The epitaxial thin film growth model without slope selection is the $L^2$-gradient flow of energy with a logarithmic potential in terms of the gradient of a height function. A challenge to numerically solving the model is how to treat the nonlinear term to preserve energy stability without compromising accuracy and efficiency. To resolve this problem, we present a high-order energy stable scheme by placing the linear and nonlinear terms in the convex and concave parts, respectively, and employing the specially designed implicit–explicit Runge–Kutta method. As a result, our scheme is linear, high-order accurate in time, and unconditionally energy stable. We show analytically that the scheme is unconditionally uniquely solvable and energy stable. Numerical experiments are presented to demonstrate the accuracy, efficiency, and energy stability of the proposed scheme.

没有斜率选择的外延薄膜生长模型是 ${\mathrm{大号}}^2$高度函数的梯度具有对数势的能量梯度流。数值求解模型的一个挑战是如何处理非线性项以保持能量稳定性而又不影响准确性和效率。为了解决这个问题，我们提出了一种高阶能量稳定方案，方法是分别将线性和非线性项放在凸部和凹部中，并采用经过特殊设计的隐式-显式Runge-Kutta方法。结果，我们的方案是线性的，时间上的高阶准确度以及无条件的能量稳定性。我们通过分析表明，该方案是无条件唯一可解的且能量稳定的。数值实验表明了该方案的准确性，效率和能量稳定性。