In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a “good” decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in [15] and adopt in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.

在本文中，我们研究了多尺度问题的时间分割算法。确切的细网格空间问题通常需要一定程度的自由度降低。设计多尺度算法可在粗糙的网格上表示精细尺度的细节，从而减小问题的大小。在解决与时间有关的问题时，可以利用解决方案的多尺度分解，并通过解决较小维度的问题来进行时间拆分，这一点已在本文中进行了研究。在提出的方法中，我们考虑基于各种低维空间近似的时间分割。因为多尺度空间分裂对解空间提供了“良好”的分解，所以可以实现有效的隐式-显式时间离散化。我们在[15]中提出了最近发展的理论结果，并在本文中用于多尺度问题。数值结果表明了所提分裂算法的有效性。