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We propose an extension of the Galerkin–Eckhaus method in order to study thermoconvective instabilities not only in the weakly nonlinear régime, but also farther from the threshold. This extension consists in an appropriate choice of the basis functions used to expand the unknowns and in an increase of the number of amplitude equations taken into account. The rigid-free Rayleigh–Bénard problem with heat-conducting boundaries and the Marangoni–Bénard problem are considered in detail before generalizing to all thermoconvective instabilities. The validity of our approach is proven by comparing its results with a multiple scale method and with a finite element method. A very good agreement between all methods is found in the weakly nonlinear regime. Farther from the threshold, the agreement between our extended Galerkin–Eckhaus method and the finite element approach is also obtained with appropriate basis functions whose choice depends crucially on the value of the Prandtl number and also on the boundary conditions imposed at the top and bottom of the system.

Extended Galerkin–Eckhaus Method in Nonlinear Thermoconvection