A semi-implicit finite difference approach for the swift hohenberg equation: Stability, convergence, and pattern formation

The Swift-Hohenberg equation (SH-PDE) is a fundamental model for pattern formation in nonlinear systems with symmetry breaking instabilities. This work presents a semi-implicit finite difference scheme for solving the SH-PDE, taking advantage of a linear/nonlinear decomposition to optimise stability and computational efficiency. We rigorously establish the theoretical properties of the method, including bounds and error estimates, proving stability and convergence under appropriate conditions. Numerical experiments confirm these conclusions, demonstrating second-order spatial accuracy and first-order temporal accuracy. The method is tested under various initial conditions and nonlinearities, capturing characteristic patterns such as stripes, rolls and dots, in line with the expected behaviour of SH-PDE. These results emphasise the robustness and efficiency of the proposed approach, positioning it as a powerful tool for studying pattern formation in nonlinear systems.