Michel Mehrenberger 1 & Eric Sonnendrücker 2 ECCOMAS 2016

Transcription

1 for Vlasov type for Vlasov type 1 2 ECCOMAS 2016 In collaboration with : Bedros Afeyan, Aurore Back, Fernando Casas, Nicolas Crouseilles, Adila Dodhy, Erwan Faou, Yaman Güclü, Adnane Hamiaz, Guillaume Latu, Pierre Navaro, Maurizio Ottaviani, Hocine Sellama. 1. IRMA, University of Strasbourg and TONUS project (INRIA), France 2. Max Planck Institute for plasma physics, Garching, Germany

2 for Vlasov type

3 Physical context : the ITER project Tokamak construction at Cadarache in France Aim : gain of energy by fusion of atoms with magnetic confinement Modelisation of plasma by PDE MHD (fluid model) Long time dynamic Instabilities can destroy the machine Multi-species Vlasov-Maxwell and gyrokinetic approximation Short time dynamic Micro-instabilities can degrade confinement quality Interest of numerical simulations : Understand how heat flux due to turbulence vary with respect to the size of the plasma for Vlasov type

4 for Vlasov type f (t, x, v) solution of f (t, x, v)dxdv represents the probability of finding particules in a volume dxdv at time t at point (x, v) (position, velocity) Transport equation tf + v xf + F(t, x) vf = 0 Non linearity through F that depends on f (Poisson, Maxwell) : F = E + v B Description of the dynamic of charged particules in a plasma 1d 1d 2d 0d 3d 1d 3d 1d + µ Euler-2d drift-kinetic gyrokinetic Numerical s : PIC/eulerian/semi-Lagrangian

5 The case of 1d constant advection t l+1 = t l + t t l Characteristics are exact Lagrange interpolation : tu + a xu = 0, u = u(t, x) u l+1 i x i ui n u l+1 i ul i +1 x i x i a t x i +1 Degree 1 (linear) : x i, x i +1 Degree 3 (cubic) : x i 1, x i, x i +1, x i +2 Cubic splines interpolation Hermite interpolation : f i : x i 2, x i 1, x i, x i +1, x i +2 u(t l+1, x i ) = u(t l, x i a t) Application to 1d 1d by Strang splitting tf + v xf = 0 on t/2 Poisson, then tf + E(t, x) vf = 0 on t tf + v xf = 0 on t/2 for Vlasov type

6 Conservative splines on a mesh containing a refined zone in velocity Application to a KEEN wave simulation Small structures in velocity located here around v [1.2, 1.6] [ 6, 6] f f eq elsewhere v coarse/ v fine = 32 Corresponding uniform velocity grid : points v 0 = 6 v N = 6 for Vlasov type (f f eq)(x i, v j ) for j [16000, 22000], N = 32000

7 for Vlasov type

8 Presence of strong magnetic field B Alignment of the solution along direction of B Need to take this into account in the numerics Design of a numerical that can avoid to take too much poloidal planes without loosing precision for Vlasov type Numerical tools : PIC/eulerian/semi-Lagrangian New idea (Hariri-Ottaviani,2013) : aligned interpolation

9 for Vlasov type Interpolation along a fixed oblic direction Reconstruction of the values necessary by interpolation in θ Reconstruction in the aligned direction

10 Constant advection along b = (b θ, b ϕ) tf + vb f = 0, f (t = 0, θ, ϕ) = e (imθ+inϕ) Lagrange interpolation of odd degree d θ in θ and Standard : Lagrange of odd degree d ϕ in ϕ e (k) T ( m θ) d θ+1 T ( n ϕ) dϕ+1 2 C dθ + C dϕ t t Aligned : Lagrange of odd degree d ϕ in aligned direction b ( ) e (k) T ( m θ) d θ+1 T n + b dϕ+1 θ m ϕ b ϕ 2 G dθ C dθ + C dϕ t t Same accuracy for ϕ aligned b ϕ ϕf b f ϕstandard. for Vlasov type

11 drift kinetic model in cylinder geometry corresponds to Grandgirard et al 2006, when b θ = 0 Poloidal cut f (t, r, θ, z = 0, v = 0) Mode (m = 10, n = 9) the most unstable (aligned, LAG17) (256 proc), 4000 itérations t = 2, on helios, supercomputer, 21 heures. for Vlasov type

12 Gain of factor 4 in Gysela (gyrokinetic code, CEA Cadarache) N ϕ 48 Initialization with a bath of modes Degree 4, and cubic splines in θ and other interpolations for Vlasov type

13 Design of a split Conservative Semi-Lagrangian G(η) = f (tn+1, η i ) = f (t n, η i ) + G η (η i), f = Jf, G(η i ) = 1 η η+ η/2 η η/2 η i η i f (tn, η)d η G( η)d η, G η (η i) = G(η i + η/2) G(η i η/2) η Reconstruction of polynomial from f (t n, η k ), k = i 1,..., i + 2 Reconstruction of flux G(η i + η/2) from G(η k ), k = i 1,..., i + 2, computed from polynomial for Vlasov type

14 for Vlasov type Conclusion Robustness of conservative cubic splines on non uniform grid Validation of the aligned in a gyrokinetic context using Lagrange interpolation Robustness of Hermite interpolation in the curvilinear case Perspectives Adaptation of the geometry for gyrokinetic simulations Convergence of SL schemes