Numerical scheme for solving the nonuniformly forced cubic and quintic Swift–Hohenberg equations strictly respecting the Lyapunov functional

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