zbMATH — the first resource for mathematics

BECOOL: ballooning eigensolver with COOL finite elements. (English) Zbl 1280.76017
Summary: An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable-order Legendre polynomials. This reduces the second-order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (ballooning eigensolver using COOL finite elements) with second-order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower-order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.
76M10 Finite element methods applied to problems in fluid mechanics
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
Full Text: DOI
[1] Connor, J.W.; Hastie, R.J.; Taylor, J.B., High mode number stability of an axisymmetric toroidal plasma, Proc. roy. soc. London ser. A, 365, 1-17, (1979)
[2] Dewar, R.L.; Glasser, A.H., Ballooning mode spectrum in general toroidal systems, Phys. fluids, 26, 3038-3052, (1983) · Zbl 0538.76123
[3] Correa-Restrepo, D., Ballooning modes in three-dimensional MHD equilibria with shear, Z. naturforsch, 33a, 789-791, (1978)
[4] J. Nührenberg, R. Zille, Equilibrium and stability of low-shear stellarators, in: Proc. Joint Varenna Lausanne Int. Conf. on Theory of Fusion Plasmas, EUR11336EN, 1987, pp. 3-23.
[5] Cooper, W.A.; Singleton, D.B.; Dewar, R.L., Spectrum of ballooning instabilities in a stellarator, Phys. plasmas, 3, 275-280, (1996), (Erratum Phys. Plasmas 3 (1996) 3520)
[6] Cooper, W.A., Variational formulation of the linear MHD stability of 3D plasmas with noninteracting hot electrons, Plasma phys. contr. fusion, 34, 1011-1036, (1992)
[7] Sanchez, R.; Jimenez, J.A.; Garcia, L.; Varias, A., Compressibility effects on ideal and resistive ballooning stability in the TJ-II heliac device, Nucl. fusion, 37, 1363-1373, (1997)
[8] Nakajima, N., High-mode-number ballooning modes in a heliotron/torsatron system. II. stability, Phys. plasmas, 3, 4556-4567, (1996)
[9] Sanchez, R.; Hirshman, S.P.; Whitson, J.C.; Ware, A.S., COBRA: an optimized code for fast analysis of ideal ballooning stability of three-dimensional magnetic equilibria, J. comput. phys., 161, 576-588, (2000) · Zbl 1006.82500
[10] Ahusborde, A.; Gruber, R.; Azaiez, M.; Sawley, M.L., Physics-conforming constraints-oriented numerical method, Phys. rev. E, 75, 056704, (2007)
[11] Appert, K.; Azaiez, M.; Gruber, R., Modes of a plasma-filled waveguide determined by a numerical hp method, Commun. comput. phys., 5, 413-425, (2009) · Zbl 1364.82055
[12] Gruber, R.; Cooper, W.A.; Beniston, M.; Gengler, M.; Merazzi, S., Software development strategies for parallel computer architectures, Phys. rep., 207, 167-214, (1991)
[13] Anderson, D.V.; Cooper, W.A.; Gruber, R.; Merazzi, S.; Schwenn, U., Methods for the efficient calculation of the (MHD) magnetohydrodynamic stability properties of magnetically confined fusion plasmas, Int. J. supercomp. appl., 4, 34-47, (1990)
[14] Gruber, R.; Rappaz, J., Finite element methods in linear ideal MHD, Springer series in computational physics, (1985), Springer-Verlag Berlin
[15] Lowan, A.N.; Davids, N.; Levenson, A., Table of the zeros of the Legendre polynomials of order \(1 - 16\) and the weight coefficients for gauss’ mechanical quadrature rule, Bull. am. math. soc., 48, 739-742, (1942) · Zbl 0061.30406
[16] Hirshman, S.P.; Van Rij, W.I.; Merkel, P., Three-dimensional free boundary calculations using a spectral green’s function method, Comput. phys. commun., 43, 143-155, (1986)
[17] Nakamura, Y.; Matsumoto, T.; Wakatani, M.; Galkin, S.A.; Drozdov, V.V.; Martynov, A.A.; Poshekhonov, Yu.Yu.; Ichiguchi, K.; Garcia, L.; Carreras, B.A.; Nührenberg (née Schwab), C.; Cooper, W.A.; Johnson, J.L., Comparison of the calculations of the stability properties of a specific stellarator equilibrium with different MHD stability codes, J. comput. phys., 128, 43-57, (1996) · Zbl 0860.76077
[18] Boozer, A.H., Guiding center drift equations, Phys. fluids, 23, 904-908, (1980) · Zbl 0432.76099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.