next up previous external

Next: 7 Tables Up: 6 Numerical Tests Previous: 6.1 Multipole Expansion

6.2 Exact Expansion in spherical harmonics

In Tabs. 6-8 we plot for various combinations of r and tex2html_wrap_inline2520 the partial sums

equation1126

of the exact expansion (45) and the transformed values

equation1134

with tex2html_wrap_inline2840 , tex2html_wrap_inline2842 and tex2html_wrap_inline2844 corresponding to the recursion of the Legendre polynomials tex2html_wrap_inline2846 . As in the case of the multipole expansion the values are rounded to 16 decimal digits. The definition Eq. (37) of the tex2html_wrap_inline2584 transformation is used. Also, we plot for both sequences tex2html_wrap_inline2850 and tex2html_wrap_inline2852 the corresponding number of exact digits. As before, this number is defined as the negative decadic logarithm of the relative error as indicated also in the table headers.

Comparison of Tabs. 2 with 6, 3 with 7 and 4 with 8, that have been computed for the same point, respectively, reveals that the converged extrapolated values differ considerably. This means that in this way the difference tex2html_wrap_inline2466 can be evaluated.

Let us remark that also in this case the additional numerical effort for the extrapolation is very low in comparison to the evaluation of the coefficients of the orthogonal expansion.

In summary, it can be stated that the acceleration of the expansion in Legendre polynomials via the tex2html_wrap_inline2584 transformation leads to pronounced error reduction, as well in the case of the multipole expansion as well as in the case of the exact computation of the electrostatic potential. Put another way, for achieving a certain accuracy, considerably less multipole moments or expansion coefficients, respectively, are necessary if a problem adapted extrapolation method as the tex2html_wrap_inline2584 transformation is used.


next up previous external
Next: 7 Tables Up: 6 Numerical Tests Previous: 6.1 Multipole Expansion

Herbert H. H. Homeier (herbert.homeier@na-net.ornl.gov)