A new class of symmetric factored approximate inverses is proposed and used in conjunction with the Preconditioned Conjugate Gradient method for solving sparse symmetric linear systems. Additionally, a new hybrid two-level solver is proposed utilizing a block independent set reordering, in order to create the two level hierarchy. The Schur complement is formed explicitly by inverting the blocks created by reordering. Then, the preconditioned conjugate gradient method is used in conjunction with the symmetric factored approximate inverse to solve the reduced order linear system. Furthermore, numerical results on the performance and convergence behavior for solving various model problems are presented.
Using N-body simulations we study the structures induced on a galactic disc by repeated flybys of a companion in decaying eccentric orbit around the disc. Our system is composed by a stellar disc, bulge and live dark matter halo, and we study the system's dynamical response to a sequence of a companion's flybys, when we vary i) the disc's temperature (parameterized by Toomre's Q-parameter) and ii) the companion's mass and initial orbit. We use a new 3D Cartesian grid code: MAIN (Mesh-adaptive Approximate Inverse N-body solver). The main features of MAIN are reviewed, with emphasis on the use of a new Symmetric Factored Approximate Sparse Inverse (SFASI) matrix in conjunction with the multigrid method that allows the efficient solution of Poisson's equation in three space variables. We find that: i) companions need to be assigned initial masses in a rather narrow window of values in order to produce significant and more long-standing non-axisymmetric structures (bars and spirals) in the main galaxy's disc by the repeated flyby mechanism. ii) a crucial phenomenon is the antagonism between companion-excited and self-excited modes on the disc. Values of Q > 1.5 are needed in order to allow for the growth of the companion-excited modes to prevail over the the growth of the disc's self-excited modes. iii) We give evidence that the companion-induced spiral structure is best represented by a density wave with pattern speed nearly constant in a region extending from the ILR to a radius close to, but inside, corotation.
The manifold theory of barred-spiral structure provides a dynamical mechanism explaining how spiral arms beyond the ends of galactic bars can be supported by chaotic flows extending beyond the bar's co-rotation zone. We discuss its applicability to N-body simulations of secularly evolving barred galaxies. In these simulations, we observe consecutive 'incidents' of spiral activity, leading to a timevarying disc morphology. Besides disc self-excitations, we provide evidence of a newly noted excitation mechanism related to the 'off-centering' effect: particles ejected in elongated orbits at major incidents cause the disc center-of-mass to recoil and be set in a wobble-type orbit with respect to the halo center of mass. The time-dependent m = 1 perturbation on the disc by the above mechanism correlates with the excitation of new incidents of non-axisymmetric activity beyond the bar. At every new excitation, the manifolds act as dynamical avenues attracting particles which are directed far from corotation along chaotic orbits. The fact that the manifolds evolve morphologically in time, due to varying nonaxisymmetric perturbations, allows to reconcile manifolds with the presence of multiple patterns and frequencies in the disc. We find a time-oscillating pattern speed profile Ωp(R) at distances R between the bar's corotation, at resonance with the succession of minima and maxima of the non-axisymmetric activity beyond the bar. Finally, we discuss disc thermalization, i.e., the evolution of the disc velocity dispersion profile and its connection with disc responsiveness to manifold spirals.
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