General concepts AP-schemes benefits? Consistency with the original problem P ε for ε = O(1) Consistency with the limit problem P 0 when ε 0 Uniform stability / asymptotic parameter ε AP-Scheme derivation Identification of the limit problem Reformulation of the equations (singular limit) Implicit discretization P ε,h h 0 P ε ε 0 ε 0 P 0,h h 0 P 0 Consistency properties of.

60 Asyptotic preserving schemes Problematic statement Derivation of Asymptotic-Preserving methods Main achievements General concepts AP-schemes benefits? Consistency with the original problem P ε for ε = O(1) Consistency with the limit problem P 0 when ε 0 Uniform stability / asymptotic parameter ε AP-Scheme derivation Identification of the limit problem Reformulation of the equations (singular limit) Implicit discretization P ε,h h 0 P ε ε 0 ε 0 P 0,h h 0 P 0 Consistency properties of.

61 Asyptotic preserving schemes Problematic statement Derivation of Asymptotic-Preserving methods Main achievements General concepts AP-schemes benefits? Consistency with the original problem P ε for ε = O(1) Consistency with the limit problem P 0 when ε 0 Uniform stability / asymptotic parameter ε AP-Scheme derivation Identification of the limit problem Reformulation of the equations (singular limit) Implicit discretization

62 Derivation of the AP-Scheme Problematic statement Derivation of Asymptotic-Preserving methods Main achievements «Macro-Micro» decomposition φ = 1 L z φ(x,z) = φ(x)+φ (x,z), φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, Reformulated problem 1/2-D elliptic problem for φ x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ ( z (A z z φ ) ε x (A x x φ )+ε x Ax x φ ) = εf ε φ, z φ = 0 on Ω x Ω z, φ = 0 on Ω x Ω z, φ = 0, in Ω x.

63 Condition number Introduction Derivation of the AP-Scheme «Macro-Micro» decomposition φ = 1 L z Reformulated problem 1/2-D elliptic problem for φ Standard scheme φ(x,z) = φ(x)+φ (x,z), Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Singular perturbation problem P ε { (P ε x(a ) x xφ ε ) 1 /ε z(a Z zφ ε ) = f in Ω,( ) φ = 0 on Ω x, zφ ε = 0 on Ω z. Untractable numerically for ε 1. Model valid ε Limit problem P 0 φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, x(āx xφ0 ) = f in Ω, φ = 0 on Ω x. Condition number independant of ε Model valid for ε 1 Reformulated system φ) x(āx x = f ( ) + x A x xφ, φ = 0 on Ωx, ) z(a z zφ ) ε x(a x xφ ) + ε x (A x xφ = εf ε x(a x x φ), x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ zφ = 0 on Ω x Ω z, φ = 0 on Ω ( x Ω z (A z z φ ) ε x (A x x )+ε x Ax x φ ) z, φ = 0 in Ω x. = εf ε φ, z φ Condition = 0 on Ω x Ω z, φ number independant of ε = 0 on Ω x Ω z, ε φ Model valid ε = 0, in Ω x Condition number as a function of ε.

64 Condition number Introduction Derivation of the AP-Scheme «Macro-Micro» decomposition φ = 1 L z Reformulated problem 1/2-D elliptic problem for φ Standard scheme φ(x,z) = φ(x)+φ (x,z), Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Singular perturbation problem P ε { (P ε x(a ) x xφ ε ) 1 /ε z(a Z zφ ε ) = f in Ω,( ) φ = 0 on Ω x, zφ ε = 0 on Ω z. Untractable numerically for ε 1. Model valid ε Limit problem P 0 φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, x(āx xφ0 ) = f in Ω, φ = 0 on Ω x. Condition number independant of ε Model valid for ε 1 Reformulated system φ) x(āx x = f ( ) + x A x xφ, φ = 0 on Ωx, ) z(a z zφ ) ε x(a x xφ ) + ε x (A x xφ = εf ε x(a x x φ), x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ zφ = 0 on Ω x Ω z, φ = 0 on Ω ( x Ω z (A z z φ ) ε x (A x x )+ε x Ax x φ ) z, φ = 0 in Ω x. = εf ε φ, z φ Condition = 0 on Ω x Ω z, φ number independant of ε = 0 on Ω x Ω z, ε φ Model valid ε = 0, in Ω x Condition number as a function of ε.

65 Condition number Introduction Derivation of the AP-Scheme «Macro-Micro» decomposition φ = 1 L z Reformulated problem 1/2-D elliptic problem for φ Standard scheme AP-Scheme φ(x,z) = φ(x)+φ (x,z), Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Singular perturbation problem P ε { (P ε x(a ) x xφ ε ) 1 /ε z(a Z zφ ε ) = f in Ω,( ) φ = 0 on Ω x, zφ ε = 0 on Ω z. Untractable numerically for ε 1. Model valid ε Limit problem P 0 φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, x(āx xφ0 ) = f in Ω, φ = 0 on Ω x. Condition number independant of ε Model valid for ε 1 Reformulated system φ) x(āx x = f ( ) + x A x xφ, φ = 0 on Ωx, ) z(a z zφ ) ε x(a x xφ ) + ε x (A x xφ = εf ε x(a x x φ), x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ zφ = 0 on Ω x Ω z, φ = 0 on Ω ( x Ω z (A z z φ ) ε x (A x x )+ε x Ax x φ ) z, φ = 0 in Ω x. = εf ε φ, z φ Condition = 0 on Ω x Ω z, φ number independant of ε = 0 on Ω x Ω z, ε φ Model valid ε = 0, in Ω x Condition number as a function of ε.

66 l 2 -norm error Condition number Introduction 10 Derivation 1 of the AP-Scheme «Macro-Micro» decomposition Standard scheme AP-Scheme φ = 1 L z Discretized limit Problem ε Error between the exact solution and the Reformulated problem numerical approximations as a function of ε 1/2-D elliptic problem for φ Standard scheme AP-Scheme φ(x,z) = φ(x)+φ (x,z), Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Singular perturbation problem P ε { x(a x xφ ε ) 1 /ε z(a Z zφ ε ) = f in Ω,( ) φ = 0 on Ω x, zφ ε = 0 on Ω z. Untractable numerically for ε 1. (P ε ) Model valid ε Limit problem P 0 φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, x(āx xφ0 ) = f in Ω, φ = 0 on Ω x. Condition number independant of ε Model valid for ε 1 Reformulated system φ) x(āx x = f ( ) + x A x xφ, φ = 0 on Ωx, ) z(a z zφ ) ε x(a x xφ ) + ε x (A x xφ = εf ε x(a x x φ), x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ zφ = 0 on Ω x Ω z, φ = 0 on Ω ( x Ω z (A z z φ ) ε x (A x x )+ε x Ax x φ ) z, φ = 0 in Ω x. = εf ε φ, z φ Condition = 0 on Ω x Ω z, φ number independant of ε = 0 on Ω x Ω z, ε φ Model valid ε = 0, in Ω x Condition number as a function of ε.

67 l 2 -norm error Condition number Introduction 10 Derivation 1 of the AP-Scheme «Macro-Micro» decomposition Standard scheme AP-Scheme φ = 1 L z Discretized limit Problem ε Error between the exact solution and the Reformulated problem numerical approximations as a function of ε 1/2-D elliptic problem for φ Standard scheme AP-Scheme φ(x,z) = φ(x)+φ (x,z), Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Singular perturbation problem P ε { x(a x xφ ε ) 1 /ε z(a Z zφ ε ) = f in Ω,( ) φ = 0 on Ω x, zφ ε = 0 on Ω z. Untractable numerically for ε 1. (P ε ) Model valid ε Limit problem P 0 φ(x,z)dz, φ (x,z) = φ(x,z) φ(x), φ = 0, x(āx xφ0 ) = f in Ω, φ = 0 on Ω x. Condition number independant of ε Model valid for ε 1 Reformulated system φ) x(āx x = f ( ) + x 0 A x xφ, φ = 0 on Ωx, ) z(a z zφ ) = ε 0, x(a x xφ ) + ε x (A x xφ = εf ε x(a x x φ), x (Āx x φ) = f + x ( A x x φ ), φ = 0 on Ωx, 2/3-D elliptic (well posed) problem for φ zφ = 0 on Ω x Ω z, φ = 0 on Ω ( x Ω z (A z z φ ) ε x (A x x )+ε x Ax x φ ) z, φ = 0 in Ω x. = εf ε φ, z φ Condition = 0 on Ω x Ω z, φ number independant of ε = 0 on Ω x Ω z, ε φ Model valid ε = 0, in Ω x Condition number as a function of ε.

68 Discretization Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Where is the difficulty? The functional space of mean function is easily approximated This is not the case for the space containing the fluctuations Weak formulation of the fluctuation equation Find φ V := {ψ H 1 (Ω)/ψ = 0 on Ω x Ω z,ψ = 0} such that (A z zφ, zψ )+ε(a x xφ, xψ ) ε(a x xφ, xψ ) = ε(f,ψ ) ε(a x x φ, xψ ), ψ V,

69 Discretization Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Where is the difficulty? The functional space of mean function is easily approximated This is not the case for the space containing the fluctuations Weak formulation of the fluctuation equation Find φ V := {ψ H 1 (Ω)/ψ = 0 on Ω x Ω z,ψ = 0} such that (A z zφ, zψ )+ε(a x xφ, xψ ) ε(a x xφ, xψ )+( P,ψ ) = ε(f,ψ ) ε(a x x φ, xψ ), ψ V, ( Q,φ ) = 0, ( P, Q) W W. W := { Q H 1 0(Ω x)}.

70 Outline Introduction Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 1 Introduction 2 3 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 4

71 Optimize AP-scheme efficiency. Problematic statement Derivation of Asymptotic-Preserving methods Main achievements (a) M 1 = A (b) M O (c) M 2 (d) M 3 Mat. M 1 = A ( ) A B M 2 = B T 0 Size N x(n z + 2) N x(n z + 3) Nnz (3N z + 4)(3N x 2) (5N z + 8)(3N x 2) ( ) s A εc B ( ) 1 Mat. M 3 = Lz CT A2 0 M O = Ã B B T B T Size N x(n z + 4) N x(n z + 3) Nnz (7N z + 13)(3N x 2) (Nz 2 + 6N z + 8)(3N x 2) Figure 14 Structure (non-zero elements (Nnz)) and size of the discretization matrices for a grid size (N x,n z) = (5, 5) : (a) matrix of the singular perturbation problem (P ε ), (b) matrix of the reformulated fluctuation (φ ) equation, (c) matrix of the reformulated sparse fluctuation (φ ) equation, (d) matrix for the direct resolution of the AP scheme (φ, φ).

72 Optimize AP-scheme efficiency. Problematic statement Derivation of Asymptotic-Preserving methods Main achievements (a) M 1 = A (b) M O (c) M 2 (d) M 3 Mat. M 1 = A ( ) A B M 2 = B T 0 Size N x(n z + 2) N x(n z + 3) Nnz (3N z + 4)(3N x 2) (5N z + 8)(3N x 2) ( ) s A εc B ( ) 1 Mat. M 3 = Lz CT A2 0 M O = Ã B B T B T Size N x(n z + 4) N x(n z + 3) Nnz (7N z + 13)(3N x 2) (Nz 2 + 6N z + 8)(3N x 2) Anisotropic elliptic problems 1 AP-scheme for elliptic problems with anisotropy directions aligned with one coordinate and constant anisotropy strength [DDN10 a ]. 2 Improved reformulation to derive an AP-scheme with sparser matrices, formulation for large variations of the anisotropy strengths [BDNY12 b ] Figure 14 Structure (non-zero elements (Nnz)) and size of the discretization a. P. Degond, F. Deluzet, C. Negulescu, An Asymptotic-Preserving matrices for a grid size (N x,n z) = (5, 5) : (a) matrix of the singular perturbation problem (P ε ), (b) matrix of the reformulated fluctuation (φ scheme for strongly anisotropic problems, ) equation, (c) matrix of the reformulated sparse fluctuation (φ SIAM MMS, 8 (2010), pp ) equation, (d) matrix for the direct b. C. Besse, F. Deluzet, C. Negulescu, resolution of the AP scheme (φ, φ). C. Yang, Efficient Numerical Methods for Strongly Anisotropic Elliptic Equations, JSC (2012).

73 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Extension to heterogeneous anisotropy ratios Heterogeneous model formulation Different Schemes definition 2 φ x 2 z 1 «HM» Harmonic mean AP-Scheme zk zk 1 1 ε(z) dz = ( 1 φ ε(x, z) z ( zk zk 1 ) = f, ε(z)dz) 1, 2 «SG» Sharfetter-Gummel AP-scheme ( zk zk 1 ε(z)dz) 1 = ( zk zk 1 1 e dz) ln(ε(z)) = zk zk 1 ( ) (e ln ε(z) ) ln ( ε(z)) ) dz 2 «Non-Conservative» AP-Scheme derived as a discretization of ε 2 φ x 2 2 φ z 2 + ( ) φ ln(ε) z z = εf 1 1 ln(ε k ) ln(ε k 1 ), z ε k ε k 1

74 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Extension to heterogeneous anisotropy ratios Comparisons with the exact solution Analytic set up manufacturing (exact solution φ e ) ( ) 2π φ e (x,z) = sin x L x ( ( )) 2π 1+εcos x. L x Comparisons with the numerical approximations φ h the approximation error being defined as E h 2 = φe φ h 2 φ e 2. Anisotropy ratio definition Function of parameters ε max 1, ε min 1 and q 80 (typical values for the simulations) if 0 z L z/2 ε(z) = 1 ( ε max(1+tanh(q(0.1l z z))) 2 ) +ε min (1 tanh(q(0.1l z z))), if L z/2 z L z ε(z) = 1 ( ε max(1+tanh(q(z 0.9L z))) 2 ) +ε min (1 tanh(q(z 0.9L z))),

75 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements 10 Extension 0 to heterogeneous anisotropy ratios 10 5 Variable ε Comparisons with the exact solution Analytic set up manufacturing ε =10 5 (exact solution φ e min ) ε min =10 10 ε ( ) min = π φ e ε =10 20 min (x,z) = sin x L x ( z axis ( )) 2π Figure 15 Heterogeneous 1+εcosanisotropy x. L x ratio as a function of z. Comparisons with the numerical approximations φ h the approximation error being defined as E h 2 = φe φ h 2 φ e 2. Anisotropy ratio definition Function of parameters ε max 1, ε min 1 and q 80 (typical values for the simulations) if 0 z L z/2 ε(z) = 1 ( ε max(1+tanh(q(0.1l z z))) 2 ) +ε min (1 tanh(q(0.1l z z))), if L z/2 z L z ε(z) = 1 ( ε max(1+tanh(q(z 0.9L z))) 2 ) +ε min (1 tanh(q(z 0.9L z))),

76 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Extension to heterogeneous anisotropy ratios 1 norm condition number estimate SP model Standard AP scheme Non conservative AP scheme HM AP scheme SG AP scheme Values of ε min Relative error of 2 norm SP model Standard AP scheme Non conservative AP scheme HM AP scheme SG AP scheme Values of ε min (a) (b) Figure 16 Condition number (a) and approximation error 2-norm E2 h (b) as functions of ε min on a mesh.

77 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Extension to heterogeneous anisotropy ratios (3D) 3D setup ( ) ( ) ε 2(x,z,y) = ε(z) (x x mid )(y y mid )(z z mid )+1 / xmid 2 +ymid 2 +zmid ( ( ) ( )) 2πz 2πz φ e(x,y,z) = x 2 (L x x) 2 y 2 (L y y) 2 1+ε 2(x,y,z) sin 2 cos 2, L z L z Figure 17 Approximation error 2-norm E h 2 as functions of ε min on a (left) and (right) meshes.

78 Arbitrary anisotropy directions Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Goals Non-uniform anisotropy directions with non-adapted (Cartesian) coordinates 1 Duality based formulation : two Lagragian multipliers for the mean and fluctuation (zero mean value) functional spaces discretization [DDLNM12 1 ]. 2 Micro-macro decomposition : the number of unknowns is dramatically reduced (5 to 2) [DLNM12 2 ]. 3 Another route [BDM 3 ] using a differential characterization of the mean and fluctuation functions space (4 th order differential problem). 1. P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, CMS, 10 (2012), pp P. Degond, A. Lozinski, J. Narski, C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, JCP, 231 (2012), pp S. Brull, F. Deluzet, A. Mouton, Numerical resolution of an anisotropic non-linear diffusion problem, submitted to CMS.

79 Context1 andl2 error motivations «AP» numerical H1 error methods Arbitrary anisotropy directions Problematic statement 1 Derivation of Asymptotic-Preserving L2 error methods H1 error Main achievements Goals 1e-05 1e-06 1e-07 Non-uniform anisotropy 1e-08 directions with non-adapted 1e-08 (Cartesian) coordinates (a) (b) (c) Figure 1 Duality 18 Ellitpic based formulation anisotropic : problem two Lagragian resolution multipliers with anfor heterogeneous the mean andoscil- latingfluctuation magnetic (zero field. mean (a) plot value) of the functional magnetic spaces fielddiscretization as a function of the two dimensional [DDLNM12 space 1 ]. variable, for a frequency oscillation equal to 20. Approximation error norm for computations carried out on a uniform Cartesian mesh 2 Micro-macro decomposition : the number of unknowns is dramatically with ε = 1 (b) and ε = 10 reduced (5 to 2) [DLNM12 (c) 2 as functions of the magnetic field oscillation ]. frequency. 3 Another route [BDM 3 ] using a differential characterization of the mean and fluctuation functions space (4 th order differential problem). 1. P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, CMS, 10 (2012), pp P. Degond, A. Lozinski, J. Narski, C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, JCP, 231 (2012), pp S. Brull, F. Deluzet, A. Mouton, Numerical resolution of an anisotropic non-linear diffusion problem, submitted to CMS. 1e-05 1e-06 1e-07

80 Arbitrary anisotropy directions Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Goals Non-uniform anisotropy directions with non-adapted (Cartesian) coordinates 1 Duality based formulation : two Lagragian multipliers for the mean and fluctuation (zero mean value) functional spaces discretization [DDLNM12 1 ]. 2 Micro-macro decomposition : the number of unknowns is dramatically reduced (5 to 2) [DLNM12 2 ]. 3 Another route [BDM 3 ] using a differential characterization of the mean and fluctuation functions space (4 th order differential problem). 1. P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, CMS, 10 (2012), pp P. Degond, A. Lozinski, J. Narski, C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, JCP, 231 (2012), pp S. Brull, F. Deluzet, A. Mouton, Numerical resolution of an anisotropic non-linear diffusion problem, submitted to CMS.

81 Arbitrary anisotropy directions Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Method # rows # non zero time Mic.-Mac s Dual.Based s Goals Stand. Meth s Table 1 Micro-Macro, Duality-Based and Standard discretizations comparison ( Non-uniform anisotropy directions with non-adapted (Cartesian) coordinates grid). 1 Duality based formulation : two Lagragian multipliers for the mean and fluctuation (zero mean value) functional spaces discretization [DDLNM12 1 ]. 2 Micro-macro decomposition : the number of unknowns is dramatically reduced (5 to 2) [DLNM12 2 ]. 3 Another route [BDM 3 ] using a differential characterization of the mean and fluctuation functions space (4 th order differential problem). 1. P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, CMS, 10 (2012), pp P. Degond, A. Lozinski, J. Narski, C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, JCP, 231 (2012), pp S. Brull, F. Deluzet, A. Mouton, Numerical resolution of an anisotropic non-linear diffusion problem, submitted to CMS.

82 Arbitrary anisotropy directions Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Method # rows # non zero time Mic.-Mac s Dual.Based s Goals Stand. Meth s Table 1 Micro-Macro, Duality-Based and Standard discretizations comparison ( Non-uniform anisotropy directions with non-adapted (Cartesian) coordinates grid). 1 Duality based formulation : two Lagragian multipliers for the mean and fluctuation (zero mean value) functional spaces discretization [DDLNM12 1 ]. 2 Micro-macro decomposition : the number of unknowns is dramatically reduced (5 to 2) [DLNM12 2 ]. 3 Another route [BDM 3 ] using a differential characterization of the mean and fluctuation functions space (4 th order differential problem). 1. P. Degond, F. Deluzet, A. Lozinski, J. Narski, C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, CMS, 10 (2012), pp P. Degond, A. Lozinski, J. Narski, C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, JCP, 231 (2012), pp S. Brull, F. Deluzet, A. Mouton, Numerical resolution of an anisotropic non-linear diffusion problem, submitted to CMS.

83 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Application to plasma under large magnetic field 1 1D Euler-Lorrentz in the drift limit [DDSV09 4 ]. The two dimensional Euler-Lorrentz system is investigated in [BDD12 5 ]. A bifluid quasi neutral Euler-Lorentz model is considered in [BDDM 6 ]. 2 Application to non linear diffusion problems : simulation of the tokamak temperature evolution [MN12 7, LMN 8 ]. 4. P. Degond, F. Deluzet, A. Sangam, M-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228 (2009), pp S. Brull, P. Degond, F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasma simulations, Communications in Computational Physics (CICP), 11 (2012), pp S. Brull, P. Degond, F. Deluzet, A. Mouton, Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model, KRM (2011), pp A. Mentrelli, C. Negulescu, Asymptotic-Preserving scheme for highly anisotropic non-linear diffusion equations, JCP (2012),pp A. Lozinski, J. Narski, C. Negulescu, Highly anisotropic temperature balance equation and its asymptotic-preserving resolution, submitted to M2AN

84 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Application to plasma under large magnetic field 1 1D Euler-Lorrentz in the drift limit [DDSV09 4 ]. The two dimensional Euler-Lorrentz system is investigated in [BDD12 5 ]. A bifluid quasi neutral Euler-Lorentz model is considered in [BDDM 6 ]. Figure 19 Bifluid Euler-Lorentz under large magnetic field and small Mach number 2: the Application dimensionless to non gyro-period linear diffusion and the problems Mach number : simulation are set toof 10the 8. Electronic momentum tokamak temperature as a functionevolution of the 2D-space [MN12variable. 7, LMN 8 Left ]. : standard scheme with a time step t < ; Right : AP-scheme with a time step t > P. Degond, F. Deluzet, A. Sangam, M-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228 (2009), pp S. Brull, P. Degond, F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasma simulations, Communications in Computational Physics (CICP), 11 (2012), pp S. Brull, P. Degond, F. Deluzet, A. Mouton, Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model, KRM (2011), pp A. Mentrelli, C. Negulescu, Asymptotic-Preserving scheme for highly anisotropic non-linear diffusion equations, JCP (2012),pp A. Lozinski, J. Narski, C. Negulescu, Highly anisotropic temperature balance equation and its asymptotic-preserving resolution, submitted to M2AN

85 Problematic statement Derivation of Asymptotic-Preserving methods Main achievements Application to plasma under large magnetic field 1 1D Euler-Lorrentz in the drift limit [DDSV09 4 ]. The two dimensional Euler-Lorrentz system is investigated in [BDD12 5 ]. A bifluid quasi neutral Euler-Lorentz model is considered in [BDDM 6 ]. Figure 20 Tokamak temperature (T) computed by the anisotropic diffusion equation : 2 Application to non linear diffusion problems : simulation of the tt ε 1 tokamak (T 5/2 temperature T) ( evolution T) = 0, with [MN12 7, the derivative LMN 8 along the magnetic ]. field direction. Temperature and magnetic field lines as functions of the space variables after s. for ε = 1 (Left) and ε = (Right), with an isotrop gaussian as initial data. 4. P. Degond, F. Deluzet, A. Sangam, M-H. Vignal, An asymptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys., 228 (2009), pp S. Brull, P. Degond, F. Deluzet, Degenerate anisotropic elliptic problems and magnetized plasma simulations, Communications in Computational Physics (CICP), 11 (2012), pp S. Brull, P. Degond, F. Deluzet, A. Mouton, Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model, KRM (2011), pp A. Mentrelli, C. Negulescu, Asymptotic-Preserving scheme for highly anisotropic non-linear diffusion equations, JCP (2012),pp A. Lozinski, J. Narski, C. Negulescu, Highly anisotropic temperature balance equation and its asymptotic-preserving resolution, submitted to M2AN

86 Outline Introduction 1 Introduction 2 3 4

87 On going project Improve numerical method efficiency Hybrid numerical method coupling the AP-scheme with the Limit Problem (A. Crestetto, IODISSE Post-doctoral position). Collaboration with the HPC linear algebra team of L. Giraud (INRIA Bordeaux) in the frame of the INRIA HiePACS a project for an efficient resolution of the augmented linear systems required for the AP-scheme resolution in 3D problems. a. http ://hiepacs.bordeaux.inria.fr/software.php

88 On going project Improve numerical method efficiency Hybrid numerical method coupling the AP-scheme with the Limit Problem (A. Crestetto, IODISSE Post-doctoral position). Collaboration with the HPC linear algebra team of L. Giraud (INRIA Bordeaux) in the frame of the INRIA HiePACS a project for an efficient resolution of the augmentedproblematic linear systems : required for the AP-scheme resolution in 3D problems. Ω 1 AP-Scheme is costfull (3D) a. http ://hiepacs.bordeaux.inria.fr/software.php The limit-problem (2D) not precise in the whole domain Σ 1 Σ 2 Approximation l2-norm error Ω L Ω Coupling Strategy : Limit problem for the highest Standard altitudes scheme Ω L φ L, AP-Scheme AP-scheme for intermediate Discretized altitudeslimit Ω i Problem φ i, i = 1,2, Coupling conditions Σ i,i=1,2, Anisotropy φ i ratio Σi = (ε) φ L Σi

89 On going project Improve numerical method efficiency Hybrid numerical method coupling the AP-scheme with the Limit Problem (A. Crestetto, IODISSE Post-doctoral position). Collaboration with the HPC linear algebra team of L. Giraud (INRIA Bordeaux) in the frame of the INRIA HiePACS a project for an efficient resolution of the augmentedproblematic linear systems : required for the AP-scheme resolution in 3D problems. Ω 1 AP-Scheme is costfull (3D) a. http ://hiepacs.bordeaux.inria.fr/software.php The limit-problem (2D) not precise in the whole domain Σ 1 Σ 2 Ω L Ω 2 Coupling Strategy : Limit problem for the highest altitudes Ω L φ L, AP-scheme for intermediate altitudes Ω i φ i, i = 1,2, Coupling conditions Σ i,i=1,2, φ i Σi = φ L Σi.

90 On going project Improve numerical method efficiency Hybrid numerical method coupling the AP-scheme with the Limit Problem (A. Crestetto, IODISSE Post-doctoral position). Collaboration with the HPC linear algebra team of L. Giraud (INRIA Bordeaux) in the frame of the INRIA HiePACS a project for an efficient resolution of the augmentedproblematic linear systems : required for the AP-scheme resolution in 3D problems. Ω 1 AP-Scheme is costfull (3D) a. http ://hiepacs.bordeaux.inria.fr/software.php The limit-problem (2D) not precise in the whole domain Σ 1 Σ 2 Ω L Ω 2 Coupling Strategy : Limit problem for the highest altitudes Ω L φ L, AP-scheme for intermediate altitudes Ω i φ i, i = 1,2, Coupling conditions Σ i,i=1,2, φ i Σi = φ L Σi.

91 On going project Improve numerical method efficiency Hybrid numerical method coupling the AP-scheme with the Limit Problem (A. Crestetto, IODISSE Post-doctoral position). Collaboration with the HPC linear algebra team of L. Giraud (INRIA Bordeaux) in the frame of the INRIA HiePACS a project for an efficient resolution of the augmentedproblematic linear systems : required for the AP-scheme resolution in 3D problems. Ω 1 AP-Scheme is costfull (3D) a. http ://hiepacs.bordeaux.inria.fr/software.php The limit-problem (2D) not precise in the whole domain Σ 1 Σ 2 Ω L Ω 2 Coupling Strategy : Limit problem for the highest altitudes Ω L φ L, AP-scheme for intermediate altitudes Ω i φ i, i = 1,2, Coupling conditions Σ i,i=1,2, φ i Σi = φ L Σi.