Introduction to Block-Structured Adaptive Mesh Refinement.

In many problems in partial differential equations, one is confronted with problems having multiple length scales  and strong spatial localizations. Examples include nonlinear systems of hyperbolic partial differential equations containing complex combinations of discontinuities and smooth flow. Also included are combustion problems in which, at any given instant, burning is taking place in a small subset of the problem domain and problems with complex geometries in which localized geometric features can generate strong, localized solution gradients. Finite difference calculation using block-structured adaptive mesh refinement (AMR) is a powerful tool for computing solutions to partial differential equations involving such multiple scales. In this approach, the underlying problem domain is discretized using a rectangular grid and a solution is computed on that grid. Regions requiring additional resolution are identified by computing some local measure of the original error and covered by a disjoint union of rectangles in the domain, which are then refined by some integer factor. The solution is then computed on the composite grid.  This process may be applied recursively, and for time-dependent problems, the error estimation and regridding can be integrated with the time evolution and refinement applied in time as well as in space.