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Solving Problems on a Grid
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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.
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Introduction to Adaptive Mesh Refinement
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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.
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 the 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.
For more examples of our use of AMR in astrophysical
fluid dynamics, see the research page.
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