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

TL;DR

This paper develops and extends a numerical scheme for the Swift-Hohenberg equations, ensuring stability, accuracy, and strict Lyapunov functional preservation for various boundary conditions and forcing scenarios.

Contribution

The authors extend a finite difference scheme to include periodic boundaries, Gaussian forcings, and the quintic model, while maintaining stability and Lyapunov functional strictness.

Findings

Unconditional stability demonstrated through code verification

Second order accuracy in time and space confirmed

Numerical experiments reproduce key physical phenomena

Abstract

Computational modeling of pattern formation in nonequilibrium systems is a fundamental tool for studying complex phenomena in biology, chemistry, materials science and engineering. 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 & Pontes; 2001 DOI: 10.1016/S0895-7177(01)00151-0), 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 behaviour, wavelength filter and competition between bulk and boundary effects.