Adaptive mesh refinement in spacetime evolution

For a Cartesian three-dimensional hierarchy, halving the spacing increases the number of cells in a fixed physical region by eight. An explicit hyperbolic method often also halves the stable timestep there, adding another factor of two in work per fixed physical time. A uniform fine-grid evolution can therefore cost roughly sixteen times a coarse one under a simple equal-work-per-cell-update model. Memory growth and communication introduce different scaling factors.

A small refined patch can reduce that ideal cost, especially when subcycling lets fine levels take more steps than coarse levels. State whether covered coarse cells are still evolved, which ghost zones are counted, and whether a patch volume is a fraction of the physical domain or of an array's point count. Without those definitions, two apparently inconsistent cost estimates may simply be counting different work. Real performance also depends on parallel balance, interpolation and memory traffic.

A Courant-type restriction compares timestep with spacing divided by the largest characteristic speed that matters to the discretization. Gauge and constraint speeds can be larger than the physical light speeds in coordinates. A refinement ratio two naturally suggests two fine substeps per coarse step when the speed bounds are comparable. If those bounds vary strongly, the hierarchy needs a consistent scheduling rule rather than a purely geometric assumption.

Fine-grid boundary values typically require interpolation from coarser data at intermediate times. Restriction transfers fine information back to a coarse level, while regridding changes which points belong to each level. Each operation can introduce errors. High-order interior finite differences cannot by themselves repair a lower-order interface treatment that dominates the global solution. Conservation, constraint preservation and interpolation accuracy are related goals but are not interchangeable properties.

A wave crossing a refinement boundary can acquire phase error or spurious reflection, even when each uniform-grid scheme behaves well in isolation. Useful verification therefore includes waves at different incidence angles, moving refinement patches, and smooth gauge or exact-solution tests. Constraint norms should be examined near interfaces as well as over the full domain. An average norm can hide localized spikes that later propagate into a wave-extraction region.

Refinement criteria based on a particular variable are coordinate dependent and may miss another physically relevant scale. A coordinate horizon can be small while outgoing radiation still needs adequate resolution farther out. Error indicators, patch motion and extraction surfaces should therefore be included in a convergence study. The tiny cell-count example below isolates scheduling arithmetic; it supplies neither an error estimate nor evidence that a particular binary spacetime has been resolved.