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

Table 3

The top (bottom) table demonstrates the errors and order accuracy of the 𝑁-body simulations in Cartesian coordinates for various number of zones, 𝑁=2^𝑘 from 𝑘=5 to 9 for the 𝐷₂ (𝐷_2,2) disk. This table shows that the accuracy is only first order.

𝑁𝑘	𝐸^1	𝐸^2	𝐿^∞	𝑁𝑘−1/𝑁𝑘	𝑂^1	𝑂^2	𝑂^∞
32	1.199𝐸−1	1.532𝐸−1	3.362𝐸−1
64	6.069𝐸−2	8.141𝐸−2	1.818𝐸−1	32/64	0.9823	0.9121	0.8870
128	3.058𝐸−2	4.203𝐸−2	9.446𝐸−1	64/128	0.9889	0.9538	0.9446
256	1.535𝐸−2	2.136𝐸−2	4.813𝐸−1	128/256	0.9943	0.9765	0.9728
512	7.687𝐸−2	1.077𝐸−2	2.429𝐸−2	256/512	0.9977	0.9879	0.9866
𝑁𝑘	𝐸^1	𝐸^2	𝐿^∞	𝑁𝑘−1/𝑁𝑘	𝑂^1	𝑂^2	𝑂^∞
32	1.522𝐸−1	1.904𝐸−1	4.227𝐸−1
64	7.762𝐸−2	1.016𝐸−1	2.326𝐸−1	32/64	0.9714	0.9065	0.8617
128	3.916𝐸−2	5.252𝐸−2	1.218𝐸−1	64/128	0.9871	0.9514	0.9333
256	1.964𝐸−2	2.671𝐸−2	6.228𝐸−2	128/256	0.9949	0.9756	0.9676
512	9.839𝐸−3	1.347𝐸−2	3.149𝐸−2	256/512	0.9973	0.9875	0.9839