Thalia — A one-dimensional magnetohydrodynamic stability program using the method of finite elements

Appert, K.; Berger, D.; Gruber, R.; Troyon, F.; Roberts, Kelly

doi:10.1016/s0010-4655(84)82547-3

Cited by 1 publication

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“…Our r is closely related to the preceding works by Appert et al ( [1], [2]), where they used the combination of approximation spaces in which the divergence free condition can be realized. But, the mathematical rigorous reasoning of the success of this combination to avoid the so-called "spectral poUution" is somewhat diŸ (see Rappaz [17]).…”

Section: W Spectral Approximationmentioning

confidence: 79%

“…If we use a "global" element around r = 0 such as r = exp(s) in (-cr R(h)], there is some risk of "poUution". In practical application, for example in THALIA [2], the following "global" elements ave used around the axis: Table 1 shows the neax second order convergence of the smaUest eigenvalue of the fast spectrum. No spectral pollution is observed in this experiment.…”

Section: Iit-thlic(v~v) ~ -Iit-thllc(v~v) + Iith --ÿmentioning

confidence: 99%

“…On the other hand, References [5], [8], [9], [10], [171 contain mathematical justifications of some numerical algorithms used by physicists [1], [2], [19] in cylindrical and toroidal geometries; it is proved that the approximations have good spectral properties, in particular "non poUution'; however these analyses do not take account of the difficulty of the singularity at the magnetic axis.…”

Section: Introductionmentioning

confidence: 99%

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Spectral approximation for the linearized MHD operator in cylindrical region

Kako

Descloux

1991

Japan J. Indust. Appl. Math.

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We propose a numerical algorithm to calculate the spectrum of linearized roagnetohydrodynamic operators in the cylindrical region. The method is based on the Ritz-Galerkin approximation with special finite element basis functions. The detailed analysis is made in the presence of the magnetic axis. Ir is proved that there is no spectral pollution in this case. Some numerical examples ate included.w IntroductionThe study of the magnetohydrodynamic stability of an ideal plasma leads to the investigation of the spectrum of a linear operator. Both for theoretical and practical aspects, two major difficulties arise: a) the resolvent of the operator is not compact, b) the coefficients of the operator possess complicated singularity in the neighborhood of the magnetic axis.A mathematically rigorous characterization of the continuous spectrum of the operator has been obtained in References [4], [11] for the toroidal geometry; however it supposes that the plasma is isolated from the magnetic axis. The cylinder case, in the presence of the magnetic axis, is treated in [13].On the other hand, References [5], [8], [9], [10], [171 contain mathematical justifications of some numerical algorithms used by physicists [1], [2], [19] in cylindrical and toroidal geometries; it is proved that the approximations have good spectral properties, in particular "non poUution'; however these analyses do not take account of the difficulty of the singularity at the magnetic axis.The purpose of this paper is to propose and to discuss a numerical method for computing the spectrum of the operator for the cylinder case in the presence of the magnetic axis. The algorithm and its analysis rely strongly on the formulation of the operator given in [13]; in particular, in order to resolve the seeming singularity at the magnetic axis, the radial variable r is replaced by s = logr so that the independent variable varies over the in¡ interval (-c~, so]. In Section 2, we recall some relevant results of [13 I. In Section 3, we define the algorithm and state the main result of spectral convergence which will be proved in Section 4. Section 5 contains remarks on numerical integration and some numerical examples. The

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Section: W Spectral Approximationmentioning

confidence: 79%

Section: Iit-thlic(v~v) ~ -Iit-thllc(v~v) + Iith --ÿmentioning

confidence: 99%