HYP 2012 Thomas Penalty methods for edge plasma transport in a Auphan , LATP, June tokamak 28, 2012 Introduction Model Thomas Auphan , LATP, June 28, 2012 Penalty methods First Approach Optimal penalization Two faces In collaboration with Ph. Angot and O. Gu` es Financial support: FR-FCM and ANR ESPOIR Picture: XKCD 1 / 20
The ITER tokamak HYP 2012 Plasma : Ion and electrons soup. Thomas Auphan , LATP, June Magnetic confinement. 28, 2012 Heating. Introduction Goal: Perform the fusion reaction Model as a reliable source of energy. Penalty methods First Approach Key figures: Optimal penalization Two faces Fusion power ≈ 500 MW Fusion power Power consumption ≥ 10 Plasma duration ≥ 300 s 2 / 20
Limiter configuration HYP 2012 Thomas Auphan , LATP, June 28, 2012 Introduction Model Penalty methods First Approach Optimal penalization Two faces TORE SUPRA, Cadarache (source: CEA) 3 / 20
Wall-plasma interaction HYP 2012 Thomas Auphan , LATP, June 28, 2012 Introduction Model Penalty methods First Approach Optimal penalization TORE SUPRA, Cadarache Two faces From ccd camera (visible) (Source: CEA) Magnetic confinement not perfect ⇒ Control the interactions (limiter, divertor). ANR ESPOIR: Numerical simulation of the edge plasma using penalization methods. 4 / 20
Why penalty methods ? HYP 2012 Thomas Auphan , LATP, June 28, 2012 Non-body-fitted Cartesian-mesh. Possible use of efficient solver : pseudo-spectral, Introduction Model multiscale grids.... Penalty methods A few references for applications : First Approach Incompressible flows [ Angot , Math. Meth. Appl. Sci. , 1999] Optimal Compressible flows [ Liu, Vasilyev , JCP, 2007] penalization Pseudo spectral methods for edge plasma [ Isoardi et al. , JCP, 2010] Two faces ❲➅❤➆② ⑨❞⑨♦ ➀s❼✐➆♠➀♣❸❧⑩❡ ➉✇❺❤⑨❡➇♥❻ ⑨♦➈♥⑨❡ ⑨❝⑩❛➆♥❻ ⑨❞⑨♦ ⑨❝⑩♦➈♠➀♣❸❧❽✐⑨❝⑩❛❼t⑩❡⑩❞❻ ❄ Shadocks, from I. Rami` ere’s thesis. 5 / 20
The 1D hyperbolic system (along a magnetic field line) N = plasma density HYP 2012 ( t , x ) ∈ R + × ] − L , L [ Γ = plasma momentum Thomas M = Γ Auphan , N = ”velocity” ∂ t N + ∂ x Γ = S LATP, June 28, 2012 � Γ 2 � ∂ t Γ + ∂ x N + N = 0 ∣ Introduction N ∣=∣ M ∣ 1 ≈ 10 − 5 m Model Boundary conditions: Penalty methods limiter limiter M ( ., − L ) = − 1 + η ∣ First Approach N ∣=∣ M ∣ 1 Optimal penalization and M ( ., L ) = 1 − η x Two faces -L 0 L Initial: N (0 , . ) and Γ(0 , . ) ≈ 10 m Strictly hyperbolic 1D. Eigenvalues : M − 1 and M + 1. One incoming wave : one boundary condition admissible on each boundary. 6 / 20
A first approach [ Isoardi et al. , JCP, 2010] ∂ t N + ∂ x Γ + χ M = Γ ε N = (1 − χ ) S 0 < ε ≪ 1 HYP 2012 N Thomas � Γ 2 � + χ Auphan , ∂ t Γ + (1 − χ ) ∂ x N + N ε (Γ − M 0 N ) = 0 LATP, June 28, 2012 Numerical test : Introduction ε = 10 − 3 , δ x ≈ 1 · 10 − 3 , t ≈ 8 . 8 · 10 − 3 (stop : Model � 0 in the plasma Penalty | M n i | > 10) χ ( x ) = methods M versus x 1 in the limiter 14 First Approach Optimal 12 penalization Two faces 10 Two problems: 8 2 fields penalized. 6 Sense of 4 � � Γ 2 (1 − χ ) ∂ x N + N ? 2 0 0.390 0.395 0.400 0.405 0.410 x M versus x ⇒ Dirac measure next to the interface. 7 / 20
An optimal penalty method HYP 2012 Thomas Penalization of a single field such that M → M 0 . Auphan , LATP, June 28, 2012 ∂ t N + ∂ x Γ = S N � Γ � Γ 2 Introduction � � + χ ∂ t Γ + ∂ x N + N − N = S Γ Model ε M 0 Penalty methods Initial conditions: N (0 , . ) and Γ(0 , . )known First Approach Optimal penalization M 0 is a constant such that 0 < M 0 = 1 − η < 1. Two faces Also obtained by a method inspired from [ Fornet and Gu` es , DCDS, 2009]. Does not generates boundary layers. 8 / 20
Convergence analysis theorem I HYP 2012 Thomas Auphan , LATP, June + (χ( x )=0) 28, 2012 - (χ( x )=1) � ∂ t v + � d 0 x d in ] − T 0 , T [ × R d j =1 A j ( v ) ∂ j v = f ( v ) Introduction + (1) on ] − T 0 , T [ × R d − 1 Model Pv | x d =0 = 0 Penalty methods First Approach A j :matrices, symmetric, C ∞ , independant from ( t , x ) Optimal penalization outside a compact set. Two faces P = orthogonal projection matrix. Maximal strictly dissipative and non characteristic boundary conditions. 9 / 20
Convergence analysis theorem II HYP 2012 Penalized system : Thomas Auphan , LATP, June d A j ( v ε ) ∂ j v ε + χ 28, 2012 � ε Pv ε = f ( v ε ) in ] − T 0 , T [ × R d ∂ t v ε + (2) Introduction j =1 Model Penalty Theorem ( T 0 > 0) methods First Approach Optimal | ] − T 0 , 0[ ∈ H ∞ ∩ Lip solution of (1) on ] − T 0 , 0[ . Consider, v 0 , + penalization Two faces There exists T > 0 and ε 0 > 0 such that both the penalized ( ∀ ε ∈ ]0 , ε 0 [ ) and the BVP (1) has a smooth solution (resp. v ε on ] − T 0 , T [ × R d and v 0 , + on ] − T 0 , T [ × R d + ) such that : � v ε − v 0 , + � H s (] − T 0 , T [ × R d ∀ s ∈ N , + ) = O ( ε ) 10 / 20
Convergence analysis theorem: Sketch of proof I HYP 2012 Thomas Auphan , LATP, June 28, 2012 Formal asymptotic expansion of a continuous solution : ε ( t , x ) = � + ∞ v ε ( t , x ) ∼ U ± n =0 ε n U n , ± ( t , x ) Introduction Model Substituting the expansion and classifying : Penalty n =0 ε n ( ∂ t U n , + + . . . ) = S Inside the physical domain : � ∞ methods First Approach Optimal In the obstacle : penalization M 0 P U 0 , − + � ∞ n =0 ε n � ∂ t U n , − + · · · + M 0 P U n +1 , − � ε − 1 1 = S Two faces Computations of the terms U n , ± : by induction. 11 / 20
Convergence analysis theorem: Sketch of proof II v ε ( t , x ) = � K n =0 ε n U n , ± ( t , x ) + ε w ε ( t , x ). HYP 2012 Equation for w ε . Thomas Auphan , LATP, June Approximation of w ε by an iterative scheme ( w k ). 28, 2012 Energy estimates: Introduction Model Lemma Penalty Weighted norm : � w � 2 ,λ = � e − λ t w � 2 methods First Approach Assumptions : � w k � ∞ < R and � ∂ j w k � ∞ < R Optimal penalization (j ∈ { 0 , . . . , d − 1 } ) Two faces √ λ � w k +1 � 2 ,λ + 1 √ ε � Pw k , − � 2 ,λ ≤ C ( R ) ∀ λ > λ 0 ( R ) , √ � g � 2 ,λ λ ( w k ) bounded sequence (for some norms). Existence of w ε = lim k →∞ w k . 12 / 20
Numerical test (2nd order FV scheme) Manufactured solution : HYP 2012 � − x 2 � N ( t , x ) = exp Thomas 0 . 16( t + 1) Auphan , � π x LATP, June � − x 2 � � Γ( t , x ) = sin exp 28, 2012 0 . 8 0 . 16( t + 1) Thick blue : N(1,x), Black : Gamma(1,x), Red : M(1,x), epsilon=1e−1 Introduction N − x = N x 1.0 − x =− x Model 0.9 Penalty methods 0.8 First Approach 0.7 Optimal limiter penalization 0.6 Two faces 0.5 x 0 0.4 0.5 0.4 0.3 0.2 Computations up to 0.1 t = 1. 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 x Mesh step: δ x = 10 − 5 Continuous lines : Approximate solution ( ε = 0 . 1) Dashed lines : exact solution. 13 / 20
Numerical tests + : in the plasma, x : in the limiter, o: x-derivative in the plasma, *:x-derivative in the limiter (Delta_x= 1e-05) 0 HYP 2012 10 Thomas -1 10 Auphan , LATP, June -2 28, 2012 10 -3 L1-error for N and dN/dx 10 Introduction -4 Model 10 Penalty -5 methods 10 First Approach -6 Optimal 10 penalization Two faces -7 10 -8 10 -9 10 -7 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10 epsilon L 1 error for N and ∂ x N as a function of ε . Optimal convergence rate for N and ∂ x N : O ( ε ) 14 / 20
Numerical tests + : in the plasma, x : in the limiter, o: x-derivative in the plasma, *:x-derivative in the limiter (Delta_x= 1e-05) 0 HYP 2012 10 Thomas -1 10 Auphan , LATP, June -2 28, 2012 10 -3 L2-error for N and dN/dx 10 Introduction -4 Model 10 Penalty -5 methods 10 First Approach -6 Optimal 10 penalization Two faces -7 10 -8 10 -9 10 -7 -6 -5 -4 -3 -2 -1 0 10 10 10 10 10 10 10 10 epsilon L 2 error for N and ∂ x N as a function of ε . Non optimal rate for ∂ x N in L 2 norm : Artefact ? 14 / 20
Two interfaces and transport of N HYP 2012 Thomas Auphan , LATP, June 28, 2012 Introduction Model Penalty M 0 =− 1 M 0 = 1 − methods First Approach Optimal penalization N N Two faces -0.5 -0.1 0.1 0.5 x Concentration of N at the center ! 15 / 20
Prevent information from crossing the limiter HYP 2012 Thomas Auphan , M 0 =− 1 LATP, June M 0 = 1 − 28, 2012 N N Introduction Model -0.5 -0.1 0.1 0.5 x Penalty methods � N � N � � �� � 0 � First Approach + χ Optimal ∂ t + ∂ x α f = S penalization Γ M 0 − N Γ Γ ε Two faces α ( x ) is : Smooth. = 1 inside the plasma area and in a neighbourhood of the interface. = 0 in the central area of the limiter. 16 / 20
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