Numerical scheme for Swift-Hohenberg equation with strict implementation of lyapunov functional

In this paper, we consider a nonlinear generalized diffusion equation called the Swift-Hohenberg equation (SH) for which a Lyapunov functional is known. We develop a computationally efficient second-order in time implicit difference scheme based on the operator-splitting method. Internal iterations are used to make the scheme both nonlinear and implicit. We prove that the scheme allows strict (independent of the truncation error) implementation of a discrete approximation of the Lyapunov functional. The new scheme is used to investigate the pattern formation from random initial conditions, and spatially chaotic states are found.