A Multigrid Poisson Solver for Yin-Yang Grid.
Yasuhiro Nishida1 , Youhei Masada1 , and Akira Kageyama111

Department of Computational Science, Graduate School of System Informatics, Kobe University, Japan

Introduction:
Solar Dynamo and Simulation Strategy

The sun has a multi-layered structure, consisting of theinner core, the radiative zone, and the convective zone
from the center to the surface. Since it is mostly composedof hydrogen and helium gases at the state of plasma, theelectric conductivity is high enough to sustain electric currents. The magnetic field is then expected to be generateddue to the convective motion of the plasma in the solarinterior. We call this process as “solar dynamo” process.
It is well known that the dynamo process also works
in the liquid metal which makes up the outer core of theearth. While the earth has quasi-steady, well-organized,
dipole magnetic fields, solar magnetic fields show timevarying features due to cyclic sunspot (active region) formations. Though 11-year activity cycle of the sun is believed as a consequence of solar dynamo process, its precise
mechanism remains poorly understood in spite of theoretical and observational works for many years.
The compressible MHD equation governs the motion of
the plasma in the sun. Since it consists of eight non-linear
partial diﬀerential equations, it is impossible for us to obtain exact solutions analytically without simplifications.
We thus often attack the complex solar dynamo problem
by means of computer simulations.
When discretizing the MHD equation, numerical time
step is limited by the CFL condition, that is ∆t < ∆x/Cs ,where Cs is the sound speed, ∆t is the numerical time step,and ∆x is the grid spacing. This is a necessary condition
for the stability of explicit finite-diﬀerence methods, andstates that the numerical domain of dependence must contain the analytical domain of dependence.
Since the gap between the sound and fluid velocities
in the sun is very large (i.e., Mach number is O(10−4 )),the numerical time step ∆t is enormously smaller than thetypical time scale of the fluid motion [1]. It is thus diﬃcult
to carry out the solar dynamo simulation with a practicaltime step when solving the compressible MHD equation.
To settle this issue associated with the CFL condition,
we take a research strategy in which we remove acoustic
waves by applying “Low-Mach number approximation” tothe MHD equation [2]. The approximated MHD equation
is summarized as follows:
∂f
∂t

=

−∇ · (vf ) − ∇π + ρg ,∂ρh+ ∇ · (ρvh)
∂t=

∇·v=

∇ · κ∇T ,–»
11 ∂p
(∇
·κ∇T)
≡S.
ρcp ∂Tρ ∂p∂ρ

The symbols have their usual meanings, and π is the second order term in the low Mach number expansion of thepressure, that is,p(x, t) = p0 (t) + Mp1 (t) + M2 π(x, t) .
Since the higher order terms of the Mach number M areneglected, the numerical time step is limited by the fluid
velocity in this approximation. Though this enables us
to carry out the dynamo simulation with a practical timestep associated with fluid motions, there exists a numerical
disadvantage. Namely, we should solve a Poisson equation
for maintaining the divergence free of velocity fields.
The implementation of a fast Poisson equation solver
to the simulation code is a key for the eﬃcient dynamo
simulation with low Mach number approximation. In thispaper, we report the development of a fast Poisson equation solver with Multigrid method, and its application to“Yin-Yang grid”, which is a type of chimera grid for spherical geometry and has been adopted in our solar dynamo
simulation code [4].
2Multigrid Method and Yin-Yang Grid

The Multigrid method provides algorithms which canbe used to accelerate the convergence of basic iterative
methods, such as Jacobi , Gauss-Seidel, and SOR methods., by global correction from time to time, accomplishedby solving the target equation on coarse grid [3]. The key
idea behind multigrid method is to reduce eﬃciently longwavelength components of residuals with calculations oncoarser grids. We adopt this method in the fast Poisson
equation solver which will be implemented into our solar
dynamo simulation code.
Yin-Yang grid is a kind of overset (Chimera) grid applied to a spherical shell, as shown in Fig.1 [4]. It is recognized that the popular latitude-longitude spherical gridhas the “pole problems” that refer to two diﬀerent kindsof diﬃculty in numerical calculations; one is the coordinate singularity on the poles, and the other is the gridconvergence near the poles. It is a great advantage of theYin-Yang grid that it has neither a coordinate singularity,
nor grid convergence. Since the Yin-Yang grid has beenadopted in our dynamo simulation code, we developed thefast Poisson equation solver for the Yin-Yang grid in anticipation of its implementation into the dynamo code.

Fig. 1. Yin-Yang Grid which is adopted in our dynamo
simulation code [4].

3Test Problem for Multigrid Poisson Solver

We choose, as the test problem for our Multigrid Poissonsolver, the boundary value problem for the magnetic fieldin the vacuum outside the sphere of r ≥ 1. When assuming
the vacuum as an insulator, the magnetic field should be
written by the gradient of a potential field in r ≥ 1, thatis B = −∇ψ. From the Maxwell equations, we can derivea Laplace equation for the potential field:∇2 ψ = 0, (r > 1) .
When applying the coordinate transformation of ζ; ζ =1/r, the equation can be rewritten, in 0 ≤ ζ ≤ 1, by
„«
–»
1 ∂
∂
1
∂2
∂2
sin θ+
ψ=0.
ζ2 2 +2∂ζsin θ ∂θ∂θsin θ ∂φ2
The potential problem is now converted into the boundary
value problem defined inside a unit sphere of 0 ≤ ζ ≤ 1.
We apply our Multigrid Poisson solver to this problem.
The boundary condition of the potential field is given by
the radial component of the magnetic filed on the boundary r = ζ = 1 which is given by the analytic formulafor the dipole, quadrupole and octapole magnetic field,and sample observational data for the earth and planetary
magnetic fields. In our Multigrid poisson solver, the Jacobi
method is used as the smoother. The V-cycle is repeated
for a couple of times until we get the convergence.
[Convergence Performance]
With applying our Multigrid Poisson solver on the YinYang grid, we obtained the numerical solutions of dipole,quadrupole and octapole magnetic fields as simplest example models. We confirmed that numerical solutions could
reproduce the analytic ones, and our Multigrid Poissonsolver delivered required performance. Fig.2 demonstrates
the convergence times to obtain each numerical solution
for the comparison among the numerical schemes adopting Jacobi method with and without multigrid acceleration. This verifies that the convergence performance isimproved by Multigrid acceleration: It yields 6–29 times
shorter convergence time.
We finally applied our Multigrid Poisson solver to theboundary value problem for the potential fields of theearth. The boundary condition on the earth surface isgiven by observational data [5]. The magnetic field line weobtained for the earth is visualized in Fig.3.

Fig. 3. Visualization of magnetic potential for the Earth

4Conclusion

The origin of magnetic fields is a longstanding issue inthe solar physics and has been explored mainly by numerical simulations. The solar dynamo simulation is, however, suﬀering from the limitation on the numerical time
step due to the CFL condition: Since the gap between thesound and fluid velocities in the sun is very large, the numerical time step is enormously smaller than the typicaltime scale of the fluid motion.
We thus take a research strategy in which we remove
acoustic waves by applying “Low Mach number approximation” to the MHD equation, which enables us to carryout the dynamo simulation with a practical time step associated with fluid motions.
In this paper, we developed the Multigrid Poisson equation solver, and applied it to “Yin-Yang Grid”[4]. This isbecause it is a key element for eﬃcient dynamo simulation
with low Mach number approximation.
Our Multigrid Poisson solver was applied to the boundary value problem for the various type of magnetic fields
in the vacuum outside the unit sphere. Then we obtainedthe numerical solutions of dipole, quadrupole, octpole, andplanetary magnetic fields. It was confirmed from the test
problems that our Multigrid Poisson solver delivered therequired performance and yielded the 6–29 times shorter
convergence time. The implementation of Multigrid Poisson solver into our dynamo simulation code is beyond thescope of this paper, but is our future work.
References[1] C.J.Schrijver and G.L.Siscoe, ‘Heliophysics - Evolving Solar Activity and the Climate of Space and Earth- ’, Cambridge University Press (2010).
[2] A.S.Almgren, J.B.Bell, C.A. Rendleman, and M.Zingale,
The Astrophysical Journal, 637, 922, (2006)[3] W.L.Briggs, V.Henson, and S.F.McCormick, ‘A Multigrid
Tutorial 2nd Edition’, Society for Industrial and Applied
Mathematics, (2000).
[4] Akira Kageyama and Tetsuya Sato, ‘Yin-Yang Grid. An
Overset Grid in Spherical Geometry’, Geochem. Geophys.
Geosyst., arXiv:physics/0403123

Fig. 2. Convergence performance for diﬀerent models.

[5] Susan Macmillan and Stefan Maus ‘International Geomagnetic Reference Field - the tenth generation’, pp.11351137(2005)