Homogeneous Solution Problem

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 steady-state 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 ``3-D Kershaw'' mesh. The basis of this mesh is a 2-D mesh that was described in ##kers81 (##kers81), which had constant spacing in one dimension and varied spacing in the second dimension. The 3-D 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 3-D.

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 node-centered data, rather than cell-centered 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 3-D Kershaw mesh problem, the same problem was run on an orthogonal mesh. A contour plot of the steady-state results is shown in Figure 8. The analytical solution is linear in x, and the method

Figure 8: Contour plot of the steady-state 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 seven-point operator in the case of an orthogonal mesh.

Figure 9 shows a contour plot of the steady-state results for the

Figure 9: Contour plot of the steady-state solution to the homogeneous problem on a 3-D Kershaw mesh.

3-D 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 steady-state solution to the homogeneous problem on a 3-D 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.