Numerical solutions of hyperbolic partial differential equations(PDEs) are ubiquitous in science and engineering. Method of lines is a popular approach to discretize PDEs defined in spacetime, where space and time are discretized independently. When using explicit timesteppers on adaptive grids, the use of a global timestep-size dictated by the finest grid-spacing leads to inefficiencies in the coarser regions. Even though adaptive space discretizations are widely used in computational sciences, temporal adaptivity is less common due to its sophisticated nature. In this paper, we present highly scalable algorithms to enable local timestepping (LTS) for explicit timestepping schemes on fully adaptive octrees. We demonstrate the accuracy of our methods as well as the scalability of our framework across 16K cores in TACC's Frontera. We also present a speed up estimation model for LTS, which predicts the speedup compared to global timestepping (GTS) with an average of 0.1 relative error.

中文翻译：

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Octree网格上的可扩展本地时步

双曲型偏微分方程（PDE）的数值解在科学和工程中是普遍存在的。线法是将时空中定义的PDE离散化的一种流行方法，其中空间和时间是独立离散的。当在自适应网格上使用显式时间步长时，由最佳网格间距指示的全局时间步长的使用会导致较粗糙区域的效率低下。尽管自适应空间离散化在计算科学中得到了广泛使用，但时间自适应性由于其复杂的性质而很少见。在本文中，我们提出了高度可扩展的算法，以针对完全自适应八叉树上的显式时间步长方案启用本地时间步长（LTS）。我们在TACC的Frontera中展示了方法的准确性以及跨16K内核的框架的可伸缩性。