Journal of Applied Mathematics

Copyright © 2014 Chien-Chang Yen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This study presents a numerical convolution representation of a potential induced from a disk of surface density, which has often been investigated in the center region of galaxies. The advantage of this representation is to release the softening length for the -body method and artificial boundary conditions for the spectral methods. With the help of fast Fourier transform, the computational complexity is only , where is the number of zones in one dimension. Numerical results show an almost second order of accuracy on a Cartesian coordinate system. A comparison study also demonstrates that this method can calculate the potential for disk surface density based on the uniform grids.

1. Introduction

Let us consider the Poisson equation with singular disk surface density [1–3] where,,, and are position, gravitational constant, Dirac symbol, and density, respectively. Hereafter, we setwithout loss of generality.

One possible method of solving (1) is to discretize the partial differential equation (1) using the finite difference method [4–6]. This discretization is where,, andfor, based on the uniform mesh gridswith the mesh sizes,, andin,, anddirections, respectively. In this approach, the values of the potential on imposed boundaries should be calculated and a fully 3D calculation must be performed. Any numerical solutions of this partial differential problem involveunknowns, whereis the number of zones in one dimension. The linear complexity of this approach is at least, for example, [7]. It seems unfeasible that a thin gaseous disk problem can directly solve the partial differential equation.

The other approach for solving (1) uses the integral form. The solution to (1) can be represented in terms of the fundamental solution,, where as The difficulties encountered in the numerical approach for solving (1) or (5) are related to the extent of the domain. The spectral method uses trigonometric bases functions and the artificial periodic boundary conditions [8]. However, the potential is not periodic in reality. A direct calculation using the-body method [1] has the total amount of complexity based on the numberof mesh zones. For the-body method based on a uniform grid and using the FFT technique, the numerical complexity can be reduced fromto. In other words, this produces a fast algorithm of linear complexity. Unfortunately, this method has a drawback, which is to introduce the softening parameter to overcome the singular integration wherecontains the origin.

The proposed method does not require the artificial periodic boundary condition and softening parameter and further improves the accuracy of the-body method. This study uses the uniform and logarithmic grid generations in Cartesian and polar coordinates, respectively. The other treatment is linear approximation for the surface density on the cell. This ensures that the proposed method is of second order in Cartesian coordinates. This study uses two examples, adisk [9] and its variation, to show numerically that the method achieves second-order accuracy. A comparison study also demonstrates that the proposed method is suitable for potentials on a thin disk.

Before the end of the section, another approach for an improvement of-body method is the fast multipole methods (FMM) [1, 10] which decomposes the region into local interactions and far-field interactions for every position. The FMM is efficient for fully three-dimensional calculations under the assumption that the far field is smooth. The FFM also relies on the ability to manipulate local multipole expansions for every box in the tree hierarchy [11–13]. The proposed presentation based on the uniform grids is without the assumption of smoothness in the far-field region and does not require extra efforts on the tree hierarchy.

This paper is organized as follows. Sections 2.1 and 2.2 present the numerical methods for Cartesian and spherical coordinates, respectively. Section 4.1 presents the order of accuracy of the proposed methods as verified by a family of finite disks[9]. Section 4.2 presents a comparison with the-body calculation. Finally, we conclude this study in Section 5.

2. Methods

2.1. Cartesian Coordinates

Assume that the numerical computation domainfor some number, which contains the support of the surface density, and further discretize the region uniformly as follows. Given a positive integer, define,,, and, where. Next, define the center of the cellasand, where. Hence, the domainis discretized into thecells.

We recall the-body method with fast Fourier transform (FFT). Represent the potential functionof (1) as Where  According to (2), the potential is Calculate (8) using a numerical approach for the planeassuming that the support of the surface density is compact. Based on a restriction to the calculation of potentials on theplane, derive the formulae for all. Equation (8) is equal to The potential atfor the-body method is approximated by That is, The approximated integral is of the first order and the fundamental function    is singular whenever. The-body method applies a softening parameter technique forwith, to avoid the singularity.

To improve the calculation of (11), approximate the surface densityonin (8) linearly by whereandandare constant in the cell. The error of the discretization is. If a higher order accuracy is required, additional terms in the Taylor expansion in (13) can be considered.

Let By substituting (13) into (9), the potential can be approximated by where

Fortunately, the integrals of (14) can be obtained analytically with the help of the following simple formulae: The valueis then equal to where the notation. The calculations ofandare split into two parts by the identity and, respectively. This leads to

The forms of,, andin (16) are a type of convolution and the computational complexity can be reduced towith the help of FFT in theanddirections. In thedirection, the computational complexity is. This implies that the total complexity of this method isandfor the planeand the whole 3D space, respectively.

2.2. Spherical and Polar Coordinates

While the method appears to be second-order-accurate for discretization on a Cartesian grid, it loses second-order accuracy using spherical polar coordinates. An algorithm for solving the Poisson equation in spherical polar coordinates has been used extensively in protostellar collapse and fragmentation calculations, where often a thin disk formed at the central core of the collapsing cloud [14]. The algorithm implemented in [15] is subsequently modified, see [16], to deal with filamentary density distributions. This method in [16] employed finite differences for the radial dependence of the potential and spherical harmonics for the angular dependence, resulting in a second-order method through convergence testing.

We are going to develop our approach for spherical coordinates, which can be reduced to polar coordinates (). The singular integral disappears, but the high order of accuracy is not attained because there is no explicit form for the integral of an elliptic function. Although the proposed calculation of potential has some drawbacks, it is still better than-body calculations and the method in [17].

The support of the surface density is compact, contained in a region  for some number. Confine the discretization on the radial region in logarithmic form and the azimuthal region uniformly as follows to achieve a fast calculation. Given a positive integer, define,,,,,,, andwhere. The effect ofis to refine the mesh. Here, the proposed method for polar coordinates is of first order because a singular integration occurs (see below). Furthermore, the region    is discretized into thecells,for. For, the cellsdo not cover the origin and extra cellsshould be included. To simplify notation, denote,.

The potential functionof (1) under the assumptionin spherical coordinates can be expressed as where

The surface densityonin (20) can be linearly approximated by whereandandare constants in the cell. Equation (22) is the Taylor expansion in two dimensions and it follows that the error of the discretization is.

In this section, let Similarly for,, andon the central region . The termin (24) is for the formulation of a convolution type. According to (20) and (22), where Equation (27) can then be rewritten as