Poisson FFT Solver under the Dirichlet Boundary Condition

Poisson FFT Solver under the Dirichlet Boundary Condition
Masayoshi Kiguchi
RIST, Kinki University
3-4-1 Kowakae, Higasi-Osaka
(Received January 9, 2007)
Abstract
In order to clear the misconception that FFT is not applicable to solve the Poisson equation with
Dirichlet boundary condition, we have study the FFT algorithm and find it useful in the simulation of star
formation.
1 Introduction
The time consuming part of the simulation for the star formation is in the calculation of gravity. Although
the most severe neck is in the calculation of radiative transfer in the region with optical depth of about one, if
we compromise to use the adequate diffusion approximation, we can get not so wrong solutions and we can
be free from the restriction of computation time. In contrast, as the calculation of gravity is too fundamental,
if we solve it sloppy, we get improper solutions and we are led to a wrong result. We need to find gravity
carefully.
The gravity in the simulation of star formation is described by the Poisson equation with Dirichlet boundary
condition. this condition is essential especially for the case of local refinement calculation. It is not
allowed to apply the cyclic boundary condition to this problem.
In this case, the conjugate gradient method (van der Vorst, 1981) or multi-grid method (Brandt, 1977) or
esoteric craftsmanship SOR method has been used in many codes. If we can use Fast Fourier Transformation,
FFT, (Cooley and Tukey, 1965) in this problem, we can get a sure solution with computational complexity
of N log N free from the problem of convergence. Regardless of this merits, this method has not been used.
When I asked to an expert of simulation, his answer was that it is difficult to fit this method to the problems.
In order to clear this misconception, I will confirm the effectiveness of the FFT methods to solve the Poisson
equation with Dirichlet boundary conditions.
2 Algorithm
Solvability of Poisson equation by FFT is based on the following fact: the solution φ of Poisson equation
△φ =4πGρ (1)
under the Dirichlet boundary condition is given by the superposition of two terms, φ = φ poisson + φlaplace,
where φpoisson is the solution of Laplace equation
△φ =0 (2)
under the given Dirichlet boundary condition, and φlaplace is the solution of Poisson equation (1) under the
boundary condition φ =0.

To solve the Poisson equation numerically, we make a cubic lattice system from N 3 =(2 n ) 3 grid points
in the regin −1 ≤ x, y, z ≤ 1 and find the value of φix,iy,iz at the grids point ,0 ≤ ix,iy,ix

another surfaces. We can solve (2) to obtain φi by separation of variables. The potential φi, for which the
boundary value in jz =0plane is ψjxjy, is given by
3 Result
φjxjyjz =
N−1
kx,ky,kz=1
sinh(π
˜ψkxky
k2 x + k2 y(N − kz)/N)
sin( πjxkx
N )sin(πjyky
), (9)
N
˜ψkxky =( 2
N )2
N−1
jx,jy=1
π k 2 x + k2 y
ψjxjy sin( πjxkx
N )sin(πjyky
). (10)
N
We have compared the calculated solution with the analytic solution for the potential of the uniform density
ellipsoid. The external potential of uniform ellipsoid (Chandrasekhar, 1969) with the axis length of a 1 ≥
a2 ≥ a3 and the density ρ is given by
1
− φ(x1,x2,x3)
πGρa1a2a4
1
=
2
2 a1 − a2 (F (φ, k) −
3
1
(a 2 1 − a2 3 )
(x 2 1
k2 (F (φ, k) − E(φ, k)) + x22 k2 (Π(φ, k2 ,k) − F (φ, k)) + x 2 3
where F is the elliptic integral of Legendre first kinds
E is of second kinds
and Π is of third kinds
The internal potential is given by
×( (a2 1 − x2 1 )
k 2
Π(φ, n, k) =
F (φ, k) =
E(φ, k) =
φ
0
− φ(x1,x2,x3)
πGρa1a2a3
φ
0
φ
0
(Π(φ, 1,k) − F (φ, k)), (11)
dθ
1 − k 2 sin 2 θ , (12)
1+k 2 sin 2 θdθ, (13)
1
(1 − n sin 2 1
dθ. (14)
θ) 1 − k2 2
sin θ
=
2
(a 2 1 − a 2 3) a 2 1 − a 3 3
(F (φ, k) − E(φ, k)) + (a22 − x22 )
k2 (Π(φ, k 2 ,k) − F (φ, k)) + (a 2 3 − x23 )(Π(φ, 1,k) − F (φ, k))).
(16)
We have used Intel 3.2GHz Xeon processor and icc compiler. In the following, we show the result of a
computation. In this computation, Poisson equation is solved in the regin −1 ≤ x, y ≤ 1, −0.1 ≤ z ≤ 0.1.
The axis lengths of ellipsoid is a1 = a2 =0.5, a3 =0.05. Grid number is 64 3 and the computation time was
0.6795 s The numerical error of the potential is the same order to the numerical error of the integration of the
mass density.
(15)

References
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phi
0
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z
r
Numerical
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Analytic
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Figure 1: Comparison of potential φ between numerical calculation and analytic solution
[1] Brandt, A., 1977, Math. Comput. 31, 333.
[2] Chandrasekhar, S., 1969, “Ellipsoidal Figures of Equilibrium”.
[3] Cooley, J. and Tukey, J., 1965, Mathematics of Computation 19, 297.
[4] van der Vorst, H. A., 1981, Journal of Computational Physics, 44, 1.