Adaptive Mesh Refinement
Adaptive mesh refinement concentrates grid resolution where the solution varies rapidly and coarsens it where the solution is smooth.
Resolution where it is needed
Many problems have features (shocks, boundary layers, current sheets, fronts) that occupy a small fraction of the domain but demand fine resolution, while the rest is smooth. A uniform fine grid wastes enormous effort on the smooth regions. Adaptive mesh refinement (AMR) dynamically adds resolution where an error indicator flags rapid variation and removes it where the solution is well resolved, tracking features as they move.
AMR can reduce the number of degrees of freedom by orders of magnitude for problems with localized structure, making otherwise intractable resolutions feasible.
Block-structured and tree-based approaches
Two families dominate. Block-structured AMR overlays progressively finer rectangular patches on a coarse base grid, organized in nested levels; it keeps efficient structured-grid kernels but needs careful flux matching at level boundaries. Cell-based or tree-based AMR refines individual cells recursively (quadtrees in 2D, octrees in 3D), giving fine-grained adaptivity at the cost of more complex data structures.
Refinement criteria and conservation
Refinement is driven by error estimators, gradient or curvature indicators, or physics-based tags. At coarse-fine interfaces, fluxes must be corrected so the scheme remains conservative; time-step subcycling advances fine levels with smaller steps to respect stability limits without penalizing the coarse grid.
- Error indicators or gradient sensors flag cells to refine
- Block-structured (patches) vs tree-based (cells) organization
- Flux correction preserves conservation across levels
- Subcycling in time matches step size to local resolution
AMR is widely used in magnetohydrodynamic and transport simulations where thin, moving structures must be captured accurately without paying for uniform fine resolution across the whole device volume.