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 τ 2 ∂ (Δ 2 + 1) u ∂ 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 ℓ ∞ (0 , T ; ℓ 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. [ABSTRACT FROM AUTHOR]