An asymptotic preserving scheme in the drift limit for the Euler-Lorentz system. Stéphane Brull, Pierre Degond, Fabrice Deluzet, Marie-Hélène Vignal

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1 1 An asymptotic preserving scheme in the drift limit for the Euler-Lorentz system. Stéphane Brull, Pierre Degond, Fabrice Deluzet, Marie-Hélène Vignal IMT: Institut de Mathématiques de Toulouse

2 1. Introduction. Plan 2 2. An AP scheme in the drift-fluid limit for the Euler-Lorentz 2.1. Drift limit for Euler-Lorentz Classical scheme for Euler-Lorentz AP scheme for Euler-Lorentz. 3. Numerical results The anisotropic elliptic problem Case Case 2.

3 1. Introduction 3

4 General topic I. 4 Numerical modelisation of a device such that an important physical parameter, ǫ, is : - very small in one part of the domain : ε 1, - an order 1 parameter elsewhere : ε = O(1), We do not want to describe the scale ε. Starting from a model P ε : Valid everywhere Classical schemes stable et consistant iff ε is resolved by the mesh very huge cost.

5 General topic II. 5 A possible solution Use P ε where ε = O(1). Use an asymptotic model where ε 1 : Problems : P = lim ε P ε. Reconnection of P ε and P. Moving interface : difficult numerical pb in 2D or 3D.

6 Another possible solution General topic III 6 Use P ε everywhere. Discritized it with a scheme such that: it does not need to resolve the scale ε : Asymptotic stability it gives an approx. solution of P when ε : Asymptotic consistence. Asymptotically stable et consistent Asymptotic preserving scheme. ([S.Jin] kinetic hydro)

7 7 2. An AP-scheme in the drift limit for the Euler-Lorentz system.

8 2.1. Drift limit for Euler-Lorentz 8

9 Euler-Lorentz model. 9 Isothermal pressure law t n ( + (nu) =, m t (nu) + (nu u) ) +T n = q n (E + u B). n = ions density, u = electrons velocity, m = ions mass, q = ions charge, B = magnetic field. T = constant temperature, E = electric field,

10 The drift regime. Motion of a particle in an electromagnetic field B gyro-motion. E 1 E B Drift center motion Regime such that : Lorentz and pressure forces are very large Consequences: gyro-periode 1. Dynamic //B quicker than B.

11 The rescaled Euler-Lorentz system 11 Rescaled Euler-Lorentz system t ( n + (nu) =, (EL ε ) ε t (nu) + (nu u) ) +T n = n (E + u B). ε = gyro-period caract. time = (Mach number)2 = mu2 T 1. Drift Limit: ε

12 The drift-fluid limit. I 12 ε in (EL ε ) drift fluid model { t n + (nu) =, (DF) T n = n (E + u B). Decomposition of the velocity according to b = B B u = u {}}{ (I b b)u+ u b {}}{ (u b)b

13 Drift-fluid limit. II Projection of the momentum equation Perpendicular part b (T n ne = nu B) nu = Eq. explicit eq. for nu 13 b (T n ne) B Parallel part b (T n ne = nu B) b (T n ne) = Implicit eq. for nu

14 Explicit eq. for the parallel velocity. I 14 (DF) t n + (nu) =, (1) nu = b (T n ne), B (2) b (T n ne) =. (3) T b (1) T b t n + T b ( (nu)) = t (3) T b t n t (nb E) = Difference Elliptic eq. for u

15 Explicit eq. for the parallel velocity II 15 Elliptic equation for u T (b )( (nu b)) }{{} ( (nu) ) = T b ( (nu )) + t (nb E) Dual operators 1D Reformulated drift fluid model t n + (nu) =, (DF) (RDF) nu = b (T n ne), B T ( (nu) ) = RHS.

16 Reformulated Euler-Lorentz system. I 16 Euler-Lorentz ε Drift Fluid Reformulated Euler-Lorentz ε Reformulated Drift Fluid In the Euler-Lorentz model (T b ) Eq. masse (b t ) Eq. vitesse ε tt(nu 2 ) T(b )( (nu b) = RHS

17 Reformulated Euler-Lorentz system II 17 Reformulated Euler-Lorentz system (ELR ε ) t n + (nu) =, ( ( ε t (nu) + (nu u) ) +T n ) = n (E + u B), ε tt(nu 2 ) T(b )( (nu b) = RHS. Formally equivalent to Euler-Lorentz

18 2.2. Classical scheme for Euler-Lorentz 18

19 Classical scheme I 19 Si n m et u m known approx. at time t m n m+1 n m + (nu) m =, t ( (nu) m+1 (nu) m ) ε + (nu u) m + T n m t = n m+1 (E + u B) m+1. Stable and consistent iff t = O( ε)

20 Discrete reformulation Classical scheme II 2 (T b ) Mass. eq. (b t discrete)velocity eq. ε (nu //) m+1 2(nu // ) m + (nu // ) m 1 t 2 T(b )( (nu // b) m 1 ) = RHS explicit scheme conditional stability ε = we lose u m+1 // consistancy pb

21 2.3. AP scheme for Euler-Lorentz 21

22 AP schemes: general methodology. P ε singulary perturbed problem P = lim ε P ε Step 1: Reformulation Identify P Lift P into P ε ie find R ε P ε st R ε is a regular perturbation of P ε as ε. 22 Step 2: Discretization Identify P ε into P ε,h (h = min( t, x)) s.t. P,h := lim ε P ε,h is a scheme for P. Find R ε,h a regular perturbation form as ε.

23 AP scheme I If n m and u m known approx. at time t m n m+1 n m + (nu) m+1 =, ( t (nu) m+1 (nu) m ) m ε + (nu u) t +T ( n) m+1/2 = n m E m+1 + (nu B) m+1, ( n) m+1/2 = ( n) m+1 // + ( n) m. 23 Stable and consistent

24 Discrete reformulation AP scheme II 24 (T b ) Eq. masse (b t discret)velocity eq. ε (nu //) m+1 2(nu // ) m + (nu // ) m 1 t 2 T(b )( (nu // b) m+1 ) = RHS Implicit scheme unconditional stability ε = we keep u m+1 // consistent drift limit

25 3. Numerical results. 25

26 3.1. The elliptic anisotropic problem. 26

27 Approximation of the wave equation. 27 The wave equation can be written: ε t (nu)m+1 T t ( ((nu ) m+1) ) = RHS Anisotropic diffusion problem. Case 1) b constant // to the mesh and (nu ) m+1 satisfies Dirichlet bc. P.Degond, F.Deluzet, A.Sangam, M.H.Vignal. Case 2) b non // to the mesh and variable, (nu ) m+1 satisfies Neumann bc: (b ν) (nu m+1 ) =. S.Brull, P.Degond, F.Deluzet, M.H.Vignal.

28 3.2. Case 1 [P.Degond, F.Deluzet, A.Sangam, M.H.Vignal]. 28

29 (EL ε ) The Model t ( n + (nu) =, ε t (nu) + (nu u) ) +T n 29 = n (E + u B). Resolved case : t x max u ± T/ε Non resolved case : t x max u

30 Parameters. I 3 T = 1, E = (,, 1), B = (, 1, ), ε = 1 6 ou 1, x = y = 1/1. z E 1 x B 2D Domain. 1 y (nu) // = (nu) y (nu) = (nu) x (nu) z

31 Parameters II 31 Initial conditions Sol. of the Drift Fluid Model. (n,nu) = (1, (,, )) (n,nu)(x,t) = (1, ( 1, 1, )) Boundary conditions 1 A y ( 1,1 + ε, ε) 1 + ε 1 ( 1 + ε,1 + ε,) ( 1 + ε,1, ε) x ε A ( 1,1,) 1 A

32 ε = 1 6, Non resolved case 32 Density Drift limit Class. scheme AP scheme (nu) x Drift limit Class. scheme AP scheme (nu) y = (nu) // 5 x, y=.5 Drift limit Class. scheme AP scheme x, y=.5 (nu) z x, y=.5 Drift limit Class. scheme AP scheme x, y=.5

33 ε = 1, Resolved case class.scheme AP scheme.1 class.scheme AP scheme Density (nu) x (nu) y = (nu) // x, y=.5 class.scheme AP scheme x, y=.5 (nu) z x, y=.5 class.scheme AP scheme x, y=.5

34 3.3. Case 2 [S.Brull, P.Degond, F.Deluzet, M.H.Vignal]. 34

35 The elliptic problem. 35 φ ε = nu in the following way (b ) ( (bφ ε )) + εφ ε = f ε, in Ω, (b ν) (bφ ε ) = on Ω where f ε is the right-hand side and ν is the unit outward normal at x Ω. Pb: Ill posed if ε =. Anisotropic elliptic problems. [Degond-Deluzet-Negulescu ] submitted.

36 Decomposition of the solution I. 36 V = {φ L 2 (Ω)/ (bφ) L 2 (Ω)}, K = {φ V / (bφ) = on Ω}, W = {h L 2 (Ω), /(b )h L 2 (Ω)}, = {h W / (b ν)h = on Ω}. W φ ǫ is decomposed as φ ε = p ε + q ε, p ε K, q ε K. p ε part of φ ε constant along the field lines. q ε fluctuation.

37 Decomposition of the solution II. 37 The anisotropic elliptic problem reads (b ) ( (bq ε )) + ε(p ε + q ε ) = f ε, in Ω, (b ν) (bq ε ) =, in Ω, p ε K and q ε K. In the variational form: (bq ε ) (bψ) dx + ε Ω Ω (p ε + q ε )ψdx = Ω f ε ψdx.

38 Equation for p ε K. 38 The problem for p ε Find p ε K and g ε W s.t. (b b g ε ) = (f ε b), p ε = 1 ε (fε + b g ε ).

39 Numerical results for p ε with ε = Exact solution for p Approached solution for p

40 Error on p ε for ε = Error for p x

41 Error on p ε in function of ε 41 2 Error on p in function of epsilon 15 Error on p Value of epsilon in log scale

42 Equation for q ε K. 42 q ε can be written q ε = b h ε. h ε is solution to the bilaplacian problem ((b b) ) ( b b h ε ) + ε ((b b) h ε ) = (bf ε ), in Ω, (b ν) ((b b) h ε ) =, on Ω, (b ν)h ε =, on Ω.

43 Error for the bilaplacian problem. 43 Normalized difference between the exact and the apporached solution Y X.5 1

44 Error in function of the angle Relative error for the bilaplacian problem. Error in norm1 Error in norm 2 Error in norm infinity Relative error angle of the magnetic field

45 Orthogonal part: q ε K. 45 exact solution for the orthogonal part x approached solution for the orthogonal part x

46 Error on the orthogonal part q ε K. 46 Error for the orthogonal part x

47 3. Conclusion 47

48 Work in progress. 48 Diffusive anisotropic problem with Dirichlet boundary conditions. AP schemes for strongly anisotropic elliptic problems. [S.Brull, P.Degond, F.Deluzet, C.Negulescu ]. Full Euler. Coupling with Poisson, Maxwell.

49 49 THANKS FOR YOUR ATTENTION!