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The method will exactly preserve the homogeneous solution to the diffusion
equation, which is a linear solution, even if the mesh is highly skewed. To
show this, a problem with a linear solution has been solved. The problem domain
is a cube, with reflective boundaries on four sides and source and vacuum
boundary conditions on opposite sides. The physical properties are constant
spatially and temporally, and there are no removal or source terms. The
steadystate analytic solution is linear in one dimension.
The mesh for this problem is 20×20×20, which results in 8000
nodes, 6859 cells, and 28519 unknowns. The mesh spacing is one that has
been developed by the authors and is termed a ``3D Kershaw'' mesh. The
basis of this mesh is a 2D mesh that was described in ##kers81 (##kers81),
which had constant spacing in one dimension and varied spacing in the second
dimension. The 3D Kershaw mesh has constant spacing in one dimension and
varied spacing in the second and third dimensions, which creates a mesh that
is very skewed in 3D.
Figures 8, 9, and 10
were made using GMV, a program written by Frank Ortega at LANL
(##orte95 ##orte95). Unfortunately, this program is best suited for nodecentered
data, rather than cellcentered data. This problem was partially
circumvented by treating the cell centers as node centers in a dual mesh
(see Figure 7). Due to the skewed nature of the
Figure 7:
Differences between the actual mesh and the dual mesh that is
used by the plotting package, shown on one face of the cube.
Actual Mesh 
Dual Mesh 
(Cell Nodes) 
(Cell Centers) 



mesh, the cell centers are not flush with the edges of the cube and give the
illusion of a wavy cube boundary, which is not the case.
Before running the 3D Kershaw mesh problem, the same problem was run on an
orthogonal mesh. A contour plot of the steadystate results is shown in
Figure 8. The analytical solution is linear in x, and the method
Figure 8:
Contour plot of the steadystate solution to the homogeneous problem
on an orthogonal mesh.

reproduces this exactly, as is seen from the straight contour lines. This was
expected because the method reduces to the standard sevenpoint operator in the
case of an orthogonal mesh.
Figure 9 shows a contour plot of the steadystate results for the
Figure 9:
Contour plot of the steadystate solution to the homogeneous problem
on a 3D Kershaw mesh.

3D Kershaw mesh. The contour lines remain linear, even though the mesh is
highly skewed. A random cutplane through the cube (see Figure 10) shows that
Figure 10:
Contour plot on a random cutplane of the steadystate solution to
the homogeneous problem on a 3D Kershaw mesh.

the contour lines are linear on the interior of the cube and highlights the
skewed nature of the mesh. Indeed, calculations exhibit linearity of the
solution down to machine precision.
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Up: Results
Previous: SecondOrder Demonstration
Michael L. Hall