We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficient linear numerical scheme with the help of an artificial regularization term of the form $A\tau^k\frac{\partial}{\partial t}\mathcal{L}^{p(k)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $\tau$ is the uniform time step-size. The exponent $p(k)$ is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $\mathcal{L}$ together with the desired temporal order of accuracy $k$. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results.

中文翻译：

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具有Lipschitz非线性的梯度流的能量稳定任意阶ETD-MS方法

我们提出了一种方法，该方法可为具有适当Lipschitz连续非线性的一类梯度流构建时间精确的数值方案的有效高阶。该策略有几个要素：指数时差（ETD），多步（MS）方法，稳定化思想和插值技术。在人工有效的正则项形式$ A \ tau ^ k \ frac {\ partial} {\ partial t} \的帮助下，将它们合成为有效的线性数值方案中的通用$ k ^ {th} $阶。 mathcal {L} ^ {p（k）} u $其中$ \ mathcal {L} $是流的正定线性部分，$ \ tau $是统一时间步长。指数$ p（k）$由与$ \ mathcal {L} $相关的Lipschitz非线性项的强度以及所需的精度时间$ k $明确确定。为了验证我们的理论分析，采用四阶ETD-MS时间离散化和傅里叶伪谱空间离散化研究了没有斜率选择模型的薄膜外延生长。我们关于收敛性和能量稳定性的数值结果与我们的理论结果一致。