A Numerical Convolution Representation of Potential for a Disk Surface Density in 3D

Table 4

The errors of the potential on the 𝑧-axis and order accuracy of the proposed method on Cartesian coordinates for various zones, 𝑁=2^𝑘 from 𝑘=5 to 9 for the 𝐷₂ disk. This table shows that the accuracy is nearly second order.

𝑁𝑘	𝐸^1(𝑧)	𝐸^2(𝑧)	𝐸^∞(𝑧)	𝑁𝑘−1/𝑁𝑘	𝑂^1	𝑂^2	𝑂^∞
32	1.175𝐸−2	1.129𝐸−2	1.518𝐸−2
64	2.998𝐸−3	3.093𝐸−3	5.230𝐸−3	32/64	1.971	1.868	1.537
128	7.672𝐸−4	7.963𝐸−4	1.541𝐸−4	64/128	1.966	1.958	1.958
256	1.910𝐸−4	2.000𝐸−4	4.171𝐸−4	128/256	2.006	1.993	1.993
512	4.775𝐸−5	5.006𝐸−5	1.085𝐸−4	256/512	2.000	1.988	1.998