AbstractPseudospectral methods are used for the computation of the time‐dependent convective flows which arise in shallow cavities filled with low‐Prandtí‐number liquids when submitted to a horizontal temperature gradient. In similar situations several former numerical results have been shown to disagree about the determination of the threshold of oscillations and about the subsequent supercritical regimes. Two different tau–Chebyshev methods based on the vorticity–streamfunction formulation and using multistep time schemes are considered. Their results are discussed to assess the validity of the solutions. The physical problems concern rectangular cavities which involve either a rigid or a stress‐free top wall and either conducting or insulating horizontal walls. Aside from the prediction of the onset of oscillations, which is discussed in the various situations with respect to the results of linear and non‐linear analyses and to other computational results, the present study exhibits some bifurcation sequences and a hysteresis cycle at moderate Grashof numbers which are associated to the occurrence of mult
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