Numerical solution of the Swift-Hohenberg equation in two dimensions

Abstract

The Swift-Hohenberg equation with either a stochastic or a constant forcing term has been solved numerically in two spatial dimensions. The parameters that enter the equation have been chosen to match the experiments on Rayleigh-Bénard convection by Meyer et al. [C.W. Meyer, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. 59 (1987) 1577]. Our numerical results for the convective heat current as a function of time fit the experiments well (the fitting parameter is the amplitude of the forcing term). We find a value of F = 5.52 × 10 −6 for the stochastic case, compared to F th = 1.06 × 10 −10, the value obtained from fluctuation theory. The structure of the convective pattern also depends on the type of forcing considered. A constant forcing induces a roll-like pattern that reflects the geometry of the sidewalls. A stochastic forcing is seen to induce a random, cellular pattern.