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.
翻译：我们提出了一个方法,用适当的利普西茨连续非线性来为某一类梯度流制定高效的高顺序、准确的及时数字计划。战略有若干要素:指数时间差异(ETD)、多步骤(MS)方法、稳定理念和内插技术。它们被合成,以开发一个通用的美元(k ⁇ th})和时间效率高的线性数字计划,同时使用一个人工正规化术语($A\Tã ⁇ k\frac),形式为$A\t ⁇ k\k\fracxde-party t ⁇ mathcal{L ⁇ p(k)}u$,其中美元是流动的确定线性正数部分,美元是统一的时间级(MS)方法、多步制(MS)方法、稳定度和内插值技术。在利普西茨非线性术语相对于$\mathcal{L}的强度和预期的准确时间顺序。为了验证我们的理论分析,对不使用斜度选择模型的薄片片类增长,将用第四级ETD-MS的离层结果与我们空间的离光谱化数据结果加以研究。
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