A Galerkin spectral method for fourth-order boundary value problems.

摘要

In the present work we develop a Fourier-Galerkin spectral technique for solving coupled fourth-order boundary value problems arising in continuum mechanics. The set of so-called beam functions are used as a basis together with the harmonic functions. All the necessary expansion formulas for the application of the technique (derivatives, unity) are derived, including the crossexpansions between the two systems of functions.; As a featuring example we treat the convective flow of a viscous liquid in an infinite vertical slot subject to harmonic gravity modulation (g-jitter). We show that the rate of convergence of the spectral solution series is fifth-order algebraic for both linear and nonlinear problems of fourth order. 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. The technique is used in the investigation of the g-jitter flow, and the parametric stability diagrams are produced.; Whilst the spatial approximation of the algorithm is tested by means of solving three fourth order model problems, the temporal approximation is assessed by comparing it to the finite difference operator-splitting conservative scheme of Christov and Homsy in [6]. For a limited range of the governing parameters, we derive asymptotic expansions for the sought functions and find these to be in good agreement with our numerical results.