In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of contrast in high-contrast cases). By treating the component containing fast dynamics (or dependent on the contrast) implicitly and the component containing slow dynamics (or independent of the contrast) explicitly, we construct partially explicit time discretization schemes, which can reduce computational cost. The derived stability conditions are contrast-independent, provided the continua are chosen appropriately. Additionally, we discuss possible methods to obtain an optimized decomposition of the solution space, which relaxes the stability conditions while enhancing computational efficiency. A Rayleigh quotient problem in tensor form is formulated, and simplifications are achieved under certain assumptions. Finally, we present numerical results for various coefficient fields and different continua to validate our proposed approach. It can be observed that the multicontinuum splitting schemes enjoy high accuracy and efficiency.

翻译：本文针对具有高对比度系数的抛物方程，提出了多尺度问题的多连续介质分裂格式。基于多连续介质均匀化框架，我们引入了空间光滑的宏观变量，并将多连续介质解空间分解为两个分量，以有效分离不同速度的动力学行为（或高对比度情形下对比度的影响）。通过隐式处理包含快速动力学（或依赖于对比度）的分量，并显式处理包含慢速动力学（或独立于对比度）的分量，我们构建了部分显式时间离散格式，从而降低计算成本。在恰当选择连续介质的前提下，所得稳定性条件与对比度无关。此外，我们讨论了获得解空间优化分解的可能方法，该方法可在提升计算效率的同时放宽稳定性条件。我们建立了张量形式的瑞利商问题，并在特定假设下实现了形式简化。最后，我们展示了针对不同系数场与连续介质配置的数值计算结果，以验证所提方法的有效性。结果表明，多连续介质分裂格式具有较高的精度与计算效率。