The Semi-Lagrangian method with obliqueinterpolation
M. Mehrenberger
IRMA, University of Strasbourg and TONUS project (INRIA)
Nice, INRIA Project Labs Fratres Meeting, 15-16.10.2015
Joint work including the participation of Yaman Güclü, Guillaume Latu,Maurizio Ottaviani and Eric Sonnendrücker
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 1 / 31
I. Introduction
Motivation
Exploit alignement of the structures according to the strong externalmagnetic field in plasma turbulence simulations
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 2 / 31
I. Introduction
Magnetic field
We consider here cylindrical or toroidal geometry
The external strong magnetic field reads B = Bθθ̂ + Bϕϕ̂Here
∇r = r̂ , ∇θ =θ̂
r, ∇ϕ =
ϕ̂
R
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 3 / 31
I. Introduction
Magnetic field
A general form of the magnetic field in tokamaks is
B = F0∇ϕ+∇Ψ×∇ϕ.
Poloidal flux Ψ for cylindrical or toroidal geometry is supposed to beof the form Ψ = Ψ(r)
⇒ A flux surface is a (θ, ϕ) plane with given r
Trajectories induced by the magnetic field remain in the flux surface.We get
Bθ =−Ψ′(r)
R, Bϕ =
F0
R
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 4 / 31
I. Introduction
Magnetic field
The pitch angle αpitch is the angle of B with toroidal direction ϕ̂
tan(αpitch) =BθBϕ
=−Ψ′(r)
F0
We define the rotational transform ι
ι =time to do one poloidal rotationtime to do one toroidal rotation
which is the inverse of the safety factor q = 1ι
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 5 / 31
I. Introduction
Magnetic field
We approximate ι so that it only depends on r (B we call it still ι)
ι(r) =R0
rtan(αpitch) =
R0BθrBϕ
We define the aspect ratio aratio = R0rmax
and have R0r ≥ aratio
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 6 / 31
I. Introduction
Trajectories in (θ, ϕ) plane
We have∂θrθ′(t) + ∂ϕrϕ′(t) = B(r , θ(t))
and∂θr = r θ̂, ∂ϕr = Rϕ̂
which leads torθ′(t) = Bθ, Rϕ′(t) = Bϕ
andθ′(t)ϕ′(t)
=B · ∇θB · ∇ϕ
=RBθrBϕ
= ι(r)RR0
So, we solve {θ′(t) = ι(r)
ϕ′(t) = R0R = 1
1+(r/R0) cos(θ(t))
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 7 / 31
I. Introduction
Trajectories in (θ, ϕ) plane
Example that will be used. At rpeak = rmin+rmax2 , we have R0
rpeak' 5.45.
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 8 / 31
I. Introduction
Specific form of the functions in (θ, ϕ) plane
Initial data is bath of modes g(t = 0) ' sin(mθ + nϕ) subject toadvection like equation, when |B| is dominant
∂tg + v‖B · ∇g ' 0
Gradient along B :
B · ∇g � k‖ cos(mθ + nϕ)
withk‖ = mι+ n R0
R
|k‖| small↔ No much variation of g along B direction.In plasma turbulent simulations, it is observed that
Mode numbers m and n are big (unbounded)k‖ remains small (bounded)
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 9 / 31
I. Introduction
⇒ Field aligned methods
Many codes simulating turbulence in tokamaks have now a fieldaligned strategyHow to deal with semi-Lagrangian codes (here of interest) ?
Change the grid with curvilinear transform
⇒ First attempt in [Brauenig et al, 2012]
Keep the grid fixed but use field aligned interpolation⇒ General idea developed first in [Hariri-Ottaviani, 2013] (3D fluid code)⇒ First results in Semi-Lagrangian gyrokinetic context together with
curvilinear geometry in the poloidal plane [Kwon et al., 2014]
Field aligned interpolation is a new emerging idea that has startedto be studied and concerns both fluid and kinetic applications,semi-Lagrangian or eulerian methodsAim : reduce the computational size / improve precision
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 10 / 31
I. Introduction
THE FCI approach
FCI = flux coordinate independantReduce the number of points in ϕ thanks to adhoc transformation(Ottaviani, 2009)
⇒ permits to treat X-point geometry⇒ discretization needs not to be changed
Former transformations : Reduce the number of points in θ (S.Cooley et al., 1991 ; Scott, 2001)
⇒ not able to treat X-point geometry
Here, we do not have to deal with X-point geometry, but still use thesame FCI approach and adapt it to the semi-Lagrangian framework
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 11 / 31
I. Introduction
Example of FCI approach
Use same fine cartesian grid of the poloidal plane to deal with
Hariri-Ottaviani, CPC (2013) ( c©picture from Hariri talk EFTC2015)
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 12 / 31
I. Introduction
Our "semi-FCI" approach
Semi-Lagrangian contextUse of polar coordinates (not cartesian)We do not change the discretization in θgeneralization to more complex geometry imply use of curvilineargeometry in the poloidal plane (until last closed surface) ( c©picturefrom Hariri talk EFTC2015)
full FCI also possible (on cartesian grid), but 1D interpolation in θ(see later) would be replaced by 2D interpolation.
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 13 / 31
I. Introduction
Validation of the approach
Aim of our work : propose one of such an approach and check itsvalidity on different models of increasing difficulty
Description of the method2D advection
⇒ comparison with exact solution
4D screw pinch drift kinetic model in cylindrical geometry
⇒ comparison with known linear instability rate
4D toroidal version of Gysela
⇒ comparison with standard method using different grid resolutions
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 14 / 31
II. Description of the method
The 2D Semi-Lagrangian method
c©V. Grandgirard slidesM. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 15 / 31
II. Description of the method
Oblique interpolation at feet of the characteristics
⇒ Reconstruction of the needed values through θ interpolation⇒ Reconstruction in the aligned direction
cubic splines not possible in the aligned directionWe use here Lagrange interpolation of odd degreeprevious 1D interpolation is now 2D interpolation
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 16 / 31
II. Description of the method
Oblique interpolation at feet of the characteristics
direction given by ι(r)
Use of splitting for (θ, ϕ) advectionFeet through first or second order Taylor expansionDerivative of potential also aligned computed
⇒ Hope to keep Nϕ small ; Nθ (and Nr ) remains big.⇒ Approx 16 points needed per mode
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 17 / 31
III. 2D advection
2D Translation in ι direction
Advection equation∂tg + B · ∇g = 0
Large aspect ratio limit : R = R0
⇒ Translation parametrized by ιInitial condition g(t = 0) = sin(mθ + nϕ)
L∞ error after one time step ; different time steps are usedDifferent regimes for m = −34 and ι = 1/
√2. Here k‖ = n + mι
n = 5 so k‖ ' −19.04 (old method better)n = 12 so k‖ ' −12.04 (equivalent methods)n = 23 so k‖ ' −1.04 (new method better)n = 30 so k‖ ' 5.96 (new method still better)
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 18 / 31
III. 2D advection
Error versus Nϕ/n ; Nθ = 200 (or 400 for n = 23)
Lagrange interpolation of degree 9
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 19 / 31
III. 2D advection
Error versus Nϕ/k‖Lagrange interpolation of degree d
d = 3 d = 5 d = 17
Gain/loss is∣∣∣ n
k‖
∣∣∣M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 20 / 31
IV. Cylindrical geometry
Screw pinch drift kinetic model (in Selalib)
Vlasov type equation for ions
∂t f −∂θφ
rB0∂r f +
∂r Φ
rB0∂θf + v∇‖f −∇‖Φ∂v f = 0, ∇‖ = b · ∇,
External magnetic field B = B0bΦ electric potential satisfies a Poisson type equation
Equations in cylindrical geometry
rmin = 0.1, rmax = 14.5, R0 ' 240that is ρ∗ = 1/14.5 and aspect ratio aratio ' 16.5Nr = 256, Nθ = 512, Nv = 128d = 5 cubic splines in θ and other interpolations.
Different ι
ι = 0 : standard drift kinetic model [Grandgirard et al., JCP, 2006]ι = 0.8
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 21 / 31
IV. Cylindrical geometry
Single mode excitation
Development of Ion Temperature Gradient (ITG) modesDispersion relation gives expected exponential growth rates fromsingle mode excitationIt depends on m and k‖ = (n + mι) bϕ, wherebϕ = 1√
1+(ιr/R)2' 1± 10−3.
We take initial data feq(r , v) (1 + ε(r) cos(mθ + nϕ)), with m = 15
n = 1, for ι = 0⇒ k‖ = 1n = −11 for ι = 0.8⇒ k‖ ' 1
⇒ Results should be the same, at least in the linear phase
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 22 / 31
IV. Cylindrical geometry
Time evolution of potential energy
⇒ Instability growth rates match with dispersion relation⇒ Nϕ = 32 and Nϕ = 64 are similar
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 23 / 31
IV. Cylindrical geometry
poloidal cut f (T , r , θ,0,0) for T = 4000 (top) andT = 6000 (bottom)
ι = 0,Nϕ = 32 ι = 0.8,Nϕ = 32 ι = 0.8,Nϕ = 64M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 24 / 31
IV. Cylindrical geometry
f (T , rmin+rmax2 , θ, ϕ,0) for T = 4000 (top) and T = 6000
(bottom)
ι = 0,Nϕ = 32 ι = 0.8,Nϕ = 32 ι = 0.8,Nϕ = 64
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 25 / 31
V. Toroidal geometry
GYSELA results
Equations in toroidal geometry
rmin = 4, rmax = 40, R0 = 120That is ρ∗ = 1/40 and aspect ratio aratio = 3Nr = Nθ = 256, Nv = 48d = 5, cubic splines in θ and other interpolations.Bath of modes initialization
Different toroidal discretizations :
Aligned method, with Nϕ = 32Standard method, with Nϕ = 32Standard method, with Nϕ = 64 and Nϕ = 128
shear case : ι(rmin) = 1 ≥ ι(r) ≥ ι(rmax) = 2/3
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 26 / 31
V. Toroidal geometry
Time evolution of potential energy
⇒ 4 times less points for the aligned method for similar accuracy
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 27 / 31
V. Toroidal geometry
GYSELA results poloidal (top) and (θ, ϕ) cut (bottom)of electric potential
Nϕ = 32, aligned Nϕ = 32, standard Nϕ = 128, standard
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 28 / 31
VI. Conclusion
Conclusion/Perspectives
Aligned method is validated on different situations⇒ Oblique interpolation seems to be enough, even for not straight
magnetic field lines (toroidal case)Possible extensions
Improve efficiencyReduce cost of interpolationAdhoc parallelization strategies
Go to larger size machines :ITER : Nr , Nθ ' 4000 Nϕ ' 36000 → Nϕ ' 64 ?
More general geometryworks on curvilinear geometryHexagonal meshdiscontinuous Galerkin method
Add electronscoupling fluid/kinetic ?PIC/semi-Lagrangian ?
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 29 / 31
VII. Backup
Time evolution of potential energy
no shear case shear case
⇒ 4 times less points for the aligned method for similar accuracy
M. Mehrenberger (UDS) The oblique Semi-Lagrangian method IPL FRATRES, Nice 2015 30 / 31