Adaptive Mesh Refinement
Dynamically concentrating grid resolution where the solution demands it, following features as they move and evolve.
Resolution that follows the physics
Many problems have features that occupy a small fraction of the domain but require fine resolution: shocks, steep gradients, reaction fronts, or thin current layers. Adaptive mesh refinement (AMR) places fine grids only where they are needed and coarse grids elsewhere, and moves the fine regions as the features evolve. This can reduce the number of unknowns by orders of magnitude compared to a uniformly fine grid, while capturing the same detail.
Block-structured and cell-based AMR
Two main styles exist. Block-structured AMR (the Berger-Oliger approach) overlays rectangular patches of finer grid on regions flagged for refinement, organized in a hierarchy of levels, each finer level nested within the coarser. Cell-based (tree-based) AMR refines individual cells by subdivision, tracked with a quadtree or octree. Block-structured AMR favors efficient structured-grid kernels; tree-based AMR offers more localized, flexible refinement.
- Refinement criteria: gradient magnitude, estimated error, or feature detection
- Refinement ratio: how much finer each level is than its parent (often a factor of two or four)
- Regridding: periodically re-evaluating where refinement is needed as the solution evolves
- Load balancing: redistributing the changing grid across processors for parallel efficiency
The bookkeeping challenges
AMR introduces interfaces between coarse and fine regions where fluxes must be matched to preserve conservation, and where interpolation fills fine-grid ghost cells from coarse data. Time stepping can be sub-cycled, advancing fine levels with smaller steps to respect the CFL condition on the finer grid. Getting these couplings right is what makes AMR correct rather than merely fast.
Use in simulation
AMR is essential for multi-scale problems in astrophysics, combustion, and plasma physics, where the range of scales is too large for a uniform grid. It is the spatial counterpart of the temporal step-size adaptivity discussed under error estimation and adaptivity, and it relies on the same principle: spend degrees of freedom where the error is largest.