AMR

Solving Problems on a Grid

Central to any direct computational method is the
manner in which it discretizes the continuous domain
of interest into a grid of many individual
elements. This grid may be *static*,
established once and for all at the beginning of
the computation, or it may be *dynamic*,
tracking the features of the result as the
computation progresses. If the computation has features
which one wants to track which are much smaller than
the overall scale of the problem, and which
move in time, then one must either include
many more static grids to cover the region of
interest, or adopt a dynamic scheme. The
advantages of a dynamic gridding scheme are:

1) Increased computational savings over a static grid approach.

2) Increased storage savings over a static grid approach.

3) Complete control of grid resolution, compared to the fixed resolution of a static grid approach.

Introduction to Adaptive Mesh Refinement

In a series of papers, Berger, Oliger, and Colella
developed an algorithm for dynamic gridding called
local adaptive mesh refinement. The algorithm begins
with the entire computational domain covered with a
coarsely resolved base-level regular Cartesian grid.
As the calculation progresses, individual grid cells
are tagged for refinement, using a criterion that
can either be user-supplied (i.e. mass per cell
remains constant, hence higher density regions are
more highly resolved) or based on Richardson
extrapolation.*physical
locations* and *times*
where they are required. This lets us solve problems
that are completely intractable on a uniform grid;
for example,we have used the code to model the
filamentary nature of isothermal collapse down to an
effective resolution of 131,072 cells per initial
cloud radius, corresponding to a resolution of 10^15
cells on a uniform grid.

All tagged cells are then refined, meaning that a finer grid is overlayed on the coarse one. After refinement, individual grid patches on a single fixed level of refinement are passed off to an integrator which advances those cells in time. Finally, a correction procedure is implemented to correct the transfer along coarse-fine grid interfaces, to insure that the amount of any conserved quantity leaving one cells exactly balances the amount enter the bordering cell. If at some point the level of refinement in a cell is greater than required, the high resolution grid may be removed and replaced with a coarser one.

The image to the left shows the grid structure of an AMR calculation of a shock impacting an inclined slope. Each of the boxes is a grid; the more boxes it is nested within, the higher the level of refinments. It is adapted From Colella and Crutchfield (1994).

As the image shows, the algorithm uses high resolution grids only at theFor more examples of our use of AMR in astrophysical fluid dynamics, see the research page.