Table 4: Same legend as for Table 3 but for the physical resolution ${\cal R}_R$ in the radial direction ( $\ell =h$ for all examples). $N_{\rm left}$ , $N_{\rm local}$ and $N_{\rm right}$ denote the number of cylinders populating the left-side, long-range, the local and right-side, long-range contributions respectively (see the Appendix C).
Finite-diff.

Radius This method methods

$a \rightarrow a_{\rm in}$ $\frac{a_{\rm in}}{L(N_{\rm left}-1)}\ln \frac{nha_{\rm in}}{(R-nh)(R-a_{\rm in})}$ $\frac{1}{N_{\rm left}-1}$

(inner edge)

$a \rightarrow R$ $\frac{h}{2qL} \sqrt{\frac{n^2}{1+n^2}} \left(\frac{1}{N_{\rm local}-1}\right)^{\frac{1}{2q}}$ $\frac{1}{N_{\rm local}-1}$

(field point)

examples

$(N_{\rm local},q)$

$(8,\frac{1}{2})$ $0.1 \frac{h}{L}$ 0.14

$(32,\frac{1}{2})$ $0.02 \frac{h}{L}$ 0.03

$(128,\frac{1}{2})$ $5 \times 10^{-3} \frac{h}{L}$ $8 \times 10^{-3}$

$(128,\frac{1}{4})$ $9 \times 10^{-5} \frac{h}{L}$ $8 \times 10^{-3}$

$a = R \pm \ell$ $\frac{hn(1+n^2)}{qL(N_{\rm local}-1)}$ $\frac{1}{N_{\rm local}-1}$

$a \rightarrow a_{\rm out}$ $\frac{a_{\rm out}}{L(N_{\rm right}-1)}\ln \frac{a_{\rm out}-R}{nh}$ $\frac{1}{N_{\rm right}-1}$

(outer edge)

**Table 4:** Same legend as for Table 3 but for the physical resolution ${\cal R}_R$ in the radial direction ( $\ell =h$ for all examples). $N_{\rm left}$ , $N_{\rm local}$ and $N_{\rm right}$ denote the number of cylinders populating the left-side, long-range, the local and right-side, long-range contributions respectively (see the Appendix C).
		Finite-diff.
Radius	This method	methods
$a \rightarrow a_{\rm in}$	$\frac{a_{\rm in}}{L(N_{\rm left}-1)}\ln \frac{nha_{\rm in}}{(R-nh)(R-a_{\rm in})}$	$\frac{1}{N_{\rm left}-1}$
(inner edge)
$a \rightarrow R$	$\frac{h}{2qL} \sqrt{\frac{n^2}{1+n^2}} \left(\frac{1}{N_{\rm local}-1}\right)^{\frac{1}{2q}}$	$\frac{1}{N_{\rm local}-1}$
(field point)
examples
$(N_{\rm local},q)$
$(8,\frac{1}{2})$	$0.1 \frac{h}{L}$	0.14
$(32,\frac{1}{2})$	$0.02 \frac{h}{L}$	0.03
$(128,\frac{1}{2})$	$5 \times 10^{-3} \frac{h}{L}$	$8 \times 10^{-3}$
$(128,\frac{1}{4})$	$9 \times 10^{-5} \frac{h}{L}$	$8 \times 10^{-3}$

$a = R \pm \ell$	$\frac{hn(1+n^2)}{qL(N_{\rm local}-1)}$	$\frac{1}{N_{\rm local}-1}$
$a \rightarrow a_{\rm out}$	$\frac{a_{\rm out}}{L(N_{\rm right}-1)}\ln \frac{a_{\rm out}-R}{nh}$	$\frac{1}{N_{\rm right}-1}$
(outer edge)

Source LaTeX | All tables | In the text

	1 Introduction
	2 The field: Theoretical grounds
	3 Vertical integration of the kernel associated with the field: "Density splitting method''
	4 Field values
	5 Numerical treatment of the self-gravitating potential: The "double splitting'' method
	6 Resolutions and efficiency
	7 Concluding remarks
	Appendix A: Values of $\vec{(\delta \rho)} \vec{\kappa}$ at the coincident point
	Appendix B: Density/secondary kernels test-pairs
	Appendix C: Input parameters for the radial sampling
	Appendix D: Analytical formula
	References