Numerical approach to pattern selection in a model problem for Bénard convection in finite fluid layer

Long wave-length pattern formation is studied by means of numerical integration of a fourth-order in space nonlinear evolution equation subjected to Dirichlet laeral boundary conditions. Computationally efficient implicit difference scheme and algorithm are devised emplouying the method of operator splitting.

The case of Bénard convection in Boussinesq limit is considered. For different sets of the parameters different convective planforms are found: a pattern of hexagons ($H^+$) with upward flow in their centers, hexagons ($H^-$) with downward flow in the centers, coexisting hexagons and squares ($S$), and a case where the squares are selected. In the case when the critical wave-number vanishes (the wave-length diverges) the pattern selected is of a single cell which fills the whole domain under consideration.