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## Second-Order Demonstration

In order to demonstrate that the method is second-order accurate, a problem with an analytic quartic3 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 3-D) the interior points of an orthogonal mesh by a random fraction of 20 of the inter-nodal 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 spatially-varying source which is proportional to x2 in each cell. With these conditions, the steady-state analytic answer is a quartic of the form x, y, z = a + bx + cx4.

New Method
 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 Seven-Point 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 second-order accurate method. The results from the orthogonal seven-point operator (on an orthogonal mesh) show similar behavior.

Next: Homogeneous Solution Problem Up: Results Previous: Results
Michael L. Hall