Nonlinear pattern formation near the onset of Rayleigh-Bénard convection

A two-dimensional relaxational model equation is studied numerically to investigate the role of lateral boundaries and nonlinear terms in pattern formation. The model reduces in perturbation theory to the same amplitude equation as the one derived from the three-dimensional Boussinesq equations for thermal convection. State-of-the-art numerical methods are described that solve the initial-boundary-value problem efficiently and accurately in large rectangular cells and for long times, for both rigid and periodic boundary conditions. The results of simulations for different aspect ratios, Rayleigh numbers, and initial conditions are discussed in detail. The interaction of defects, the effect of lateral boundaries on the growth and saturation of linear instabilities, and the origin of the long-time scales needed to reach a stationary state are studied. Wave-number selection is investigated using spatial Fourier analysis, and evidence is presented that the band of stable wave numbers is not uniformly occupied as a pattern evolves from random initial conditions of all length scales. These results are in good agreement with many of the observed experimental features of pattern formation in small- and large-aspect-ratio cells, and show some new features that have not yet been seen.