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Accurate derivative evaluation for any Grad-Shafranov solver. (English) Zbl 1349.76258
Summary: We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
82D10 Statistical mechanical studies of plasmas
82D40 Statistical mechanical studies of magnetic materials
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[1] Grad, H.; Rubin, H., Hydromagnetic equilibria and force-free fields, (Proceedings of the Second United Nations Conference on the Peaceful Uses of Atomic Energy, vol. 31, (1958)), 190
[2] Shafranov, V. D., On magnetohydrodynamical equilibrium configurations, Sov. Phys. JETP, 6, 545, (1958) · Zbl 0081.21801
[3] Deriaz, E.; Desprès, B.; Faccanoni, G.; Gostaf, K.; Imbert-Gérard, L.-M.; Sadaka, G.; Sart, R., Magnetic equations with freefem++: the Grad-Shafranov equation & the current hole, (ESAIM: Proceedings, vol. 32, (2011), EDP Sciences), 76-94 · Zbl 1235.76064
[4] Howell, E. C.; Sovinec, C. R., Solving the Grad-Shafranov equation with spectral elements, Comput. Phys. Commun., 185, 1415, (2014) · Zbl 1344.76063
[5] Huysmans, G. T.A.; Goedbloed, J. P.; Kerner, W., Isoparametric bicubic Hermite elements for solution of the Grad-Shafranov equation, Int. J. Mod. Phys. C, 2, 371, (1991)
[6] Lütjens, H.; Bondeson, A.; Roy, A., Axisymmetric MHD equilibrium solver with bicubic Hermite elements, Comput. Phys. Commun., 69, 287, (1992)
[7] Lütjens, H.; Bondeson, A.; Sauter, O., The CHEASE code for toroidal MHD equilibria, Comput. Phys. Commun., 97, 219, (1996) · Zbl 0922.76240
[8] Goedbloed, J. P.; Keppens, R.; Poedts, S., Advanced magnetohydrodynamics: with applications to laboratory and astrophysical plasmas, (2010), Cambridge University Press Cambridge
[9] Beliën, A. J.C.; Botchev, M. A.; Goedbloed, J. P.; van der Holst, B.; Keppens, R., FINESSE: axisymmetric MHD equilibria with flow, J. Comput. Phys., 182, 91, (2002) · Zbl 1021.76026
[10] Ling, K. M.; Jardin, S. C., The Princeton spectral equilibrium code: PSEC, J. Comput. Phys., 58, 300, (1985) · Zbl 0582.76123
[11] Pataki, A.; Cerfon, A. J.; Freidberg, J. P.; Greengard, L.; O’Neil, M., A fast, high-order solver for the Grad-Shafranov equation, J. Comput. Phys., 243, 28, (2013) · Zbl 1349.76925
[12] Lee, J. P.; Cerfon, A. J., ECOM: a fast and accurate solver for toroidal axisymmetric MHD equilibria, Comput. Phys. Commun., 190, 72, (2015) · Zbl 1344.76090
[13] Gruber, R.; Troyon, F.; Berger, D.; Bernard, L. C.; Rousset, S.; Schreiber, R.; Kerner, W.; Schneider, W.; Roberts, K. V., ERATO stability code, Comput. Phys. Commun., 21, 323, (1981)
[14] Liu, D. H.; Bondeson, A., Improved poloidal convergence of the MARS code for MHD stability analysis, Comput. Phys. Commun., 116, 55, (1999) · Zbl 1074.76646
[15] Jolliet, S.; Bottino, A.; Angelino, P.; Hatzky, R.; Tran, T. M.; Mcmillan, B. F.; Sauter, O.; Appert, K.; Idomura, Y.; Villard, L., A global collisionless PIC code in magnetic coordinates, Comput. Phys. Commun., 177, 409, (2007)
[16] Vlad, G.; Briguglio, S.; Fogaccia, G.; Zonca, F., Toward a new hybrid MHD gyrokinetic code: progresses and perspectives, (11th IAEA Technical Meeting on Energetic Particles in Magnetic Confinement Systems, Kyiv 21-23 Sept. 2009, (2009), International Atomic Energy Agency Vienna, Austria), P-25
[17] Trefethen, L. N.; Bau, D., Numerical linear algebra, (1997), SIAM · Zbl 0874.65013
[18] Jardin, S. C., Computational methods in plasma physics, (2010), Chapman & Hall/CRC Press New York · Zbl 1198.76002
[19] Evans, L. C., Partial differential equations, (1998), American Mathematical Society
[20] Trefethen, L. N., Spectral methods in MATLAB, (2000), SIAM Philadelphia · Zbl 0953.68643
[21] Klöckner, A.; Barnett, A.; Greengard, L.; O’Neil, M., Quadrature by expansion: a new method for the evaluation of layer potentials, J. Comput. Phys., 252, 332, (2013) · Zbl 1349.65094
[22] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73, 325, (1987) · Zbl 0629.65005
[23] Epstein, C. L.; Greengard, L.; Klöckner, A., On the convergence of local expansions of layer potentials, SIAM J. Numer. Anal., 51, 2660, (2013) · Zbl 1281.65047
[24] M. Rachh, A. Klöckner, M. O’Neil, Fast algorithms for ‘quadrature by expansion’ I: globally valid expansions, in preparation.
[25] Manas, Rachh, Integral equation methods for problems in electrostatics, elastostatics and viscous flow, (2015), New York University, PhD thesis
[26] Beatson, R. K.; Greengard, L., A short course on fast multipole methods, (Ainsworth, M.; Levesley, J.; Light, W.; Marletta, M., Wavelets, Multilevel Methods and Elliptic PDEs, (1997), Oxford University Press), 1-37, Freely accessible at · Zbl 0882.65106
[27] Trefethen, L. N.; Weideman, J. A.C., The exponentially convergent trapezoidal rule, SIAM Rev., 56, 385, (2014) · Zbl 1307.65031
[28] Sifuentes, J.; Gimbutas, Z.; Greengard, L., Randomized methods for rank-deficient linear systems, (2014)
[29] Solov’ev, L. S., Sov. Phys. JETP, 26, 400, (1968)
[30] Cerfon, A. J.; Freidberg, J. P., “one size fits all” analytic solutions to the Grad-Shafranov equation, Phys. Plasmas, 17, (2010)
[31] Aymar, R.; Barabaschi, P.; Shimomura, Y., The ITER design, Plasma Phys. Control. Fusion, 44, 519, (2002)
[32] Sabbagh, S. A.; Kaye, S. M.; Menard, J.; Paoletti, F.; Bell, M.; Bell, R. E.; Bialek, J. M.; Bitter, M.; Fredrickson, E. D.; Gates, D. A.; Glasser, A. H.; Kugel, H.; Lao, L. L.; LeBlanc, B. P.; Maingi, R.; Maqueda, R. J.; Mazzucato, E.; Wurden, G. A.; Zhu, W., Equilibrium properties of spherical torus plasmas in NSTX, Nucl. Fusion, 41, 1601, (2001)
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