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

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Abstract

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 \(A\tau ^2\frac{\partial (\varDelta ^2+1)u}{\partial t}\), is added to ensure the energy stability. The long-time unconditional energy stability of the algorithm is established rigorously. In addition, error estimates in \(\ell ^\infty (0,T;\ell ^2)\)-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.

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Funding

This work is funded by three Funds, namely Beijing Municipal Natural Science Foundation (No. 1192003), National Natural Science Foundation of China (No. 12371386), and National Natural Science Foundation of China (No. 12201021).

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  1. Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Ming Cui, Yiyi Niu & Zhen Xu

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  1. Ming Cui
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Cui, M., Niu, Y. & Xu, Z. A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term. J Sci Comput 99, 26 (2024). https://doi.org/10.1007/s10915-024-02490-9

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