Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed computations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.
翻译:用于建模和模拟的许多应用中都使用了时间分数的 PDE 。 许多这些应用是多尺度的,在介质特性方面差异很大。 它需要非常小的时间级大小才能进行详细的计算。 另一方面,在有小的空间网格的情况下,需要非常小的时间级大小才能使用明确的方法。 清晰的方法有许多我们在文件中讨论过的优点。 我们在本文件中为时间分数的 PDE 提出了一个部分明确的方法。 这种方法解决了粗糙的计算网格上的问题, 远大于空间的异质, 只需要略小的自由度才能得到暗中处理。 通过建造适当的空间和仔细的稳定性分析, 我们可以证明时间级的选定并不取决于差异或尺度, 因为粗微的网格大小。 因此, 人们可以在明确的方法中使用更大的时间级大小。 我们提出的方法的稳定性理论和我们的数字结果证实了稳定性结论, 并展示了方法的性能。