Abstract
In this article, we propose and analyze a global-in-time energy stable third-order accurate in time exponential time differencing multi-step (ETD-MS) numerical scheme for the Landau-Brazovskii (LB) equation, which is a generic model to describe weakly first-order order-disorder phase transition for isotropic systems. The ETD-based explicit multi-step temporal discretization is combined with the Fourier collocation spectral approximation in the numerical design, and a third-order stabilizing term is added to ensure energy stability. In turn, a global-in-time energy estimate for the ETD-MS numerical scheme is established. In more details, an a-priori assumption at the previous time steps, combined with an H2 estimate of the numerical solution at the next time step, is the key point in the analysis. Such a global-in-time energy estimate is the first such result of a third-order stabilized numerical scheme for a gradient flow. Moreover, an ℓ∞(0,T;ℓ2)∩ℓ∞(0,T;Hh2) error estimate is derived, with the help of careful Fourier eigenvalue analysis. Numerical experiments are conducted to verify the theoretical results, and several structures in block copolymers are effectively simulated for a long time in two- and three-dimensional spaces, which demonstrate the accuracy and high efficiency of the proposed numerical scheme.
•A fully discrete ETD-MS method with third-order, global-in-time energy stability is proposed and analyzed for the Landau-Brazovskii model, which is a generic model for describing weakly first-order order-disorder phase transitions for isotropic systems. Here, the global-in-time energy stability estimate for the numerical scheme is established for any final time without Lipschitz assumption and is the first such result of a third-order stabilized numerical scheme for a gradient flow.•An a-priori assumption at the previous time steps, combined with an H2 estimate of the numerical solution at the next time step, is the key point in the analysis. Such an H2 estimate recovers the uniform-in-time ‖⋅‖∞ bound of the numerical solution at the next time step, and then the value of the stabilization parameter can be theoretically justified. The uniform bound for the various global operators involved in the algorithm are derived with the help of careful Fourier eigenvalue estimates. The much sharper uniform-in-time ‖⋅‖Hh2 and ‖⋅‖∞ bound for the numerical solution at any time are obtained with the help of a theoretical justification of the energy stability analysis.•A continuous version of Dupont-Douglas type regularization term Aτ3∂ϕ∂t is added to the physical model to ensure the energy stability, this type of regularization term can be generalized to the case of higher order without any difficulty.•O(τ3)-order convergence analysis in ℓ∞(0,T;ℓ2)∩ℓ∞(0,T;Hh2) norm are provided rigorously.•This method can be effectively used for long-time simulations, and is expected to be available for many other higher order numerical schemes to the gradient flow. In particular, two dimensional (2D) and 3D periodic crystal structures in block copolymers are effectively obtained under a long-time simulation.