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We develop and test spectral Galerkin schemes to solve the coupledOrr-Sommerfeld (OS) and induction equations for parallel, incompressible MHD infree-surface and fixed-boundary geometries. The schemes' discrete bases consistof Legendre internal shape functions, supplemented with nodal shape functionsfor the weak imposition of the stress and insulating boundary conditions. Theorthogonality properties of the basis polynomials solve the matrix-coefficientgrowth problem, and eigenvalue-eigenfunction pairs can be computed stably atspectral orders at least as large as p = 3,000 with p-independent roundofferror. Accuracy is limited instead by roundoff sensitivity due to non-normalityof the stability operators at large hydrodynamic and/or magnetic Reynoldsnumbers (Re, Rm > 4E4). In problems with Hartmann velocity and magnetic-fieldprofiles we employ suitable Gauss quadrature rules to evaluate the associatedexponentially weighted sesquilinear forms without error. An alternativeapproach, which involves approximating the forms by means ofLegendre-Gauss-Lobatto (LGL) quadrature at the 2p - 1 precision level, is foundto yield equal eigenvalues within roundoff error. As a consistency check, wecompare modal growth rates to energy growth rates in nonlinear simulations andrecord relative discrepancy smaller than $ 1E-5 $ for the least stable mode infree-surface flow at Re = 3E4. Moreover, we confirm that the computed normalmodes satisfy an energy conservation law for free-surface MHD with errorsmaller than 1E-6. The critical Reynolds number in free-surface MHD is found tobe sensitive to the magnetic Prandtl number Pm, even at the Pm = O(1E-5) regimeof liquid metals.

A spectral Galerkin method for the coupled Orr-Sommerfeld and induction equations for free-surface MHD