1
L A T E X Tik Zposter Particle in Fourier Discretisation of Kinetic Equations Jakob Ameres, Eric Sonnendr ¨ ucker Numerical Methods for Plasma Physics, Technische Universit ¨ atM ¨ unchen, Max-Planck-Institut f ¨ ur Plasmaphysik Particle in Fourier Discretisation of Kinetic Equations Jakob Ameres, Eric Sonnendr ¨ ucker Numerical Methods for Plasma Physics, Technische Universit ¨ atM ¨ unchen, Max-Planck-Institut f ¨ ur Plasmaphysik Introduction • The gyrokinetic model, which approximates the Vlasov-Maxwell equa- tions by averaging over the gyro-motion, is well suited for the study of turbulent transport in tokamaks and stellarators. Gyrokinetic Particle in Cell (PIC) codes using a finite element (FEM) field description are known to conserve energy but not momentum. • Using the Vlasov-Poisson equations in periodic domains with a purely Fourier based field solver yields a Monte Carlo particle method, Particle in Fourier (PIF), conserving both energy and momentum. • Fourier filters on FEM/PIC solvers are applied since the total number of physically relevant Fourier modes remains small. In the case of PIF one directly calculates the relevant modes without computational overhead. • In the scope of a field aligned description we derive a field solver, which couples a two dimensional Fourier transform in the torus’ angular direc- tions to B-splines over the radial coordinate yielding a hybrid PIC/PIF scheme. Particle Discretisation of Vlasov-Poisson • The Vlasov equation with external magnetic Field B , div (B ) = 0, ∂f ∂t + v ·∇ x f - (E + v × B ) ·∇ v f =0 (1) with the Poisson equation for the Electric potential Φ -ΔΦ = ρ - 1, ρ = Z f dv E := -∇Φ (2) • Solve by following the characteristics (V (t),X (t)). d dt V (t)= - (E (t, X (t)) + V (t) × B (t, X (t))) , d dt X (t)= V (t) • The solution f of equation (1) is constant along any characteristic f (t =0,X (t = 0),V (t = 0)) = f (t, X (t),V (t)) ∀t ≥ 0 • Probability density g (t, x, v ) with RR g (t =0, x, v ) dx dv = 1, g (t =0, x, v ) ≥ 0 ∀(x, v ) ∂g ∂t + v ·∇ x g - (E + v × B ) ·∇ v g =0. With supp{g }⊂ supp{f }, g is used to sample from f • Let one characteristic (X (0),V (0)) be randomly distributed according to g (t =0, ·, ·) and let the (x k ,v k ) be independent and identically distributed according to g (t =0, ·, ·) for all k =1,...,N p . • Define constant weights and the particle discretisation f h of f . c k := f (t, x k (t),v k (t)) g (t, x k (t),v k (t)) = f (t =0,x k (0),v k (0)) g (t =0,x k (0),v k (0)) f (t, x, v ) ≈ f h (t, x, v )= 1 N p N p X k =1 δ (x - x k (t)) δ (v - v k (t)) c k Poisson Equation in Fourier Space • The n-th spatial Fourier mode of ρ = R f dv is ˜ ρ n (t) := Z L 0 e inx 2π L ρ(x, t) dx = Z R Z L 0 e inx 2π L ρ(x, t) dxdv • Estimate ˜ ρ n (t) from the particle discretisation f h (x, t) ˜ ρ k (t) ≈ ˆ ˜ ρ k (t) := ZZ e ikx f h (x, v, t) dxdv = 1 N p N p X k =1 c t k ZZ e ikx δ (x - x t n )δ (v - v t n ) dxdv = 1 N p N p X k =1 c t k e ikx t n • The Electric Field is determined in Fourier space by the estimated Fourier modes. E (x, t)= X n6=0 e -inx 2π L ˜ E n (t) with ˜ E n (t)= ˜ ρ n (t) in ≈ ˆ ˜ ρ n (t) in • Momentum conservation at discrete level Z E (t, x)ρ(x, t) dx ≈ 1 N p N p X l =1 N h X n=-N h ,n6=0 1 N p N p X k =1 e inx t k ic l n e -inx t l = 1 N 2 p N h X n=1 i n N p X k,l =1 e in(x t k -x t l ) - e -in(x t k -x t l ) = -2 N 2 p N h X n=1 N p X k,l =1 1 n sin n(x t k - x t l ) | {z } -sin ( n(x t l -x t k ) ) =0 Variance Reduction • To reduce the variance of the estimate for the fields the δf method [1] is used. R f 1 dx RR e ix f 2 dx RR e ix f 3 dxdv RR e ix f 4 dxdv Estimating integrals with N p = 100 randomly distributed markers, uniformly in x and normally in v and the standard Monte Carlo estimator ˆ θ . Introduction of a control variate h allows sampling the difference δf = f - h thus reducing variance. • Step function f 1 (x) := x 8 , h 1 (x)= x • For a small perturbation f 2 (x) := 1+ cos(x) the zeroth Fourier mode F{f 2 }(0) = 1 causes the relative error on the first F{f 2 }(1) = 2 to be constant in expectation only for N p ∼ 1 2 , thus depending on the amplitude of the perturbation. Removing the zeroth Fourier mode with a control variate h 2 (x) = 1 the relative error to be of order 1 2 √ N p . • A one dimensional plasma density with a small spatial perturbation of a Maxwellian background f 3 (x, v ) := (1 + cos(2πx)) 1 √ 2π e - v 2 2 . Taking the zeroth spatial Fourier mode R 1 0 f 3 (x, v ) dx = 1 · 1 √ 2π e - v 2 2 =: h 3 (x, v ), here the Maxwellian background as Control Variate yields the same variance reduction as in the previous case. • Even for a perturbed Maxwellian velocity distribution the standard Maxwellian control variate is good choice f 4 (x, v ) := (1 + x cos(2πx)) (1+ v cos(6πv )) √ 2π e - v 2 2 . The Aliasing Problem • Finite Element PIC codes based on B-Splines suffer from aliasing, which means that even under Fourier filtering high frequencies appear in a low frequency interval. • By Fourier transform get the high frequency behavior of m-th degree B-Spline S m F (S m )(ω )= sinc ω 2 m+1 = 2sin ( ω 2 ) ω ! m+1 ∈O 1 ω m+1 • Estimating the Fourier modes directly - as in PIF - yields no aliasing of the energy of other frequencies, which allows calculating the error due to aliasing for a Fourier filtered PIC simulation. By increasing the B-Spline degree aliasing is suppressed and the PIC energy estimate converges to the PIF estimate. • Bump-on-tail instability [4] , N h = 32,N p = 10 6 , f ilter =1: 10, rk 2s Discretisation of the Cylinder • Use Fourier modes (PIF) in poloidal and toroidal direction and B- Splines (PIC) for the radial component. • Mesh grading in the radial component with knots r k and grading parameter α r > 0. α r = 1 uniform spacing, α r =0.5 equiareal cells, α r = 2 resolving singularity. r k := R max k N r α r , k =1,...,N r Summary • Particle in Fourier allows both momentum and energy conserving par- ticle simulations. • Due to its slim structure PIF eases the study of stochastic methods in theory and implementation. • The aliasing problems in PIC codes, which are resolved by PIF, can be studied. • With PIF turbulent transport simulations in the poloidal plane and the cylinder have been conducted and will be extended to the full torus. Guiding Center Model (2D) • A guiding center type equation on the polar plane Ω [2] ∂ t ρ +(∇Φ) y ∂ x ρ - (∇Φ) x ∂ y ρ =0 on Ω × [0, ∞) -4Φ= γρ Φ(x, y ) = 0 on ∂ Ω • Diocotron Instability r - =4,r + = 5, r max = 10, = 10 -2 , γ = -1. ρ(t =0, r, θ )= ( 1+ cos(lθ ) for r - ≤ r ≤ r + 0 else. Drift kinetic model (3D+1V) • Drift kinetic ions with adiabatic electrons in a cylindrical domain[2, 3]. ∂f ∂t + ~v GC ·∇ ⊥ f + v k ∂f ∂φ + dv k dt · ∂f ∂v k =0 -∇ ⊥ · (n 0 (r )∇ ⊥ Φ) + n 0 (r ) T e (r ) ( Φ - ¯ Φ ) = R f dv k - n 0 (r ) ¯ Φ(r, θ, t) := 1 L ϕ R L ϕ 0 Φ(r, θ, φ, t)dϕ • Ion Temperature Gradient instability in the linear phase full f δf (with local Maxwellian as control variate) References References [1] A. Y. Aydemir. “A unified Monte Carlo interpretation of particle simulations and applications to non-neutral plasmas” . In: Physics of Plasmas (1994-present) 1.4 (1994), pp. 822–831. [2] N. Crouseilles et al. “Semi-Lagrangian simulations on polar grids: from dio- cotron instability to ITG turbulence” . Feb. 2014. url: https://hal.archives- ouvertes.fr/hal-00977342. [3] V. Grandgirard et al. “A drift-kinetic Semi-Lagrangian 4D code for ion turbu- lence simulation” . In: Journal of Computational Physics 217.2 (2006), pp. 395– 423. issn: 0021-9991. doi: 10.1016/j.jcp.2006.01.023. url: http://www. sciencedirect.com/science/article/pii/S0021999106000155. [4] T. Nakamura and T. Yabe. “Cubic interpolated propagation scheme for solving the hyper-dimensional vlasov—poisson equation in phase space” . In: Computer Physics Communications 120.2 (1999), pp. 122–154. Email: [email protected] Advisory Board Meeting, 4 th September 2015, Greifswald
