AN EFFICIENT CODE TO COMPUTE NONPARALLEL STEADY FLOWS AND THEIR LINEAR-STABILITY

A simple, fast and efficient algorithm to compute steady non-parallel flows and their linear stability in parameter space is described. The pseudo-arclength continuation method is used to trace branches of steady states as one of the parameters is varied. To determine the linear stability of each state computed, a generalized eigenvalue problem of large order is 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 systems are solved with preconditioned Bi-CGSTAB. It is shown that it is possible to deal efficiently with (discretized) problems with O(10(5)) degrees of freedom. As an application, the bifurcation structure of steady two-dimensional Rayleigh-Benard Rows in large rectangular containers (up to aspect ratio 20) is computed. We show how the results connect up with those obtained with weakly nonlinear theory and extend these into the nonlinear regime. Main aim is to investigate whether pattern selection occurs through the occurrence of saddle node bifurcations creating intervals of unique steady states. It turns out that these intervals do not exist; multiple stable states continue to exist at large aspect ratio over a large range of Rayleigh numbers. In addition, the bifurcation structure provides no answer why the 'preferred' wavelength increases with increasing Rayleigh number, as observed in experiments.