Simulation gyrocinétique des plasmas de fusion magnétique

Transcription

1 Simulation gyrocinétique des plasmas de fusion magnétique IRMA, Université de Strasbourg & CNRS Calvi project-team INRIA Nancy Grand Est Collaborators: J.P. Braeunig, N. Crouseilles, M. Mehrenberger, T. Respaud (CALVI) V. Grandgirard, G. Latu (CEA Cadarache) IHP, Paris, 5 November 2010

2 Outline The physics context: magnetic fusion and ITER Specific issues linked to geometry and gyrokinetic equation Semi-Lagrangian Vlasov simulations Conservative semi-lagrangian method

3 The ITER project The Physics context ITER being built in France: Several new research actions. INRIA Large scale initiative on magnetic fusion. Involve applied mathematicians and computer scientists in ITER. Mathematical understanding and development of reduced models Gyrokinetic simulations: GYSELA (CEA+Strasbourg) MHD simulations: JOREK (CEA+Bordeaux+Nice) Inovative numerical methods: Asymptotic Preserving, geometric.. Plasma control: EQUINOXE (CEA+Nice)

4 Turbulent transport in magnetized plasma Plasma not very collisional and far from fluid state Kinetic description necessary. Fluid and kinetic simulations of turbulent transport yield very different results. Vlasov (6D espace des phases) coupled to Maxwell or Poisson f t + v xf + q m (E + v B) v f = 0. Toroidal geometry

5 The gyrokinetic model The Vlasov-Poisson gyrokinetic model reads f t + dx dt xf + dv f = 0, dt v with B dx dt B dv dt = b J(φ) + 1 q (mv 2 b + µb B) + V B = (B + m q V b) ( µ m B + q m J(φ)) and B = B + m q V b b. The gyroaverage operator J transforms the guiding center distribution to the distribution at the particle position which allows to take into account the finite Larmor radius effects.

6 Conservativity The Physics context We have the relations (B dx dt ) = (b J(φ)) + 1 (b µ B) q On the other hand = J(φ) b + µ q B b (B dv v dt ) = b (µ B + J(φ)). q Hence the phase-space divergence vanishes, which leads to the conservatity.

7 Numerical solution of the gyrokinetic model Difficulties: defined in phase-space (5D). Appearance of small scales Two approaches Full f. The gyrokinetic equation is solved for the full distribution function. δf. The perturbation with respect to the equilibrium function is solved, involving in some cases linearized equations. Two types of numerical methods: Particle methods: often preferred for higher dimensions. Often good qualitative results at lower cost. Numerical noise and slow convergence difficult to get precise results in some situations. Methods based on phase-space grids: Good grid resolution necessary for qualitatively correct results. No numerical noise, but diffusion. Easier to parallelize on a large number of processors.

8 The classical backward semi-lagrangian Method f conserved along characteristics Find the origin of the characteristics ending at the grid points Interpolate old value at origin of characteristics from known grid values High order interpolation needed Typical interpolation schemes. Cubic spline (Cheng-Knorr) Cubic Hermite with derivative transport (Nakamura-Yabe)

9 A history of semi-lagrangian schemes for the Vlasov equation Cheng-Knorr (JCP 1976): split method for 1D Vlasov-Poisson ES-Roche-Bertrand-Ghizzo (JCP 1998) : general semi-lagrangian framework for Vlasov type equations. Nakamura-Yabe (CPC 1999): CIP semi-lagrangian scheme with Hermite interpolation. Filbet-ES-Bertrand (JCP 2001): PFC semi-lagrangian positive and conservative with Lagrange interpolation. Nakamura-Tanaka-Yabe-Takizawa (JCP 2001): Conservative 1D split CIP semi-lagrangian. N. Besse - ES (JCP 2003) : SL schemes on unstructured grids. Crouseilles-Respaud-ES (CPC 2009) : Forward SL method. Crouseilles-Mehrenberger-ES (JCP 2010): conservative semi-lagrangian spline PSM + new classes of positive filters. Qiu-Christlieb (JCP 2010): conservative SL WENO schemes.

10 Convergence of semi-lagrangian schemes Filbet (SINUM 2001): PFC method for Vlasov-Poisson N. Besse (SINUM 2003): semi-lagrangian method with linear interpolation for Vlasov-Poisson. Campos Pinto - Mehrenberger (Numer. Math. 2008): adaptative SL method for Vlasov-Poisson N. Besse - Mehrenberger (Math of Comp 2008): split SL method for Vlasov-Poisson for symmetric high order interpolation Lagrange and spline interpolation. N. Besse (SINUM 2008): SL method with Hermite interpolation and propagation of gradients. Respaud - ES (Numer math, in revision): Forward SL for Vlasov-Poisson with linear interpolation.

11 Problem with non conservative Vlasov solver When non conservative splitting is used for the numerical solver, the solver is not exactly conservative. Does generally not matter when solution is smooth and well resolved by the grid. The solver is still second order and yields good results. However: Fine structures develop in non linear simulations and are at some point locally not well resolved by the phase space grid. In this case a non conservative solvers can exhibit a large numerical gain or loss of particles which is totally unphysical. Lack of robustness.

12 Vortex in Kelvin-Helmholtz instability diagt1kh2.plot u 1:3 diagt16kh2.plot u 1:3 diagt17kh2.plot u 1: diagt1kh2.plot u 1:4 diagt16kh2.plot u 1:4 diagt17kh2.plot u 1:

13 Computation of the origin of the characteristics Transport equation f t + a f = 0, Characteristics dx dt = a Computation of the origin of the characteristics : Explicit solution if a does not depend on x Else, numerical algorithm needed.

14 Splitting for exact computation of characteristics In many cases splitting can enable to solve a constant coefficient advection at each split step. E.g. separable Hamiltonian H(q, p) = U(q) + V (p). Vlasov equation in canonical coordinates reads f t + ph q f q H p f = 0. Split equations then become f t + pv q f = 0, f t qu p f = 0, where U does not depend on p and V does not depend on q characteristics can be solved explicitly. Vlasov-Poisson falls into this category with q = x, p = v, H(x, v) = 1 2 mv2 + qφ(x, t).

15 First order computation of characteristics Consider general case. Characteristics defined by dx dt = a(x, t). Backward solution: X n is known and a n known on the grid. Standard procedure to derive first order numerical method for EDO. Integrate on one time step and use quadrature formula for integral (left or right rectangle). X n+1 X n = t a n (X n ) or X n+1 X n = t a n+1 (X n+1 ). No explicit solution: Fixed point procedure needed in first case (e.g. Newton). Predictor-corrector method on a needed in second case.

16 Second order method for Vlasov-Poisson (1/2) Technique can be used for solving separable Hamiltonian Vlasov equations without splitting. We shall consider Vlasov-Poisson to illustrate this case. At time t n : f n and E n are known at grid points. f n+1 and E n+1 need to be computed. Need to solve characteristics backward in time from t n+1 to t n dv dt = E(X (t), t), dx dt = V. Main problem E n+1 needed and not known.

17 Second order method for Vlasov-Poisson (2/2) A second order in time algorithm predictor-corrector algorithm is defined as follows: 1 Predict Ē n+1 using continuity equation. 2 For all grid points x i = X n+1, v j = V n+1 compute V n+1/2 = V n+1 t 2 Ē n+1 (X n+1 ), X n = X n+1 tv n+1/2, V n = V n+1/2 t 2 E n (X n ). Interpolate f n at point (X n, V n ). 3 Yields first approximation of f n+1 (x i, v j ) = f n (X n, V n ) that can be used to correct Ē n+1. Iterate until Ē n+1 does not vary anymore. At most a couple iterations required.

18 A two step second order method Solve characteristics defined by dx dt = a(x, t). Centered quadrature on two time steps: X n+1 X n 1 = 2 t a n (X n ), X n+1 +X n 1 = 2X n +O( t 2 ). Use fixed point procedure to compute X n 1 such that X n+1 X n 1 = t a n ( Xn+1 + X n 1 ). 2 Problem: compute f n+1 from f n 1. Even and odd order time approximations become decoupled after some time. Artificial coupling needs to be introduced.

19 A one step predictor-corrector second order method Solve characteristics defined by dx dt = a(x, t). Centered quadrature on one time step: X n+1 X n = t a n+ 1 2 (X n+ 1 2 ), X n+1 +X n = 2X n O( t 2 ). Now a n+ 1 2 is unknown. Predictor-corrector procedure needed. Use fixed point procedure to compute X n such that X n+1 X n = t ā n+ 1 X n+1 + X n 2 ( ). 2 Both predictor-corrector and fixed point iterations needed.

20 The forward semi-lagrangian method f conserved along characteristics Characteristics advanced with same time schemes as in PIC method. Leap-Frog Vlasov-Poisson Runge-Kutta or Cauchy-Kovalevsky for guiding-center or gyrokinetic Values of f deposited on grid of phase space using convolution kernel. Identical to PIC deposition scheme but in whole phase space instead of configuration space only. Similar to PIC method with reconstruction introduced by Denavit (JCP 1972).

21 Discrete distribution function Function projected on partition of unity basis for conservativity. Good choice is cubic B-splines. f is reprojected on mesh at each time step. Between t n and t n+1 f h (x, v, t) = i,j w i,j B(x X (t; x i, v j, t n ))B(v X (t; x i, v j, t n )). Weight w i,j associated to the particle starting from grid point (x i, v j ) at t n is coefficient of spline satisfying interpolation conditions f h (x k, v l, t n ) = i,j w i,jb(x k x i )B(v l v j ). Projection on phase space mesh is obtained with formula f n+1 (x k, v l ) = i,j w i,j B(x k X (t n+1, x i, v j, t n ))B(v l V (t n+1, x i, v j, t n

22 Time advance for Vlasov-Poisson As opposite to BSL, trajectories are advanced forward in time. Advection field is known at initial time. Standard ODE algorithms can be applied. For Vlasov-Poisson, we have z(t n ) = (x n, v n ), and E(t n, z n ) = E(t n, x n ). Separable Hamiltonian. Natural scheme is Verlet algorithm, which is second order accurate in time Step1 : v n+ 1 2 v n = t 2 E(tn, x n ), Step2 : x n+1 x n = t v n+1/2, Step3 : v n+1 v n+ 1 2 = t 2 E(tn+1, x n+1 ).

23 Time advance for guiding-center Standard ODE solvers can be used in the general case, e.g. Runge-Kutta methods Second-order Runge Kutta method Step1 : X n+1 X n = te (t n, X n ) Step2 : Compute E (t n+1, X n+1 ) Step3 : X n+1 X n = t 2 [ ] E (t n, X n ) + E (t n+1, X n+1 ) Problem: Electric field needs to be computed at each stage. Computation time blows up when order (i.e. number of stages) are increased

24 Landau damping The Physics context 1 f 0 (x, v) = ( cos(kx)) e v2 2, L = 4π. 2π -6-8 Runs/thek.dat u 1: *x

25 Bump on tail (BSL (top) vs. FSL (bottom)) Electric energy time=0 time=20 time=30 Electric energy time=40 time=70 time= time time Electric energy time=0 time=20 time=30 Electric energy time=40 time=70 time= Simulation gyrocinétique 0 des plasmas de fusion magnétique

26 Bump on tail: total energy (BSL (left) vs. FSL (right))

27 Energy conservation for Kelvin-Helmoltz instability thdiago2.dat u 1:3 thdiago4.dat u 1:3../CG_BSL/thdiag.dat u 1:3 thdiag.dat u 1:

28 FSL vs. BSL The Physics context very promising. Accurate description of whole phase-space in particular tail of distribution function and small perturbations. A little more diffusive than BSL. Better with smaller time steps. Some advantages: classical explicit EDO solver can be used, in particular high order if needed. No need for predictor-corrector or fixed point algorithm. Can benefit from charge conserving PIC algorithms (Villasenor-Buneman).

29 Conservative semi-lagrangian method Start from conservative form of Vlasov equation f t + (f a) = 0. f dx dv conserved along caracteristics V Three steps: High order polynomial reconstruction. Resolution Projection Splitting conservative 1D equations. Filtering for positivity and suppressing oscillations.

30 Link between classical and conservative semi-lagrangian methods For constant coefficient advections we can show that C-Lag(2d) SL-Lag(2d+1) PSM SPL Consequences: 1 Classical semi-lagrangian method is conservative for split schemes reducing to constant coefficients advection for each split step. 2 PFC (Filbet-ES-Bertrand) corresponds to classical semi-lagrangian method with cubic Lagrange interpolation. 3 PSM corresponds to classical semi-lagrangian method with cubic spline interpolation. Note that Collella and Woodward Finite Volume scheme is also algebraically equivalent for constant coefficient advections.

31 Motivation for curvilinear coordinates Due to confinement by large magnetic field particle travel a lot faster along magnetic field lines than across. Look at typical turbulence simulation shows that large scale structures form along field line and small scale structures across. Significant anisotropy of flow For optimal resolution anisotropic mesh would be needed. Simplest idea used in most gyrokinetic codes: use curvilinear coordinates locally aligned with magnetic field lines.

32 L1 and L norms of relative error for oblic advection (travel 10 times the domain) function of the angle between advection direction and mesh lines direction:

33 4D drift-kinetic model From left to right: BSL, PSM and PSM with entropic flux limiter. 4D simulation of 128x256x128x64 cells, time =2000.

34 Semi-Lagrangian method has proved to be a good choice for full-f gyrokinetic simulations. 5D gyrokinetic code efficiently running on several thousands of processors. What option to best handle anisotropy and long time accuracy at minimal cost? Field aligned curvilinear coordinates first step. Best choice? Unstructured mesh. Not easy... NURBS for description of field lines and geometry of Tokamak.