Galerkin technique based on beam functions in application to the parametric instability of thermal convection in a vertical slot

A Fourier–Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.