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**An efficient code to compute non-parallel steady flows and their linear stability.**
*(English)*
Zbl 0848.76056

An algorithm which is not only simple, but fast as well as efficient has been presented. It is used to compute steady non-parallel flows and their linear stability in parameter space. The pseudo-arclength continuation method is used to trace branches of steady states as one of the parameters is varied. In order to determine the linear stability of each state computed, a generalized eigenvalue problem of large order is to be solved. Only a prescribed number of eigenvalues, those closest to the imaginary axis, are calculated by a combination of a complex mapping and the simultaneous iteration technique. The underlying linear system are solved with preconditioned Bi-CGSTAB. The method is applied efficiently to problems up to \(O(10^5)\) degrees of freedom. The authors use the algorithm to compute the bifurcation structure of steady two-dimensional Rayleigh-Bénard flows in large rectangular containers up to aspect ratio 20. The results compare qualitatively well with those of weakly nonlinear theory.

The authors’ hypothesis about the selection induced by saddle node bifurcation fails in this particular case. The bifurcation results do not clarify the experimentally observed increase of wavelength of pattern with increasing Rayleigh numbers. The numerical method is deficient in that the test functions used to detect simple bifurcation points are not robust and have to be reconsidered for each application. Further, the linear solver works very well for linear systems which are diagonally dominant, but for systems which are not, the amount of fill-in in the preconditioning matrix and thereby the required CPU time and use of memory increase. Notwithstanding these limitations, the numerical approach can be used to study new problems like stability of large scale ocean circulation etc..

The authors’ hypothesis about the selection induced by saddle node bifurcation fails in this particular case. The bifurcation results do not clarify the experimentally observed increase of wavelength of pattern with increasing Rayleigh numbers. The numerical method is deficient in that the test functions used to detect simple bifurcation points are not robust and have to be reconsidered for each application. Further, the linear solver works very well for linear systems which are diagonally dominant, but for systems which are not, the amount of fill-in in the preconditioning matrix and thereby the required CPU time and use of memory increase. Notwithstanding these limitations, the numerical approach can be used to study new problems like stability of large scale ocean circulation etc..

Reviewer: S.C.Rajvanshi (Chandigarh)

### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76E15 | Absolute and convective instability and stability in hydrodynamic stability |

### Keywords:

parameter space; pseudo-arclength continuation method; generalized eigenvalue problem; complex mapping; simultaneous iteration technique; preconditioned Bi-CGSTAB; bifurcation structure; two-dimensional Rayleigh-Bénard flows; rectangular containers
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\textit{H. A. Dijkstra} et al., Comput. Fluids 24, No. 4, 415--434 (1995; Zbl 0848.76056)

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