In this paper, we consider the classical wave equation with time-dependent, spatially multiscale coefficients. We propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with a backward Euler scheme in time. We show optimal convergence rates in space and time beyond the assumptions of spatial periodicity or scale separation of the coefficients. Further, we propose an adaptive update strategy for the time-dependent multiscale basis. Numerical experiments illustrate the theoretical results and showcase the practicability of the adaptive update strategy.

翻译：在本文中,我们考虑传统的波等式,以时间为依存、空间多尺度系数。我们建议一种完全独立的多尺度计算法,本着空间局部正方形分解的精神,及时采用后向尤勒办法。我们显示了超出空间周期或系数比例分离假设的空间和时间的最佳趋同率。此外,我们提出了基于时间的多尺度适应性更新战略。数字实验说明了理论结果,并展示了适应性更新战略的实用性。