A non-iterative and unconditionally energy stable method for the Swift–Hohenberg equation with quadratic–cubic nonlinearity

Most implicit methods for the Swift–Hohenberg (SH) equation with quadratic–cubic nonlinearity require costly iterative solvers at each time step. In this paper, a non-iterative method for obtaining approximate solutions of the SH equation which is based on the convex splitting idea is presented. By regularizing the cubic–quartic function in the energy for the SH equation and adding an extra linear stabilizing term, we arrive at a non-iterative convex splitting method, where the operator involved is linear and positive and has constant coefficients. We further prove the unconditional energy stability of the method. Numerical examples illustrating the accuracy, efficiency, and energy stability of the proposed method are provided.