Semi-Lagrangian kinetic and gyrokinetic simulations

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1 Semi-Lagrangian kinetic and gyrokinetic simulations M. Mehrenberger IRMA, University of Strasbourg and TONUS project (INRIA) Marseille, Equations cinétiques, Collaborators : B. Afeyan, A. Back, F. Casas, E. Chacon Golcher, A. Dodhy, N. Crouseilles, E. Faou, P. Glanc, V. Grandgirard, A. Hamiaz, P. Helluy, S. Hirstoaga, G. Latu, E. Madaule, L. Mendoza, P. Navaro, E. Sonnendrücker, M. Ottaviani, C. Prouveur, F. Rozar, C. Steiner M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 1 / 40

2 Outline Overview of recent works around kinetic and gyrokinetic simulations 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) 2. 2D conservative Semi-Lagrangian scheme (11-13) 3. Curvilinear and hexagonal meshes (14-17) 4. Gyroaverage operator on polar grids (18-33) 5. Field aligned interpolation (34-37) 6. Conclusion/perspectives/references (38-40) M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 2 / 40

3 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Strang splitting Since Cheng-Knorr (1976), most Vlasov codes use Strang splitting. Transport in x over t f (t, x, v)+v@ x f (t, x, v) =0, t! t + Update of E through the Poisson equation Transport in v over t t 2 Transport in x over t f (t, x, v)+e(t, x)@ v f (t, x, v) t f (t, x, v)+v@ x f (t, x, v) =0, t! t + Splitting is parametrized by s = 3, begin_with_x = true, a 1 = 1/2, a 2 = 1, a 3 = 1/2. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 3 / 40 t 2

4 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Abstract form Hamiltonian is defined as H(f )=T(f)+U(f), with Z v 2 Z 1 H(f ) = f (x, v)dxdv E(f )(x) 2 dx. (0,2 ) R Vlasov-Poisson can be written as (0,2 t f { H f (f ), f } = 0, with Poisson bracket {f, g} x v v x g Equivalent to d 8 G, G(f )=[H, G](f ) dt with Poisson bracket for functionals Z H [H, G] = f (f ){ G (f ), f }dxdv = f (0,2 ) R [G, H]. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 4 / 40

5 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Relation between T and U We have the relation (in 1D 1D) [[T, U], U] =2U This implies RKN condition [[[T, U], U], U] =0. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 5 / 40

6 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Derivation of high order methods We write p = e b1 U e a1 T e b 2 U e b2 U e a1 T e b 1 U Using for example the Baker-Campbell-Haussdorff formula, we get where (T +U)+R( ) p = e R( ) = 2 p 21 [T, U]+ 3 (p 31 [[T, U], T ]+p 32 [[T, U], U]) + 4 (p 41 [[[T, U], T ], T ]+p 42 [[[T, U], U], T ] +p 43 [[[T, U], U], U]) + O( 5 ) p ij are polynomials in the parameters a i, b i. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 6 / 40

7 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Use of modified potentials We can incorporate the flow associated to [[T, U], U]. p = e b 1 U+c 1 3 [[T,U],U] e a 1 T e a1 T e b 1 U+c 1 3 [[T,U],U] ) Achievement of given order with less stages further reduction of e Ci, when [[T, U], U] =2U, with C i = b i U + c i 2 [[T, U], U]+d i 4 W 5,1 + e i 6 W 7,1 = (b i + 2c i 2 + 4d i 4 8e i 6 )U with W 5,1 = [U, [U, [T, [T, U]]]] W 7,1 = [U, [T, [U, [U, [T, [T, U]]]]]]. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 7 / 40

8 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) 6-th order time splitting s = 11, a 1 = t 2 ( ), a 2 = , a 3 = t 2 ( ) +4 t 4 ( ), a 4 = , a 5 = t 2 ( ) +4 t 4 ( ) 8 t 6 ( ), a 6 = , together with a 6+i = a 6 i, i = 1,...,5 and begin_with_x = false. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 8 / 40

1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Application to a KEEN wave test problem FIGURE: Comparison of the time evolution of the first 5 harmonics t = 0.

9 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Application to a KEEN wave test problem FIGURE: Comparison of the time evolution of the first 5 harmonics t = (Strang) comparable to t = 0.5 (Sixth-order scheme). gain of factor 10 at least M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM 14 9 / 40

10 1. Sixth order splitting scheme for Vlasov-Poisson (3-10) Application to Landau damping test problem t = 2 M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

11 2. 2D conservative Semi-Lagrangian scheme (11-13) CSL2D : Conservative Semi-Lagrangian Vlasov solver Typical 2D advection block, here in @ t r g g = 0. r r Unknowns are mean values ḡ i,j = 1 A i,j RA i,j g h Degree 2 reconstruction of g h with finite difference of order p for edge and point values Update of mean values by intersection of backward volume f method for radial boundary conditions M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

12 2. 2D conservative Semi-Lagrangian scheme (11-13) CSL2D : diocotron instability test case FIGURE: Time evolution of relative total mass (left) and of energy (right) M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

13 2. 2D conservative Semi-Lagrangian scheme (11-13) CSL2D : drift kinetic simulation Better mass (left) and energy (right) conservation w.r.t classical BSL2D (Backward semi-lagrangian) Here on grid, with t = 8. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

14 3. Curvilinear and hexagonal mesh (14-17) Semi-Lagrangian scheme on curvilinear grids FIGURE: Representation of a mapped mesh in two dimensions M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

15 3. Curvilinear and hexagonal mesh (14-17) Guiding center simulation on curvilinear mesh Examples of meshes : Colella mesh, D-shaped mesh M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

16 3. Curvilinear and hexagonal mesh (14-17) Semi-Lagrangian schemes on hexagonal mesh r 3 r 2 r 1 FIGURE: Representation of an hexagonal mesh M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

17 3. Curvilinear and hexagonal mesh (14-17) Circular advection test problem M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

18 4. Gyroaverage operator on polar grids (18-33) Gyroaverage operator definition The gyroaverage J (f ) of a polar fonction f is defined as J (f )(r, )= 1 Z 2 f (~x 2 G + ~ )d. 0 where ~x G = r(cos( ), sin( )) and ~ = (cos( ), sin( )). This gyroaverage process consists in computing an average on the Larmor circle. It tends to damp any fluctuation which develops at sub-larmor scales. Let consider a uniform polar mesh (r, ) 2 [r min, r max ] [0, 2 [ with N r N cells, our goal is to approximate the operator f j,k 2 R (Nr +1) N 7! J (f ) j,k 2 R (Nr +1) N. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

19 4. Gyroaverage operator on polar grids (18-33) Method based on Padé approximation FIGURE: The zero-th order Bessel function J 0 (k ) compare to its Padé approximation 1/ 1 +(k ) 2 /4. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

20 4. Gyroaverage operator on polar grids (18-33) Method based on interpolation M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

21 4. Gyroaverage operator on polar grids (18-33) Benchmark with the classical 5D Cyclone DIII-D case FIGURE: Linear growth rate versus k i for the CYCLONE DIII-D base case : Padé vs interpolation M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

22 4. Gyroaverage operator on polar grids (18-33) Derivation of the quasi-neutrality equation Electrons follow a Boltzmann distribution : ( h i) n e = n 0 exp n 0 1 T e 1 ( h i). T e The ion density n i can be written at the first gyrokinetic order : Z n i (x) = J (f + g)(x, v z ) d dv z, = p 2µ with the correction : g(x, ):=@ µ F M (r, )[ (x) J ( )(x)]. Under the assumption of quasi-neutral limit n i = n e : Z n 0 ( J 2 n p 0 ( )) exp( µ/t T i R + 2µ i )dµ ( h i) = n i n 0. T e M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

23 4. Gyroaverage operator on polar grids (18-33) QN solver using Padé := 1 Z Jp T i ZR 2 ( )e µ/t i dµ = Jp 2 ( )e µ dµ. + 2µ R + 2µTi In Fourier space : \J ( )(k) =J 0 ( k ) b (k) ) b (k) = 0 ( k 2 T i ) b (k) where the function 0 is defined by : Z 0(k 2 ):= exp( R + A development of 0 leads to x 2 /2)J 2 0 (kx)xdx. r? (n 0 r? ) + n 0 T e ( h i) = n i n 0. This equation can be solved by FFT in and finite differences in r. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

24 4. Gyroaverage operator on polar grids (18-33) QN solver using Padé in a simplified model We also consider a simplified case of the quasi-neutrality equation where the integrals in µ are replaced by the evaluation at one value µ = µ 0 : n 0 T i ( J 2 p 2µ0 T i ( )) + n 0 T e ( h i) = In Fourier space, \ [ J 2 p 2µ0 T i ( )](k) = Z J p 2µ 0 (f )(x, v z )dv z n 0. 1 J 2 0 ( k p 2µ 0 T i ) b(k). The development : 1 J0 2 ( k p 2µ 0 T i )= k 2 µ 0 T i + O( k 4 ). leads to the equation : µ 0? + 1 ( h i) = 1 Z J T e n p (f )(x, v 2µ z )dv z 0 0 n 0. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

25 4. Gyroaverage operator on polar grids (18-33) QN solver using interpolation Matrix formulation of the linear operator : 7! n Z 0 ( Jp 2 ( )) exp( µ/t T i R + 2µ i )dµ. Use of the method with interpolation for the computation of the gyroaverage Quadrature in µ using Gauss-Legendre method Precomputation in Fourier space. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

26 4. Gyroaverage operator on polar grids (18-33) Numerical comparison with analytical solutions Homogeneous Dirichlet conditions on r min and r max. m,` (r, )= J m ( m,`)y m r Y m ( m,`)j m r r max where m,` is the `th zero of y 7! J m (y)y m y r min verifies Then : Z +1 0 r max p2µ J p 2µ ( )(r m,` 0, 0 )=J 0 r max ( J 2 p 2µ ( ))e µ dµ + = m,` r max e im Y m (y)j m y r min r max (r 0, 0 ). 2 m,` rmax 2!!. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

4. Gyroaverage operator on polar grids (18-33) Numerical comparison with analytical solutions FIGURE: Comparison between the analytical solution of the quasi-neutrality equation (in red) and the

27 4. Gyroaverage operator on polar grids (18-33) Numerical comparison with analytical solutions FIGURE: Comparison between the analytical solution of the quasi-neutrality equation (in red) and the solution obtained with Padé (left - in green) and with the interpolation method (right - in green) for ` = 1, m = 1. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

28 4. Gyroaverage operator on polar grids (18-33) Numerical comparison with analytical solutions FIGURE: Comparison between the analytical solution of the quasi-neutrality equation (in red) and the solution obtained with Padé (left - in green) and with the interpolation method (right - in green) for ` = 1, m = 5. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

29 4. Gyroaverage operator on polar grids (18-33) Application to gyrokinetic simulations The distribution function f (t, r,,z, v) is solution of J 2µ r J p! 0 2µ t r f f + r r v@ z z J 2µ0 v f = 0. for (r,,z, v) 2 [r min, r max ] [0, 2 ] [0, L] [ v max, v max ]. Coupling with : the quasi-neutrality equation solved with the interpolation method : 1 T i ( J 2 p 2µ0 T i ( ))+ 1 T e ( h i) = 1 n 0 Z J p 2µ 0 (f )(x, v z )dv z or its Padé approximation solved by FFT in and finite diff. in r : µ 0? + 1 ( h i) = 1 Z J T e n p (f )(x, v 2µ z )dv z n n 0 M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

4. Gyroaverage operator on polar grids (18-33) Numerical results FIGURE: f (r,,z = 0, v k = 0), 128 256 32 128, t = 2, interpolation (left), Padé (right),

30 4. Gyroaverage operator on polar grids (18-33) Numerical results FIGURE: f (r,,z = 0, v k = 0), , t = 2, interpolation (left), Padé (right), top : µ = 0.1 and T = 3000, bottom : µ = 0.2 and T = M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

31 4. Gyroaverage operator on polar grids (18-33) Numerical results FIGURE: f (r,,z = 0, v k = 0), , t = 2, µ = 0.7, T = 5000 (top), T = 7000 (bottom), with interpolation method (left) and Padé (right). M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

32 4. Gyroaverage operator on polar grids (18-33) Numerical results F IGURE: f (r,, z = 0, vk = 0), t = 2, µ = 1, T = 7000, interpolation (top), Padé (bottom). Mesh (from left to right) : ; ; and M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

4. Gyroaverage operator on polar grids (18-33) Numerical results FIGURE: Time evolution of R r max R 2 r min (r,,0)rdrd, mesh : 32 32 32 64, 0 comparison

33 4. Gyroaverage operator on polar grids (18-33) Numerical results FIGURE: Time evolution of R r max R 2 r min (r,,0)rdrd, mesh : , 0 comparison Padé/interpolation for different µ (0, 0.1, 0.2, 0.5, 0.7, 1). M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

34 5. Field aligned interpolation (34-37) Field aligned drift kinetic model Vlasov equation reads using v = v k for shorter @ r f + + v f + vb z f rb 0 rb 0 r r + b v f = 0, together with quasi-neutral equation (without zonal flow) r 2 + r r n r + 1 n 0 r + 1Te = 1n0 Z f f eq dv. = b /(2 r) b z /L, L = z max z min = 2 R. M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

35 Numerical method 5. Field aligned interpolation (34-37) Oblic constant advection with oblic interpolation constant advection in on a stencil around feet of characteristic for a given z = z k Lagrange interpolation of degree 2d + 1 in field aligned direction M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

36 5. Field aligned interpolation (34-37) Problem of visualization vs for representing the mode (30, 23) M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

37 5. Field aligned interpolation (34-37) = 0.8, m = 30, n = 23, T = 4000, 6000, 8000 M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

38 Conclusion Conclusion/perspectives/references (38-40) High order splitting scheme for Vlasov Poisson can outperform Strang splitting scheme Better energy (and mass) conservation property for conservative semi-lagrangian scheme Curvilinear grid for dealing with different geometries is an option Hexagonal grid is promising Interpolation method for gyroaverage operator leads to better accuracy First comparisons for quasi neutrality equation Validity of field aligned interpolation approach M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

39 Perspectives Conclusion/perspectives/references (38-40) Robust and efficient numerical methods for the poloidal plane Gyroaverage New Padé approximants Comprehension of continuous model and properties field aligned interpolation Test with shear safety factor Extension to 2D Hermite Explore "spaghetti" interpolation M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

40 References Conclusion/perspectives/references (38-40) Selalib : semi-lagrangian library. Afeyan, Casas, Crouseilles, Dodhy, Faou, M., Sonnendrücker, EPJD Vlasovia Steiner, M., Crouseilles, Grandgirard, Latu, Rozar, EPJD Vlasovia Crouseilles, Glanc, Hirstoaga, Madaule, M., Pétri, EPJD Vlasovia Mendoza, Prouveur, M., Sonnendrücker, Selhex project, CEMRACS Hamiaz, M., Back, Inria report, Hariri-Ottaviani, CPC M. Mehrenberger (UDS) Semi-Lagrangian (gyro)kinetic simulations Equations cinétiques, CIRM / 40

Advective and conservative semi-lagrangian schemes on uniform and non-uniform grids

Advective and conservative semi-lagrangian schemes on uniform and non-uniform grids M. Mehrenberger Université de Strasbourg and Max-Planck Institut für Plasmaphysik 5 September 2013 M. Mehrenberger (UDS