Computation of pedestal and stellarator neoclassical effects using a new spectral speed grid
Matt Landreman, MIT PSFC
Thanks to Michael Barnes, Peter Catto, Darin Ernst, Felix Parra, Istvan Pusztai
First part of work: J Comp Phys (2013) http://dx.doi.org/10.1016/j.jcp.2013.02.041
Outline
• New spectral discretization scheme for v or v.
• Application 1: Pedestal global Fokker-Planck code.
• Application 2: Stellarator Fokker-Planck code.
2
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.
3
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).
4
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.
5
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.• Want grid points clustered at smallish v.
6
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.• Want grid points clustered at smallish v.• Accurate integrals for density/momentum/pressure, which differ by
a factor of v = (2– Non-analyticity of at v = 0 can destroy spectral convergence (Barnes,
Dorland, & Tatsuno, PoP 17, 032106 (2010)).
7
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.• Want grid points clustered at smallish v.• Accurate integrals for density/momentum/pressure, which differ by
a factor of v = (2– Non-analyticity of at v = 0 can destroy spectral convergence (Barnes,
Dorland, & Tatsuno, PoP 17, 032106 (2010)).
• Modal vs. collocation
8
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.• Want grid points clustered at smallish v.• Accurate integrals for density/momentum/pressure, which differ by
a factor of v = (2– Non-analyticity of at v = 0 can destroy spectral convergence (Barnes,
Dorland, & Tatsuno, PoP 17, 032106 (2010)).
• Modal vs. collocation
9
,0Laguerre: m m m y
i j i jdy L y L y y e 22 /2
0
ˆ 2 / th
nm
j j thj
f f L e
Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).• Should work well for Maxwellian-like functions.• Want grid points clustered at smallish v.• Accurate integrals for density/momentum/pressure, which differ by
a factor of v = (2– Non-analyticity of at v = 0 can destroy spectral convergence (Barnes,
Dorland, & Tatsuno, PoP 17, 032106 (2010)).
• Modal vs. collocation
10
,0Laguerre: m m m y
i j i jdy L y L y y e
2
,0New polynomials: x
i j i jd P x P x e 2/
0
ˆ / thn
j j thj
f f P e
22 /2
0
ˆ 2 / th
nm
j j thj
f f L e
These new non-standard polynomials lead to an integration and differentiation scheme.
11
2
0
ˆ n
xj j
j
f f P x e
/ 2 /x T m
2
0 x
i j ijdx P x P x e
These new non-standard polynomials lead to an integration and differentiation scheme.
12
/ 2 /x T m
2
0 x
i j ijdx P x P x e
Locationsof zeros:(can scaleto smaller x)
First 10 modes:
2
0
ˆ n
xj j
j
f f P x e
These new non-standard polynomials lead to an integration and differentiation scheme.
13
/ 2 /x T m
2
0 x
i j ijdx P x P x e
Locationsof zeros:(can scaleto smaller x)
• Gaussian integration
• Spectral differentiation: Weideman & Reddy, ACM Trans. Math. Software 26, 465 (2000).
• I use collocation method, but could also use a modal approach.
• Grid points at polynomial zeros.
• Can add a point at x=0 if desired.
2
0
ˆ n
xj j
j
f f P x e
New scheme outperforms others at both integration and differentiation
14
New scheme outperforms others on some physics applications
15
Number of speed grid points
Rel
ativ
e er
ror i
n Sp
itzer
resi
stiv
ity
1D problem: Spitzer resistivity 1 || MeEC f fT
3|| 1/ E e d f
New spectral scheme may or may not work well for your problem
• Pros:– Spectrally accurate integration and differentiation.– Very small # of points needed.– Can be exactly conservative:
(Barnes, Abel, Dorland et al, PoP 16, 072107 (2009))
– Grid points localized to small v.
16
0
0Ad A
New spectral scheme may or may not work well for your problem
• Pros:– Spectrally accurate integration and differentiation.– Very small # of points needed.– Can be exactly conservative:
(Barnes, Abel, Dorland et al, PoP 16, 072107 (2009))
– Grid points localized to small v.• Cons:
– Differentiation matrix is dense (though diagonal is a great preconditioner for Krylov solvers.)
– So far, seems unstable for time-dependent problems, even with implicit time-advance!
17
0
0Ad A
Application 2: stellarator Fokker-Planck code SFINCS
24
Stellarator Fokker-Planck Iterative Neoclassical Conservative Solver
1 1|| 1 1
ME FP m
f f ff C f
b v v
1 1 , , ,f f
Application 2: stellarator Fokker-Planck code SFINCS
25
11 12 13
21 22 23
|| ||31 32 33
ln ln
ln
Transport matrix
d p e d d TL L L d T d d
d TL L Ld
V B E BL L L
q
10-2
10-1
100
101
102
10-4
10-2
100
102
*
-L11 (Particle diffusivity)
Fokker-Planckpitch-angle scatteringmomentum-conserving model
10-2
10-1
100
101
102
10-4
10-2
100
102
*
L12=L21 (Thermodiffusion)
10-2
10-1
100
101
102
-1.5
-1
-0.5
0
0.5
*
L13=L31 (Bootstrap/Ware)
10-2
10-1
100
101
102
10-1
100
101
102
*
-L22 (Heat diffusivity)
10-2
10-1
100
101
102
-3
-2
-1
0
1
*
L23=L32 (Bootstrap/Ware)
10-2
10-1
100
101
102
10-2
100
102
104
*
L33 (Conductivity)
Application 2: stellarator Fokker-Planck code SFINCS
26
11 12 13
21 22 23
|| ||31 32 33
ln ln
ln
Transport matrix
d p e d d TL L L d T d d
d TL L Ld
V B E BL L L
q
For ion neoclassical physics in LHD, momentum-conserving model collision operator compares well to full Fokker-Planck operator.
Summary• New spectral discretization scheme for v or v gives rapid
convergence with # of grid points.
– Very useful for the time-independent collisional problems I’ve considered. Other applications?
– Matlab & fortran source code for generating grid, integration weights, & differentiation matrices available at http://web.mit.edu/landrema/www/software/
– J Comp Phys (2013) http://dx.doi.org/10.1016/j.jcp.2013.02.041
• Scheme implemented in global Fokker-Planck code for tokamak pedestals.
– Strong poloidal asymmetries arise in flow.
• Scheme implemented in stellarator Fokker-Planck code.27
Extra slides
28
Zeros of polynomials = Grids for Gaussian integration
29
New polynomials
Laguerre (dots)Associated Laguerre, m=1/2 (crosses)
30
2
First 10 new polynomial modes: xjP x e
22First 10 Laguerre polynomial modes: xjL x e
