Abstract
Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials and engineering sciences. The pursuit for theoretical descriptions of some among those physical problems led to the Swift-Hohenberg equation (SH3) which describes pattern selection in the vicinity of instabilities. A finite differences scheme, known as Stabilizing Correction (Christov and Pontes, 2002), developed to integrate the cubic Swift-Hohenberg equation in two dimensions, is reviewed and extended in the present paper. The original scheme features Generalized Dirichlet boundary conditions (GDBC), forcings with a spatial ramp of the control parameter, strict implementation of the associated Lyapunov functional, and second-order representation of all derivatives. We now extend these results by including periodic boundary conditions (PBC), forcings with Gaussian distributions of the control parameter and the quintic Swift-Hohenberg (SH35) model. The present scheme also features a strict implementation of the functional for all test cases. A code verification was accomplished, showing unconditional stability, along with second-order accuracy in both time and space. Test cases confirmed the monotonic decay of the Lyapunov functional and all numerical experiments exhibit the main physical features: highly nonlinear behavior, wavelength filter and competition between bulk and boundary effects. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
NSTL
Elsevier