Adaptive mesh refinement (AMR) algorithms

The basic adaptive refinment strategy used in AMRClaw Description and Detailed Contents is to refine on logically rectangular patches. A single Level 1 grid covers the entire domain (usually — if it is too large it may be split into multiple Level 1 grids). Some rectangular portions of this grid are covered by Level 2 grids refined by some refinement factor R in each direction (anisotropic refinement is now allowed too — see Specifying AMRClaw run-time parameters in Regions of each Level 2 grid may be covered by Level 3 grids, that are further refined (perhaps with a different refinement ratio). And so on.

For the hyperbolic solvers in Clawpack the time step is limited by the Courant number (see Section cfl), and so if the spatial resolution is refined by a factor of R in each direction then the time step will generally have to be reduced by a factor R as well.

The AMR code thus proceeds as follows:

  • In each time step on the Level 1 grid(s), the values in all grid cells (including those covered by finer grids) are advanced one time step. Before this time step is taken, ghost cells around the boundary of the full computational domain are filled based on the boundary conditions specified in the library routine bcNamr.f (where N is the number of space dimensions). Check the Makefile of an application to see where this file can be found.

  • After a step on the Level 1 grid, R time steps must be taken on each Level 2 grid, where R denotes the desired refinement ratio in time from Level 1 to Level 2.

    For each of these time step, ghost cell values must be filled in around all boundaries of each Level 2 grid. This procedure is defined below in Ghost cells and boundary conditions for AMR.

  • After taking R steps on Level 2 grids, values on the Level 1 grid are updated to be consistent with the Level 2 grids. Any cell on Level 1 that is covered by a Level 2 grid has its q value replaced by the average of all the Level 2 grid cells lying within this cell. This gives a cell average that should be a better approximation to the true cell average than the original value.

  • The updating just described can lead to a change in the total mass calculated on the Level 1 grid. In order to restore global conservation, it is necessary to do a conservation fix up. (To be described…)

This style of AMR is often called Berger-Oliger-Colella adaptive refinement, after the papers of Berger and Oliger [BergerOliger84] and [BergerColella89].

The Fortran code in $CLAW/amrclaw is based on code originally written by Marsha Berger for gas dynamics, and merged in Clawpack in the early days of Clawpack development by MJB and RJL. The algorithms used in AMRClaw are described more fully in [BergerLeVeque98].

Ghost cells and boundary conditions for AMR

Consider a Level k > 1 grid for which we need ghost cells all around the boundary at the start of each time step on this level. The same procedure is used at other levels.

  • Some Level k grids will be adjacent to other Level k grids and so any ghost cell that is equivalent to a Level k cell on some other grid has values copied from this this grid.

  • Some ghost cells will be in the interior of the full computational domain but in regions where there is no adjacent Level k grid. There will be a Level k-1 grid covering that region, however. In this case the ghost cells are obtained by space-time interpolation from values on the Level k-1 grid.

  • Some ghost cells will lie outside the full computational domain, where the boundary of the Level k grid lies along the boundary of the full domain. For these cells the subroutine bcNamr (where N is the number of space dimensions) is used to fill ghost cell values with the proper user-specified boundary conditions, unless periodic boundary conditions are specified (see below).

For many standard boundary conditions it is not necessary for the user to do anything beyond setting appropriate parameters in (see Specifying classic run-time parameters in Only if user-specified boundary conditions are specified is it necessary to modify the library routine bcNamr.f (after copying to your application directory so as not to damage the library version, and modifying the Makefile to point to the new version).

There some differences between the bcNamr.f routine and the bcN.f routine used for the single-grid classic Clawpack routines (which are found in $CLAW/classic/src/Nd/bcN.f). In particular, it is necessary to check whether a ghost cell actually lies outside the full computational domain and only set ghost cell values for those that do. It should be clear how to do this from the library version of the routine.

If periodic boundary conditions are specified, this is handled by the AMRClaw software along with all internal boundaries, rather than in bcNamr.f. With AMR it is not so easy to apply periodic boundary conditions as it is in the case of a single grid, since it is necessary to determine whether there is a grid at the same refinement level at the opposite side of the domain to copy ghost cell values from, and if so which grid and what index corresponds to the desired location.

Choosing and initializing finer grids

Every few time steps on the coarsest level it is generally necessary to revise modify the regions of refinement at all levels, for example to follow a propagating shock wave. This is done by

  1. Flagging cells that need refinement according to some criteria.

  2. Clustering the flagged cells into rectangular patches that will form the new set of grids at the next higher level.

  3. Creating the new grids and initializing the values of q and also any aux arrays for each new grid.

Clustering is done using and algorithm developed by Berger and Rigoutsis [BergerRigoutsis91] that finds a nonoverlapping set of rectangles that cover all flagged points and balances the following conflicting goals:

  • Cover as few points as possible that are not flagged, to reduce the number of grid cells that must be advanced in each time step.

  • Create as few new grids as possible, to minimize the overhead associated with filling ghost cells and doing the conservation fix-up around edges of grids.

A parameter cutoff can be specified (see Specifying AMRClaw run-time parameters in to control clustering. The algorithm will choose the grids in such a way that at least this fraction of all the grid points in all the new grids will be in cells that were flagged as needing refinement. Usually cutoff = 0.7 is used, so at least 70% of all grid cells in a computation are in regions where they are really needed.

Initializing the new grids at Level k+1 is done as follows:

  • At points where there was already a Level k+1 grid present, this value is copied over.

  • At points where there was not previously a Level k+1 grid, bilinear interpolation is performed based on the Level k grids.

Flagging cells for refinement

The user can control the criteria used for flagging cells for refinement.

See AMR refinement criteria for details.

For more details