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Two variational forms of the MHD ballooning equation. (English) Zbl 0804.76058
Summary: Two alternative variational formulations of the magnetohydrodynamic (MHD) ballooning equation are presented and compared. When discretized using finite elements, the two forms are shown to differ by a term of order \((\Delta\chi)^ 2\), where \(\chi\) is the azimuthal coordinate. Convergence studies indicate that this is reflected in quadratic convergence of the stability limit boundaries and that one form is generally pessimistic and the other optimistic with respect to these limits. The optimistic form has superior convergence properties with respect to errors in the equilibrium solution and, in contrast to the pessimistic form, cannot predict unphysical instabilities when the pressure gradient vanishes. On the other hand, the pessimistic form has a slightly better convergence rate in \(\Delta\chi\). The two forms together provide a much more reliable check for ballooning stability than either one alone.
MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
Software:
ERATO
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