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

Table 1

The errors and order accuracy of the proposed method on Cartesian coordinates for various number of zones, 𝑁=2^𝑘 from 𝑘=5 to 9 for the 𝐷₂ (up table) and 𝐷_2,2 (down table) disks. This table shows that the accuracy is nearly second order.

𝑁𝑘	𝐸^1	𝐸^2	𝐸^∞	𝑁𝑘−1/𝑁𝑘	𝑂^1	𝑂^2	𝑂^∞
32	5.617𝐸−3	5.836𝐸−3	1.309𝐸−2
64	1.573𝐸−3	1.555𝐸−3	3.435𝐸−3	32/64	1.836	1.908	1.930
128	3.990𝐸−4	4.022𝐸−4	8.841𝐸−4	64/128	1.979	1.951	1.958
256	1.025𝐸−4	1.023𝐸−4	2.322𝐸−4	128/256	1.961	1.975	1.928
512	2.584𝐸−5	2.578𝐸−5	5.614𝐸−5	256/512	1.988	1.988	2.048
𝑁𝑘	𝐸^1	𝐸^2	𝐸^∞	𝑁𝑘−1/𝑁𝑘	𝑂^1	𝑂^2	𝑂^∞
32	8.008𝐸−3	7.331𝐸−3	1.822𝐸−2
64	2.129𝐸−3	2.046𝐸−3	5.179𝐸−3	32/64	1.911	1.841	1.815
128	5.344𝐸−4	5.415𝐸−4	1.374𝐸−3	64/128	1.994	1.918	1.914
256	1.396𝐸−4	1.374𝐸−4	3.468𝐸−4	128/256	1.937	1.979	1.986
512	3.516𝐸−5	3.462𝐸−5	8.734𝐸−5	256/512	1.989	1.989	1.989