Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively managed, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider both the sequential solver and all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.

翻译：求解多尺度扩散问题通常计算代价高昂，这源于高对比度系数带来的空间和时间离散化挑战。为解决这一问题，本文提出了一种部分显式时间分裂格式。通过适当构造多尺度空间，空间多尺度特性得到有效处理，并且已证明时间步长与对比度无关。为提升模拟速度，我们针对多尺度流动问题提出了一种并行算法，该算法利用了部分显式时间分裂格式。其思想是首先使用粗时间步长演化部分显式系统，然后在每个粗时间区间上用精细传播子校正解，对此我们考虑了顺序求解器和全一次性求解器。该过程随后迭代执行直至收敛。我们分析了所提算法的稳定性和收敛性。数值实验表明，所提算法对于高对比度问题实现了高数值精度，并在较少的迭代次数内收敛。迭代次数随粗区间数量增加而保持稳定，从而通过并行处理显著提高了计算效率。