Boundary conditions

Boundary conditions are imposed each time step by filling ghost cells adjacent to the edge of each grid patch. See Chapter 4 of [LeVeque-FVMHP] for more details.

Boundary conditions are set by the library routines:

• $CLAW/classic/src/Nd/bcN.f for the classic code (N = 1, 2, 3).

• $CLAW/amrclaw/src/Nd/bcNamr.f for the amrclaw code (N = 2, 3).

Several standard choices of boundary condition procedures are provided in these routines, and can be selected at each boundary by setting the input parameters bc_lower and bc_upper in each dimension (see Specifying classic run-time parameters in setrun.py) to one of the following:

• 1 or ‘extrap’ : extrapolation (non-reflecting outflow)

  In this case values from the grid cell adjacent to the boundary are copied into all ghost cells moving in the direction normal to the boundary. This gives a fairly good approximation to a non-reflecting or outgoing boundary condition that lets waves pass out of the boundary without reflection, particularly in one space dimension. In more than one direction this is not perfect for waves that hit the boundary at an oblique angle.

• 2 or ‘periodic’ : periodic boundary conditions

  In this case ghost cell values are set by copying from interior cells at the opposite boundary so that periodic boundary conditions are perfectly imposed. Normally periodic boundary conditions would be imposed by setting this value for both bc_lower and bc_upper in some dimension, but this is not required.

• 3 or ‘wall’ : solid wall boundary conditions are imposed for systems where the second component of q is the x velocity or momentum in one dimension (and where the third component of q is also the y velocity/momentum in more dimensions, etc.) This is true, for example, if the acoustics equations are solved with components q = (p, u, v) or shallow water equations with q = (h, hu, hv).

  In this case the normal velocity/momentum at a wall is reflected about the boundary (copied to a ghost cell from the cell equally far from the boundary on the interior side) while all other components are extrapolated.

  Reflecting boundary conditions can also often be used on a line of symmetry of a solution in order to reduce the computational domain to be only half of the physical domain.

  Note that this option does not work on a mapped grid… Add pointer to modified version

If none of the above boundary conditions are desired, the user can modify the subroutine bcN so that setting the appropriate component of bc_lower or bc_upper to 0 will execute code added by the user. In this case it is best to put the modified version of bcN.f in the application directory and modify the Makefile to point to the modified version. See User-defined boundary conditions below.

Boundary conditions for adaptive refinement

When AMR is used, any interior patch edges (not at a domain boundary) are filled automatically each time step, either by copying from adjacent patches at the same level or by interpolating (in both space and time) from coarser levels if needed.

The user must still specify boundary conditions at the edges of the computational domain. The same set of choices for standard boundary conditions as described above are implemented in the library routine bcNamr.f, and so specifying these boundary conditions requires no change to setrun.py when going from Classic Clawpack to AMRClaw. However, if special boundary conditions have been implemented in a custom version of bcN.f then the same procedure for setting ghost cells will have to be implemented in a custom version of bcNamr.f. This routine is slightly more complicated than the single-grid Classic version, since one must always check whether each ghost cell lies outside the computational domain (in which case the custom boundary condition procedure must be applied) or lies within the domain (in which case ghost cell values are automatically set by the AMR code and the user bcNamr routine should leave these values alone.

Boundary conditions for GeoClaw

For tsunami modeling or other geophysical flows over topography the computational domain has artificial boundaries that are placed sufficiently far from the region of interest that any flow or waves leaving the domain can be ignored and there should be no incoming waves. Extrapolation boundary conditions are then appropriate. If the ocean is truncated at some point then these generally have been found to give very small spurious reflection of outgoing tsunami waves. Extroplation boundary conditions can also be used on dry land (where the depth h is zero).

In some cases reflecting boundary conditions might be more appropriate, e.g. along the walls of a wave tank.

The library routine $CLAW/geoclaw/src/2d/shallow/bc2amr.f is modified from the amrclaw version only by extrapolating the depth at the boundaries into ghost cells.

Boundary conditions for clamshell grids on the sphere

In 2D AMRClaw and GeoClaw, an additional option is available for bc_lower and bc_upper that is implemented in the library routines:

• 4 or ‘sphere’ : sphere boundary conditions

  Must set bc_lower[0:2] = bc_upper[0:2] = 4 (i.e. at all 4 boundaries)

  These boundary conditions are similar to periodic boundary conditions, but for the clamshell grid introduced in [CalhounHelzelLeVeque] for solving problems on the sphere using a single logically rectangular grid. This is best envisioned by folding a rectangular piece of paper in half, gluing the edges together, and inflating to a sphere. See the animations on the website for the original paper See also [BergerCalhounHelzelLeVeque] for further examples.

User-defined boundary conditions

If none of the boundary conditions described above is suitable at one or more boundaries of the domain, then you will have to modify the library routine to implement the desired boundary condition. See Chapter 4 of [LeVeque-FVMHP] for hints on how to specify the ghost cell values each time step.

Suppose you need to specify different boundary conditions at the boundary xlower, for example. Then in setrun.py you should set bc_lower[0] = 0 and modify the library boundary condition routine to insert your desired boundary conditions at the point indicated in the code, where it says:

c     # user-specified boundary conditions go here in place of error output

in the section marked left boundary. The details of how this is done differ a bit between the classic and AMR codes and also depend on the number of space dimensions. Examine the way other boundary conditions are implemented and follow the model in your own code.

TODO: Give some hints on how things work in AMR code – must check which ghost cells extend outside the physical domain and which are filled automatically from adjacent grid patches or by interpolation from coarser patches if they are interior to the domain.