Bisymmetric normal modes in soft-centred and realistic galactic discs
  We test methods for the determination of unstable modes in stellar discs: a point collocation scheme in the action sub-space, a scheme based on expansion of the density and potential on the biorthonormal basis, and a finite element method. Using models of galaxies with low and high mass concentration to the center, the existence of two different kinds of spectra of unstable modes is demonstrated. Characteristic features of methods and obtained spectra are discussed.   Despite ignoring any constraint on the continuity or differentiability of the perturbed DF, the collocation scheme is reliable for obtaining spectra and patterns in both kinds of models. The method based on the expansion of the perturbed potential and surface density over a biorthogonal basis was not applicable to a model with high mass concentration. The finite element method successfully used in various fields of science and engineering is currently sensitive to the presence of resonant orbits due to the choice of interpolation functions for the distribution function. 
Evgeny Polyachenko, Andreas Just
astro-ph.GA