Newton–Krylov continuation of periodic orbits for Navier–Stokes flows

Umbría

2017

Phys. Rev. E

This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddlenode or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center-symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, it is found out which of the bifurcations at the limits of the intervals of stable periodic orbits are sub or supercritical.PACS numbers: 47.15.-x, 47.20.-k

Section: Discussionmentioning

confidence: 82%

Section: Mathematical Formulationmentioning

confidence: 99%

Periodic orbits in tall laterally heated rectangular cavities

Umbría

2017

Phys. Rev. E

“…where the coefficients α i , β j and γ 0 do not depend on n, and are listed, for instance, in [6]. It is well known that, if the step number k satisfies that k ≤ 6, and the time steps are constant or the ratios ∆t n /∆t n−1 are kept adequately bounded (see e.g.…”

Section: Time Integration Methodsmentioning

confidence: 99%

“…The exclusive use of time integrations is not sufficient to establish the origin of the laminar flows and their dependence on the parameters, specially in the case of subcritical or multicritical transitions. In these situations pseudo-arclength continuation methods [5,6,7], and the linear stability analysis of the time dependent solutions [8,9] provide powerful instruments to clarify the dynamics. In addition, the computation of Liapunov exponents is used to assess if a system is chaotic.…”

Section: Introductionmentioning

confidence: 99%

A comparison of high-order time integrators for thermal convection in rotating spherical shells

García

Journal of Computational Physics

García-Archilla

et al. 2010

Self Cite

A numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.

“…where the coefficients α i , β j and γ 0 do not depend on n, and are listed, for instance, in [24]. It is well known that, if k 6 and the time steps are constant (see e.g.…”

Section: Implicit-explicit Schemesmentioning

confidence: 99%

Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells

García

Bonaventura

Journal of Computational Physics

et al. 2014

We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicitexplicit (IMEX) multi-step methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earth's outer core conditions.