Dynamically adaptive numerical techniques for solving differential equations provide a means for concentrating effort to computationally demanding regions. In the case of hierarchical AMR methods, this is achieved by tracking regions in the domain that require additional resolution and dynamically overlaying finer grids over these regions. Techniques based on AMR start with a base coarse grid with the lowest acceptable resolution that covers the entire computational domain. As the solution progresses, regions in the domain with high solution error and requiring additional resolution are flagged and finer grids are overlaid on the flagged regions of the coarse grid. Refinement proceeds recursively so that regions on the finer grid requiring higher resolution are similarly flagged, and even finer grids are overlaid on these regions. The resulting grid structure is a dynamic adaptive grid hierarchy. The figure below shows adaptive grid hierarchy for the classic Berger AMR formulation.

The Berger-Oliger Algorithm [(1984) Journal of Computational Physics, 53, 484-512]

• Suited for solving hyperbolic partial differential equations on structured computational domains
• Refinement factor is the same in both space and time: nested time marching on all levels (efficient)
• Leads naturally to a hierarchic tree of grids
• Fine grids have a separate identity from the underlying coarse grid; relation children to parent is maintained
• Uses a prolongation operator to populate Grid Function on a newly created grid from the underlying coarser grid
• Uses a restriction operator to inject newly computed function values of a fine grid to the underlying coarser grid