A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

In this article, we propose and analyze an energy stable, linear, second-order in time, exponential time differencing multi-step (ETD-MS) method for solving the Swift–Hohenberg equation with quadratic–cubic nonlinear term. The ETD-based explicit multi-step approximations and Fourier collocation spectral method are applied in time integration and spatial discretization of the corresponding equation, respectively. In particular, a second-order artificial stabilizing term, in the form of \(A\tau ^2\frac{\partial (\varDelta ^2+1)u}{\partial t}\) $A{\tau }^2\frac{\partial ({\Delta }^2+1)u}{\partial t}$ , is added to ensure the energy stability. The long-time unconditional energy stability of the algorithm is established rigorously. In addition, error estimates in \(\ell ^\infty (0,T;\ell ^2)\) ${\ell }^{\infty }(0,T;{\ell }^2)$ -norm are derived, with a careful estimate of the aliasing error. Numerical examples are carried out to verify the theoretical results. The long-time simulation demonstrates the stability and the efficiency of the numerical method.