Computation of pedestal and stellarator neoclassical ......stellarator neoclassical effects using a...
Transcript of Computation of pedestal and stellarator neoclassical ......stellarator neoclassical effects using a...
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.
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Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.
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Desirable features of a discretization for v or v
• Accurate integration and differentiation on the same grid or modes.• Domain: [0, ).
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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.
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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.
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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)).
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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
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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
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,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
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,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.
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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.
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/ 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.
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/ 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
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New scheme outperforms others on some physics applications
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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.
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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!
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0
0Ad A
Application 2: stellarator Fokker-Planck code SFINCS
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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
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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
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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
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Zeros of polynomials = Grids for Gaussian integration
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New polynomials
Laguerre (dots)Associated Laguerre, m=1/2 (crosses)
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2
First 10 new polynomial modes: xjP x e
22First 10 Laguerre polynomial modes: xjL x e