Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is $dt=H^2/{\kappa }_{max}$, where ${\kappa }_{max}$ is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

许多多尺度问题具有高对比度，这表示为介质属性之间的非常大的比率。众所周知，这种对比会给多尺度方法和域分解方法的设计带来许多挑战。这些问题在空间多尺度和域分解方法的设计中得到了一定程度的分析。然而，其中一些问题仍然存在于时间相关问题中，因为对比度会影响时间尺度，特别是对于显式方法。例如，在抛物线方程中，时间步长为$d吨=H^2/{\kappa }_{米\mathrm{一种}X}$， 在哪里 ${\kappa }_{米\mathrm{一种}X}$是最大的扩散。在本文中，我们通过设计分裂算法在抛物线方程的背景下解决这个问题。所提出的分裂算法以隐式方式处理主要的多尺度模式，而其余的则以显式方式处理。这些算法的与对比度无关的稳定性需要特殊的多尺度空间设计，这是本文的主要目的。We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. 这可以节省计算量，因为显式方法中的时间步长会受到对比度的不利影响。我们讨论了所提出算法的一些理论方面。给出了数值结果。