When Is Local Time Stepping Worthwhile? A Cost Analysis for Nested Adaptive Mesh Refinement

Abstract

In deep hierarchies of adaptive mesh refinement, the update work that subcycling avoids is set mainly by the dimension of what is refined, not of the space. Subcycling (local time stepping) does less work than global stepping, advancing each level with the finest step, provided its extra coordination cost stays below an overhead budget $β=1-1/\Sid$ as a fraction of the global-stepping work. Here $\Sid$ is the ideal speedup that the cell counts give for equal update costs. When each level follows a feature of box-counting dimension $δ$ and the counts grow geometrically, the budget of a code that advances every cell tends to $(r-1)/(r^{δ+1}-1)$ for refinement ratio r. A point leaves a budget near one, while a surface in three dimensions leaves 1/7 at r=2 and 1/21 at r=4, values implied by the planar estimates of Dursi and Zingale. Published comparisons in discontinuous Galerkin and octree codes with local time stepping imply overheads of 0.1 to 5 percent of the work, below 1/7 and comparable with 1/21. In paired runs of one AMReX code on one core, a cost model fixed in advance and fitted on points and filled regions predicted the speedups of curves and surfaces with a median error of 7 percent (22 percent on the saving); post hoc, the update counts alone did as well except for points. The overhead, as defined here, was negative in all 17 configurations, but when computed from speedups like the published values, it ranged from -0.11 to 0.19, and no configuration came near break-even. When an update costs the same under both schedules, one profiled run can show that subcycling pays.