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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 spec...

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Bibliographic Details
Published in:Journal of scientific computing 2024-04, Vol.99 (1), p.26, Article 26
Main Authors: Cui, Ming, Niu, Yiyi, Xu, Zhen
Format: Article
Language:English
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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02490-9