An energy stable method for the Swift-Hohenberg equation with quadratic-cubic nonlinearity

We present temporally first- and second-order accurate methods for the Swift-Hohenberg (SH) equation with quadratic-cubic nonlinearity. In order to handle the nonconvex, nonconcave term in the energy for the SH equation, we add an auxiliary term to make the combined term convex, which yields a convex-concave decomposition of the energy. As a result, the first- and second-order methods are unconditionally uniquely solvable and unconditionally stable with respect to the energy and pseudoenergy of the SH equation, respectively. And the Fourier spectral method is used for the spatial discretization. We present numerical examples showing the accuracy and energy stability of the proposed methods and the effect of the quadratic term in the SH equation on pattern formation.