We discuss how to combine exponential time differencing technique with multi-step method to develop higher order in time linear numerical scheme that are energy stable for certain gradient flows with the aid of a generalized viscous damping term. As an example, a stabilized third order in time accurate linear exponential time differencing (ETD) scheme for the epitaxial thin film growth model without slope selection is proposed and analyzed. An artificial stabilizing term \(A\tau ^3\frac{\partial \Delta ^3 u}{\partial t}\) is added to ensure energy stability, with ETD-based multi-step approximations and Fourier pseudo-spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long-time energy stability and an \(\ell ^{\infty }(0,T; \ell ^2)\) error analysis are provided, based on the energy method. In addition, a few numerical experiments are presented to demonstrate the energy decay and convergence rate.

中文翻译：

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梯度流的能量稳定高阶线性ETD多步法：在薄膜外延中的应用

我们讨论了如何将指数时差技术与多步方法相结合，以在广义线性阻尼项的帮助下，开发出对某些梯度流能量稳定的时间线性数值方案的高阶方法。例如，提出并分析了没有斜率选择的外延薄膜生长模型的时间精确线性指数时间微分（ETD）方案的稳定三阶。通过基于ETD的多步逼近和傅里叶伪谱方法，添加了人工稳定项\（A \ tau ^ 3 \ frac {\ partial \ Delta ^ 3 u} {\ partial t} \）以确保能量稳定性。分别应用于演化方程的时间积分和空间离散化。长期的能量稳定性和\（\ ell ^ {\ infty}（0，T; \ ell ^ 2）\）基于能量方法提供了误差分析。此外，提出了一些数值实验来证明能量衰减和收敛速度。