The new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f(x)=0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f(x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume.

1.
L. L.
Lao
,
H.
St. John
, and
R. D.
Stambaugh
,
Nucl. Fusion
25
,
1611
1622
(
1985
).
2.
L.
Spitzer
,
Phys. Fluids
1
,
253
264
(
1958
).
3.
A.
Dinklage
,
C. D.
Beidler
,
P.
Helander
,
G.
Fuchert
,
H.
Maaßberg
,
K.
Rahbarnia
,
T.
Sunn Pedersen
,
Y.
Turkin
,
R. C.
Wolf
,
A.
Alonso
 et al,
Nat. Phys.
14
,
855
860
(
2018
).
4.
M. D.
Kruskal
and
R. M.
Kulsrud
,
Phys. Fluids
1
,
265
274
(
1958
).
5.
R.
Chodura
and
A.
Schlüter
,
J. Comput. Phys.
41
,
68
88
(
1981
).
6.
F.
Bauer
,
O.
Betancourt
, and
P.
Garabedian
,
A Computational Method in Plasma Physics
, Springer Series in Computational Physics, edited by
W.
Beiglbock
,
H.
Canannes
, and
S. A.
Orszag
(
Springer-Verlag
,
1978
).
7.
U.
Schwenn
,
Comput. Phys. Commun.
31
,
167
199
(
1984
).
8.
A.
Bhattacharjee
,
J. C.
Wiley
, and
R. L.
Dewar
,
Comput. Phys. Commun.
31
,
213
225
(
1984
).
9.
T. C.
Hender
,
B. A.
Carreras
,
L.
Garcia
, and
J. A.
Rome
,
J. Comput. Phys.
60
,
76
96
(
1985
).
10.
S. P.
Hirshman
and
J. C.
Whitson
,
Phys. Fluids
26
,
3553
3568
(
1983
).
11.
S. A.
Lazerson
,
J.
Loizu
,
S.
Hirshman
, and
S. R.
Hudson
,
Phys. Plasmas
23
,
012507
(
2016
).
12.
O.
Betancourt
,
Commun. Pure Appl. Math.
41
,
551
568
(
1988
).
13.
K.
Harafuji
,
T.
Hayashi
, and
T.
Sato
,
J. Comput. Phys.
81
,
169
192
(
1989
).
14.
A.
Reiman
and
H.
Greenside
,
Comput. Phys. Commun.
43
,
157
167
(
1986
).
15.
S. P.
Hirshman
,
R.
Sanchez
, and
C. R.
Cook
,
Phys. Plasmas
18
,
062504
(
2011
).
16.
S. R.
Hudson
,
R. L.
Dewar
,
G.
Dennis
,
M. J.
Hole
,
M.
McGann
,
G.
von Nessi
, and
S.
Lazerson
,
Phys. Plasmas
19
,
112502
(
2012
).
17.
R. C.
Grimm
,
J. M.
Greene
, and
J. L.
Johnson
, “
Computation of the magnetohydrodynamic spectrum in axisymmetric toroidal confinement systems
,” in
Methods in Computational Physics: Advances in Research and Applications
, edited by
S.
Fernbach
,
M.
Rotenberg
, and
J.
Killeen
(
Academic Press, Inc
.,
New York
,
1976
), Vol.
16
, pp.
253
280
.
18.
W. D.
D'haeseleer
,
W. N. G.
Hitchon
,
J. D.
Callen
, and
J. L.
Shohet
,
Flux Coordinates and Magnetic Field Structure
, Springer Series in Computational Physics, edited by
R.
Glowinski
,
M.
Holt
,
P.
Hut
,
H. B.
Keller
,
J.
Killeen
,
S. A.
Orszag
, and
V. V.
Rusanov
(
Springer-Verlag
,
1991
).
19.
A. H.
Boozer
,
Rev. Mod. Phys.
76
,
1075
(
2005
).
20.
J. P.
Boyd
,
Chebyshev and Fourier Spectral Methods
(
Dover Publications, Inc
.,
New York
,
2000
).
21.
F.
Zernike
,
Mon. Not. R. Astron. Soc.
94
,
377
384
(
1934
).
22.
T.
Sakai
and
L. G.
Redekopp
,
J. Comput. Phys.
228
,
7069
7085
(
2009
).
23.
T.
Matsushima
and
P. S.
Marcus
,
J. Comput. Phys.
120
,
365
374
(
1995
).
24.
W. T. M.
Verkley
,
J. Comput. Phys.
136
,
100
114
(
1997
).
25.
V.
Lakshminarayanan
and
A.
Fleck
,
J. Mod. Opt.
58
,
545
561
(
2011
).
26.
J.
Loomis
, “
A Computer Program for Analysis of Interferometric Data
,” in
Optical Interferograms—Reduction and Interpretation
, edited by
A.
Guenther
and
D.
Liebenberg
(
ASTM International
,
West Conshohocken, PA
,
1978
), pp.
71
86
.
27.
V. L.
Genberg
,
G. J.
Michels
, and
K. B.
Doyle
,
Proc. SPIE
4771
,
276
286
(
2002
).
28.
J. P.
Boyd
and
F.
Yu
,
J. Comput. Phys.
230
,
1408
1438
(
2011
).
29.
W. H.
Press
,
B. P.
Flannery
,
S. A.
Teukolsky
, and
W. T.
Vetterling
,
Numerical Recipes: The Art of Scientific Computing
(
Cambridge University Press
,
1989
).
30.
H. R.
Lewis
and
P. M.
Bellan
,
J. Math. Phys.
31
,
2592
2596
(
1990
).
32.
D. K.
Lee
,
J. H.
Harris
,
S. P.
Hirshman
, and
G. H.
Neilson
,
Nucl. Fusion
28
,
1351
1364
(
1988
).
33.
R. L.
Dewar
and
S. R.
Hudson
,
Physica D
112
,
275
280
(
1998
).
34.
J. M.
Carnicer
and
C.
Godes
,
Numer. Algorithms
66
,
1
16
(
2014
).
35.
D.
Ramos-Lopez
,
M. A.
Sanchez-Granero
,
M.
Fernandez-Martinez
, and
A.
Martinez-Finkelshtein
,
Appl. Math. Comput.
274
,
247
257
(
2016
).
36.
L. N.
Trefethen
,
Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations
(
Cornell University
,
1996
).
37.
K.
Levenberg
,
Q. Appl. Math.
2
,
164
168
(
1944
).
38.
D.
Marquardt
,
SIAM J. Appl. Math.
11
,
431
441
(
1963
).
39.
J. J.
Moré
, “
The levenberg-marquardt algorithm: Implementation and theory
,” in
Numerical Analysis
, edited by
G. A.
Watson
(
Springer
,
1978
), pp.
105
116
.
40.
S. P.
Hirshman
and
J.
Breslau
,
Phys. Plasmas
5
,
2664
2675
(
1998
).
41.
D. W.
Dudt
and
E.
Kolemen
(
2020
). “
DESC stellarator equilibrium solver
,” DESC. https://github.com/ddudt/DESC.

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