Poisson

Direct Numerical Solution of Poisson's Equation in Cylindrical r; z  Coordinates
Edward H. Chao Stephen F. Paul, and Ronald C. Davidson , Plasma Physics Laboratory Princeton University Princeton, New Jersey 08543 Kevin S. Fine Department of Physics University of California at San Diego La Jolla, California 92093 July 22, 1997
Abstract
A direct solver method is developed for solvingPoisson's equation numerically for the electrostatic potential r; z  in a cylindrical region r Rwall; 0 z L. The method assumes the charge density r; z  and wall potential r = Rwall; z  are speci ed, and @ =@z = 0 at the axial boundaries z = 0; L.
Corresponding author: email: chao@pppl.gov, Tel. 609-243-2856, Fax 609-2432418 PACS Numbers: 02.60.Cb, 52.25.Wz Keywords: Poisson, Cylindrical
1

Many calculations in plasma physics require a rapid solution of Poisson's equation

r = ,4 ;
2

1

where is the electrostatic potential, and is the charge density. This paper was motivated by the need to determine the potential variation in an axisymmetric Malmberg-Penning trap con ning a pure electron plasma 1 3 . Hughes 4 has previously described a direct solver for Poisson'sequation in cylindrical r; z coordinates, but the potentials on axis r = 0 and at some radius r = r must be known. Often, the axial potential is not known a priori. Trunec 5 has also developed a direct Poisson solver in cylindrically symmetric geometry without requiring knowledge of the axial potential. Both Hughes and Trunec utilize a Fourier transform in the axial direction, but Trunecapproximates the radial solution using the basis functions for cubic splines, while Hughes nds a radial solution using only the nite-di erence form of the radial di erential equation. Trunec's approach allows for unequal grid spacing in the radial direction, but the benchmarking results suggest that the spline approximation introduces more error than Hughes' method of solving the nite-di erenceequations directly. The purpose of this Note is to extend Hughes' solver so that it does not require knowledge of the axial potential. For @=@ = 0; Poisson's equation in cylindrical r; z coordinates is
0

@ + 1 @ + @ = ,4 ; @r r @r @z 2
2 2 2 2

2

where is the electrostatic potential, and is the known charge density. The potential is assumed to be speci ed at radius r = Rwall, and @ =@z= 0 at the axial boundaries z = 0; L. The latter assumption is appropriate for the applications of interest, but can easily be modi ed to describe the case where = 0 at the axial boundaries or the case of periodic boundary conditions by using a sine or Fourier transform instead of a cosine transform. We begin the analysis by applying a discrete cosine transform in the axial ^ direction toPoisson's equation. The cosine transform uses cosines z only to form a complete set of basis functions in the interval from 0 to 2, and guarantees that the solution will have zero derivative at the axial boundaries 6 . The cosine transform is de ned by

Fk =
with inverse

N ,1 X j =0

fj cos

kj +  N ;
1 2

3

kj +  2 N, fj = N 0 Fk cos N :
1 X 1 2

k=0

4

Here, theprime on the summation symbol means that the k = 0 term has a coe cient of multiplying 2=N F . We consider the r; z plane covered by a uniform mesh with constant spacing
r and
z in the r and z directions:
1 2 0

z = i + 1 

z ; i = 0; 1; :::; NZ , 1; 2 r = j

r ; j = 0; 1; :::; NR;
3

5

wall where
z = NLZ and
r = RNR . The cosine transform can be written as

2 NZ ,0 ~ rcos kz : r; z = N k
z NZ Z k
X 1 =0

6

Substituting Eq. 6 into Poisson's equation 2 yields

@ ~k + 1 @ ~k , 1 k @r r @r
z NZ
2  2 2

!2

~k = ,4 ~k :

7

where has been similarly transformed. The next step is to write these equations in nite-di erence form. Away from the axis j  1, Eq. 7 becomes ~k;j , 2 ~k;j + ~k;j, ~k;j , ~k;j, +
r 2j
r 1 k ,
N ~k;j...