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 extend 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 unconditional stability of these algorithms require 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.
翻译:许多多尺度问题有着很大的反差,表现为介质属性之间的比例非常大。 反差在设计多尺度的方法和域分解方法时提出了许多挑战。 在设计空间多尺度和域分解方法时,分析这些问题到某些范围的问题。 但是,其中一些问题仍然在时间上随时间而存在问题,因为反差影响时间尺度,特别是明确的方法。例如,在抛物线方程中,时间步骤是$dt=H2\\\kappa ⁇ mauxmax}$($@kappa ⁇ max}$),这是最大的差异。在本文中,我们通过设计一个分离算法来解决这个问题。提议的分离算法以隐含的方式处理占支配地位的多尺度模式,而其余则以明确的方式处理。这些算法的无条件稳定需要特殊的多尺度空间设计,这是文件的主要目的。 我们表明,如果适当选择多尺度的空间,我们就可以在对比时实现无条件的稳定。 提出计算法的阶乘法, 将会产生反差的偏差。 我们提出的计算方法会影响。