307 research outputs found
An improved method for solving quasilinear convection diffusion problems on a coarse mesh
A method is developed for solving quasilinear convection diffusion problems
starting on a coarse mesh where the data and solution-dependent coefficients
are unresolved, the problem is unstable and approximation properties do not
hold. The Newton-like iterations of the solver are based on the framework of
regularized pseudo-transient continuation where the proposed time integrator is
a variation on the Newmark strategy, designed to introduce controllable
numerical dissipation and to reduce the fluctuation between the iterates in the
coarse mesh regime where the data is rough and the linearized problems are
badly conditioned and possibly indefinite. An algorithm and updated marking
strategy is presented to produce a stable sequence of iterates as boundary and
internal layers in the data are captured by adaptive mesh partitioning. The
method is suitable for use in an adaptive framework making use of local error
indicators to determine mesh refinement and targeted regularization. Derivation
and q-linear local convergence of the method is established, and numerical
examples demonstrate the theory including the predicted rate of convergence of
the iterations.Comment: 21 pages, 8 figures, 1 tabl
Basins of attraction in nonsmooth models of gear rattle
This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the system's behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit)
Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow
Magnetorotational dynamo action in Keplerian shear flow is a three-dimensional, non-linear magnetohydrodynamic process whose study is relevant to the understanding of accretion processes and magnetic field generation in astrophysics. Transition to this form of dynamo action is subcritical and shares many characteristics of transition to turbulence in non-rotating hydrodynamic shear flows. This suggests that these different fluid systems become active through similar generic bifurcation mechanisms, which in both cases have eluded detailed understanding so far. In this paper, we build on recent work on the two problems to investigate numerically the bifurcation mechanisms at work in the incompressible Keplerian magnetorotational dynamo problem in the shearing box framework. Using numerical techniques imported from dynamical systems research, we show that the onset of chaotic dynamo action at magnetic Prandtl numbers larger than unity is primarily associated with global homoclinic and heteroclinic bifurcations of nonlinear magnetorotational dynamo cycles. These global bifurcations are found to be supplemented by local bifurcations of cycles marking the beginning of period-doubling cascades. The results suggest that nonlinear magnetorotational dynamo cycles provide the pathway to turbulent injection of both kinetic and magnetic energy in incompressible magnetohydrodynamic Keplerian shear flow in the absence of an externally imposed magnetic field. Studying the nonlinear physics and bifurcations of these cycles in different regimes and configurations may subsequently help to better understand the physical conditions of excitation of magnetohydrodynamic turbulence and instability-driven dynamos in a variety of astrophysical systems and laboratory experiments. The detailed characterization of global bifurcations provided for this three-dimensional subcritical fluid dynamics problem may also prove useful for the problem of transition to turbulence in hydrodynamic shear flows
A geometric Newton method for Oja's vector field
Newton's method for solving the matrix equation runs
up against the fact that its zeros are not isolated. This is due to a symmetry
of by the action of the orthogonal group. We show how
differential-geometric techniques can be exploited to remove this symmetry and
obtain a ``geometric'' Newton algorithm that finds the zeros of . The
geometric Newton method does not suffer from the degeneracy issue that stands
in the way of the original Newton method
- …