Specifying the location of the boundary points

CoordBase also provides a way for specifying the discretisation of the boundary that is independent of the actual boundary thorns. This defines the locations of the boundary points and thus the extent of the computational grid.

Each face of the grid is specified independently. The specification does not depend on the resolution, so that it need not be adapted when the resolution is changed. Figure A3.1 shows a domain with several interior grid points and one boundary point.

Figure A3.1: A domain (rectangle) with three interior points (empty circles) and one boundary point (full circle).

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While the physical boundary has a width of zero, the computational grid can have more than one boundary points. This boundary size is defined by the integer parameter boundary_size_x_lower for the lower $x$ boundary (and similarly for the other boundaries). Figure A3.2 shows an example where there are two boundary points. When it is necessary to increase the number of boundary points (e.g., to accommodate a larger stencil), then this is the only parameter that needs to be changed.

Figure A3.2: A domain (rectangle) with three interior points (empty circles) and two boundary points (full circles). This is an exterior boundary, i.e. the boundary points are located outside of the physical domain.

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Depending on the physical setup -- and depending on the personal taste -- the boundary points should be located either inside or outside of the physical boundary. The boolean parameter boundary_internal_x_lower specifies whether the boundary points extend inwards at the lower $x$ face. Figure A3.3 shows the same example, but with internal boundary points.

Figure A3.3: A domain (rectangle) with two interior points (empty circles) and two boundary points (full circles). This is an interior boundary, i.e. the boundary points are located inside of the physical domain.

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Depending on the physical setup -- and depending on the personal taste -- the boundary points should either be staggered about the physical boundary, or the last boundary point should be located exactly on the physical boundary. This is specified by the boolean parameter boundary_staggered_x_lower. Figures A3.4 and A3.5 show exampled of external and internal staggered boundary points.

Figure A3.4: A domain (rectangle) with three interior points (empty circles) and two boundary points (full circles) which are staggered about the physical boundary. (This is an exterior boundary.)

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Figure A3.5: A domain (rectangle) with two interior points (empty circles) and two boundary points (full circles) which are staggered about the physical boundary. (This is an interior boundary.)

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Finally, the integer parameter boundary_shiftout_x_lower can be used to shift the boundary points outwards (or inwards with negative values) by multiples of the grid spacing. Figure A3.6 shows an example of an exterior, non-staggered boundary with a shiftout of one.

Figure A3.6: A domain (rectangle) with four interior points (empty circles) and two boundary points (full circles) which are shifted outwards by one grid point. (This is an exterior, non-staggered boundary.)

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The following table gives examples for common situations:

Boundary condition	internal?	staggered?	shiftout	example
reflection symmetry, not staggered	no	no	1	figure A3.6
reflection symmetry, staggered	no	yes	0	figure A3.4
periodicity, closed boundary	no	no	1	figure A3.6
periodicity, open boundary	no	no	0	figure A3.2
periodicity, staggered boundary	no	yes	0	figure A3.4

For other boundary conditions such as Dirichlet or Robin, one can choose these parameters freely.