... rings

The distinction between tori, discs and rings usually implies some hierarchy in the aspect ratio (thickness to radius ratio) and axis ratio (outer edge to inner edge ratio) of systems. Here, it is employed only to outline that the application range of the present method encompasses these three kinds of systems.  

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... time

It is worthwhile to revise or to clarify this kind of argument. The "large'' computing time i) is based on the assumption that the number of source points needed to perform the volume integral equals the number of field points, as imposed in finite-different methods; ii) does not account for the determination of boundary conditions; and iii) does not consider the accuracy (see Sect. 6.3).  

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... tri-dimensional

In the paper, "tri-dimensional'' only means that matter occupies a volume. Regarding the axial symmetry, the problem can however be viewed as a "bi-dimensional'' problem.  

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... kinds

For nominal accuracy, special functions must be computed from numerical libraries. It is possible to use approximations in the form of truncated k-expansions (e.g. Byrd & Freeman 1971) or from fitting functions (Abramowitz & Stegun 1964; Bonazzola & Schneider 1974). This enables a significant reduction of the execution time, but high accuracy can generally not be reached on field and potential values.  

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... surface

The additional symmetry with respect to the mid-plane Z=0 is introduced for convenience only. Two equations can be considered: z⁺(a) for the top surface and z^-(a) for the bottom. Here, we have set z⁺=-z^-=H(a).

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... value

There can be sub-domains where H=0 (for instance, to model discs with gaps). In such a case, the method must be applied to each sub-domain.

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... cylinder

By cylinder, we mean a topologically bi-dimensional object with zero radial extent. To be rigorous (i.e. for dimensional reasons), we should introduce a lateral surface density $\sigma$ to refer to the field due to a topologically bi-dimensional object, and perform the conversion $\sigma \rightarrow \rho da$ at the level of the radial integration.  

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... high-order

This scheme is 5-order accurate for irregular spacing, and 6-order accurate for regular spacing.  

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... model

Note that $\rho _1$ cannot represent any physical situation since the density depends on the field point. This is not crucial at this level since we need an exact solution.  

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... cylinder

For non-coincident cylinders, the Poisson kernel is regular suggesting that the splitting methods must be applied only to cases where k=1. This is not true. For field points located very close to the coincident cylinder, kernel components both exhibit a sharp peak at $z \approx Z$. Such peaks are expected and have finite amplitude, but it is difficult to reach high accuracy from a straightforward computation of Eq. (12). This difficulty can be circumvented by using the splitting method for nearby cylinders, as will be done here.  

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... components

Note that $\breve{\kappa}_R$ is discontinuous when going through the coincident cylinder, as soon as $\rho \ne 0$ at the field point (see for instance Fig. 7), but the amplitude of the discontinuity is known. Thus, each radial integration must numerically be separated into two parts, according to

$$\int_{a_{\rm in}}^{a_{\rm out}}{\breve{\kappa}_R{\rm d}a} \ri... ...\kappa}_Rda}+\int_{R}^{a_{\rm out}}{\breve{\kappa}_R{\rm d}a}.$$ (30)

Such an integral splitting must also be considered for $\breve{\kappa}_Z$ (as well as for $\breve{\kappa}_\Psi$; see Sect. 5).

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... discs

The diameter of discs in active galaxies reach typically the parsec, and for circumstellar discs, it can be as large as 10³ AU. In both cases, one finds $\frac{a_{\rm out}}{a_{\rm in}} \sim 10^5$. Further, there is evidence that such discs flare and have a decreasing density profile.  

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... technique

In Hachisu's paper, $\Psi$ was computed from the Green function with a systematic cutoff in the series, and the polar-type grid contained at least $129 \times 129$ points in the half-plane $z \ge 0$. Here, we do not take advantage of the mid-plane symmetry, points being distributed vertically in the range [-H,H].

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... method

To get a computer-independent quantity, $\eta$ should be multiplied by the "speed'' of the actual computer. Values reported in the following are obtained with a 1 GHz personal computer. 

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... experimentally

For this comparative study, we use the simple precision NETLB library HWSCYL(R3.10) routine which finds the solution $\Psi$ of the axi-symmetric Poisson equation from cyclic reduction based on a 5-point finite differences (Adams et al. 1980). Boundary conditions are computed from the double splitting method with maximum accuracy.

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