A fully discrete stable discontinuous Galerkin method for the thin film epitaxy problem without slope selection
A fully discrete stable discontinuous Galerkin method for the thin film epitaxy problem without slope selection
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DOI：
10.1016/j.jcp.2014.09.025
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Yinhua Xia
中科院分区：
文献类型：
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作者：
Yinhua Xia
在本文中，我们发展了一个能量稳定的全离散间断伽辽金（DG）有限元方法的薄膜外延问题。基于直线法，首先构造并证明了空间半离散DG格式的能量稳定性。为了避免显式时间积分方法严格的时间步长限制，采用一阶凸分裂方法得到无条件稳定的全离散DG方法，该方法是求解该非线性问题的线性隐式格式。证明了全离散凸分裂DG格式的能量稳定性。为了提高时间精度，采用了光谱延迟校正（SDC）方法，在时间和空间上都达到了高阶精度。结合凸分裂方法，SDC方法在数值试验中具有线性可解、高阶精度和稳定性。这些优点确保了所得到的完全离散DG方案对于执行薄膜外延模型的长时间模拟是有效的。数值实验的精度和长时间模拟表明了该方法的能力和效率。
In this paper, we develop an energy stable fully discrete discontinuous Galerkin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.