A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation
A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation
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DOI：
10.4208/cicp.oa-2017-0074
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发表时间：
2018
影响因子：
3.7
通讯作者：
Ruihan Guo;Yan Xu
中科院分区：
文献类型：
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作者：
Ruihan Guo;Yan Xu
本文建立了修正相场晶体(MPFC)方程的一阶和二阶凸分裂以及一阶线性能量稳定的全离散局部间断Galerkin(LDG)方法。其中，一阶线性格式基于不变能量二次化方法。MPFC方程是一个有阻尼的波动方程，为了保持能量的稳定性，需要引入一个伪能量，这都增加了与相场晶体(PFC)方程相比构造数值方法的难度。由于显式时间推进方法的严格时间步长限制，我们引入了一阶和二阶半隐式格式，并证明了它们是无条件能量稳定的。为了提高时间精度，采用了半隐式谱延迟校正(SDC)方法和一阶凸分裂格式相结合。MPFC方程的数值模拟往往需要很长的时间才能达到稳态，而自适应时间步长方法是必要的，也是至关重要的。隐式时间层上的格式是线性或非线性的，我们用多重网格求解器来求解。通过精度和长时间仿真的数值实验，验证了所提方法的能力和效率，以及自适应时间步进策略的有效性。AMS科目分类：65M60、35L75、35G25
In this paper, we will develop a first order and a second order convex splitting, and a first order linear energy stable fully discrete local discontinuous Galerkin (LDG) methods for the modified phase field crystal (MPFC) equation. In which, the first order linear scheme is based on the invariant energy quadratization approach. The MPFC equation is a damped wave equation, and to preserve an energy stability, it is necessary to introduce a pseudo energy, which all increase the difficulty of constructing numerical methods comparing with the phase field crystal (PFC) equation. Due to the severe time step restriction of explicit time marching methods, we introduce the first order and second order semi-implicit schemes, which are proved to be unconditionally energy stable. In order to improve the temporal accuracy, the semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting scheme is employed. Numerical simulations of the MPFC equation always need long time to reach steady state, and then adaptive time-stepping method is necessary and of paramount importance. The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver. Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods, and the effectiveness of the adaptive time-stepping strategy. AMS subject classifications: 65M60, 35L75, 35G25