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In order to demonstrate that the method is secondorder accurate, a problem
with an analytic quartic^{3}
solution is solved. The problem which is chosen is described in detail in
##more92 (##more92). The problem domain is a cube, with a random mesh obtained
by moving (in 3D) the interior points of an orthogonal mesh by a random
fraction of 20 of the internodal distance, in a random direction. There are
reflective boundaries on four sides, and vacuum boundaries on two opposite
sides. The properties are constant spatially and temporally, and there is a
spatiallyvarying source which is proportional to x^{2} in each cell. With
these conditions, the steadystate analytic answer is a quartic of the form
x, y, z = a + bx + cx^{4}.
New Method
Table 2:
Results from the SecondOrder Accuracy Test.
Problem Size (cells) 

Error Ratio 
5×5×5 
1.0248
×10^{2} 

10×10×10 
2.6190
×10^{3} 
3.91 
20×20×20 
6.6082
×10^{4} 
3.96 
40×40×40 
1.6530
×10^{4} 
4.00 
Orthogonal SevenPoint Solution
Problem Size (cells) 

Error Ratio 
5×5×5 
1.0202
×10^{2} 

10×10×10 
2.6205
×10^{3} 
3.92 
20×20×20 
6.5952
×10^{4} 
3.97 
40×40×40 
1.6515
×10^{4} 
3.99 
The results from running this problem for different mesh sizes are given in
Table 2. It can be seen that the error is reduced by a factor
of four each time the mesh spacing is reduced by a factor of two, which
indicates a secondorder accurate method. The results from the orthogonal
sevenpoint operator (on an orthogonal mesh) show similar behavior.
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Michael L. Hall