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and Boundary Condition at the Origin
The second consequence of the convergence of the meridians is that differential equations
are singular at r = 0. For example, the Laplace operator is
‚2 1 ‚2
1‚
≡ 2+ 2
(18.2)
+
r ‚r r2 ‚θ2
‚r
Thus, the coef¬cients of the ¬rst radial derivative and second polar derivative are both
singular at r = 0. And yet, when the Laplacian is written in Cartesian coordinates, it is
constant coef¬cient:
‚2 ‚2

2
(18.3)
+2
‚x2 ‚y
18.4. POLAR COORDINATES: PARITY THEOREM 383

A simple function like f (x, y) = x2 + y 2 has a Laplacian which is a constant. What could
be less singular!
The resolution of the paradox is that the singularities of the Laplace operator, when
written in polar coordinates, are only “apparent”. The solution to a differential equation is
usually smooth and in¬nitely differentiable at the pole.
To avoid evaluating differential equation coef¬cients where they are in¬nite, the spec-
tral grid usually excludes the origin. This is only a trivial complication, however; the inter-
esting issue is: How does one impose boundary conditions at the origin?
For ¬nite difference methods, one must normally impose explicit, numerical boundary
conditions, based on careful analysis of the behavior of the solution near the origin, at the
grid point nearest r = 0. For spectral methods, however, the singularity actually simpli-
¬es the algorithm. To a spectral method, the boundary condition at the origin is behavioral
rather than numerical: the correct behavior is that the solution should be analytic at the
origin even though the coef¬cients of the differential equation are not. Since the terms of
a spectral series are individually analytic, it follows that a Chebyshev or similar series in
radius automatically satis¬es the boundary condition at r = 0, and no additional explicit
conditions are necessary. If the differential equation were not singular at r = 0, it would
be necessary (for a second order equation) to impose an explicit numerical boundary con-
dition at the origin. Thus, the singularity of the coordinate has ironically simpli¬ed the use
of a spectral method.
For best results, one must respect the parity symmetry at the origin which is explained
in the next two sections. However, one only needs to constrain the spectral series through
a numerical boundary condition at the outer boundary of the disk. If one applies polar
coordinates to an in¬nite domain, then both boundary conditions are behavioral, and no
conditions need to be explicitly imposed on the spectral series.


18.4 Polar Coordinates: Parity Theorem
Theorem 35 (Polar Coordinates: Parity in Radius) In polar coordinates where r is radius and
θ is angle, expand an arbitary function as a Fourier series in θ:

(18.4)
f (r, θ) = fm (r) cos(mθ) + gm (r) sin(mθ)
m=0

Suppose that f (r, θ) is a SCALAR or the z-velocity in cylindrical coordinates or is the product of
r with the radial or tangential components of a vector in polar coordinates. Then if the function is
analytic at r = 0, continuity of the function and its derivatives demands the following:

(i) fm (r) and gm (r) have m-th order zeros at r = 0.

(ii) If m is EVEN, then fm (r) and gm (r) are both about r = 0 and their power series
SYMMETRIC
contain only EVEN powers of r.

(iii) If m is ODD, then fm (r) and gm (r) are both about r = 0 and their power
ANTISYMMETRIC
series contain only ODD powers of r.


Proof: A geometric argument for spherical coordinates is given in Sec. 18.8 below. This
visual argument applies also to polar coordinates because latitude and longitude become a
local polar coordinate system in a small neighborhood of the north and south poles. Here,
we present a different line of proof, couched as an informal, heuristic argument. Because
CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY
384

there seems to have been much confusion about this theorem in the literature and multiple
rederivations of its conclusions, we give this argument in some detail, but the rest of this
section can be skipped without loss of continuity. A careful, rigorous proof is given in
Eisen, Heinrichs &Witsch(1991).
First, note that any function f (x, y) which is analytic at and near x = y = 0 can be
locally approximated, to any desired accuracy, by a bivariate polynomial of suf¬ciently
high order. (Indeed, a double Taylor series will suf¬ce). Substituting x = r cos(θ) and
y = r sin(θ), it follows that a bivariate polynomial in r cos(θ) and r sin(θ) will suf¬ce as a
proxy for f (r, θ) to any desired accuracy. The monomials that appear in this polynomial
approximation have properties described by the following two lemmas.

Lemma 1 (Polar Wavenumbers of Cartesian Powers) Let “polar wavenumber m” denote the
Fourier terms cos(mθ) or sin(mθ). When a monomial

Mjk ≡ xj y k (18.5)

in powers of the Cartesian coordinates x and y is converted into polar coordinates and expanded as
a Fourier series in the polar angle θ, polar wavenumber m will only appear in the expansion of the
monomial if the total degree of the monomial is greater than or equal to m, that is, if

d≡j+k ≥m (18.6)

Proof: A monomial of total degree d can be written, expressing the sine and cosine as
complex exponentials, in the form
d
(’i)k
x y ≡r (exp(iθ) ± exp(’iθ) )
jk j+k
(18.7)
2d n=1

The crucial fact is that each exponential in the product is proportional to polar wavenumber
one, that is, to exp(±iθ). To obtain a term of wavenumber 7, for example, for example, it is
necessary to multiply such wavenumber one terms together at least 7 times. This requires
that the number of factors in the product is at least 7. Since the number of factors is the
total degree d, it follows that the total degree of the monomial must be at least 7. The same
reasoning applies to arbitrary wavenumber m.
The factor of rd in front of the product in Eq. (18.7) then immediately implies the fol-
lowing.

Lemma 2 (Radial Degree of Cartesian Powers) A Cartesian monomial xj y k is proportional to
rd where d ≡ j + k is the total degree of the monomial.
Because of Lemma 1, it follows that a Cartesian monomial will contain terms like cos(mθ) or
sin(mθ) only if it is proportional to rd where d ≥ m.

Together, these lemmas show that the terms proportional to either cos(mθ) or sin(mθ)
must also be proportional to rm , and thus the coef¬cients of these terms in the Fourier
series of f (r, θ) must have an m-th order zero at r = 0.
The parity of the Fourier coef¬cients with respect to r = 0 follows from a third lemma.

Lemma 3 (Monomial Parity) A monomial xj y k , when expanded as a Fourier series in polar
angle θ, is a trigonometric polynomial of degree j + k which contains contains only those polar
wavenumbers of the same parity as the degree d = j +k. That is, if d is odd, then the Fourier series of
xj y k contains only {cos(dθ), sin(dθ), cos([d’2]θ), sin([d’2]θ), cos([d’4]θ), sin([d’4]θ), . . . , }.
Odd polar wavenumbers m are thus multiplied only by odd powers of r, and even powers only
by even.
18.5. RADIAL BASIS SETS AND RADIAL GRIDS 385

Proof: The product
d
(’i)k
x y ≡r (exp(iθ) ± exp(’iθ) ) ,
jk j+k
(18.8)
2d n=1

can be expanded into terms of the form exp(inθ) where n depends on how many factors
of exp(iθ) and how many factors of its complex conjugate are combined in a given term.
The term of highest polar wavenumber is exp(idθ) when all d exponentials exp(iθ) are
combined. The crucial point is that when exp(iθ) is replaced by its complex conjugate
exp(’iθ), which is the other term inside each factor in the product, the degree j is lowered
by two. Thus, all wavenumbers j in the expansion of the product will be of the same parity,
either even or odd, as the total degree d.
The radial dependence of the monomial xj y k is rd where d is the sum of j and k. If
d is even, then rd is symmetric with respect to r = 0. (We have already seen that if d
is even, then all the Fourier coef¬cients of the monomial are of even polar wavenumber,
too.). Similarly, if d is odd, then its radial factor rd is antisymmetric (i. e., odd parity) with
respect to r = 0. Thus odd powers of r are paired only with cos(mθ) and sin(mθ) where m
is odd and even powers of r are paired only with even m.
The parity theorem has major implications for the choice of grids and basis functions as
explained in the next section.


18.5 Radial Basis Sets and Radial Grids
On a ¬nite domain in radius, which can be always be normalized to r ∈ [0, 1] by a linear
change-of-coordinate, there are several options (where m denotes the angular wavenum-
ber):
1. Bessel Functions: Jm (jm,j r) cos(mθ), sin(mθ)
2. Polar Robert Functions: rm Tj (r), j = 0, 1, 2, . . .
3. Shifted Chebyshev Polynomials of Linear Argument:
Tj (2r ’ 1), j = 0, 1, 2, . . .
4. Shifted Chebyshev Polynomials of Quadratic Argument:
Tj (2r2 ’ 1), (m even), r Tj (2r2 ’ 1), (m odd), j = 0, 1, 2, . . .
5. Unshifted Chebyshev Polynomials of Appropriate Parity:
T2j (r) cos(2mθ), T2j’1 (r) cos([2m ’ 1]θ], j = 0, 1, 2, . . .
6. One-Sided Jacobi Polynomials:
0,|m|
(2r2 ’ 1) cos(mθ), sin(mθ)
r m Pj
The “cylindrical harmonics” are

cos(mθ)
Jn (r, θ) ≡ Jm (jm,n r)
m
(18.9)
sin(mθ)

where there are two basis functions, one proportional to cos(mθ) and the other to sin(mθ),
for all pairs of integers (m, n) with m > 0 and where jm,n denotes the n-th zero of the m-th
Bessel function Jm (r). The cylindrical harmonics are popular in engineering because these
are the eigenfunctions of the Laplace operator in polar or cylindrical coordinates, which
made it easy in the old pre-computer days to apply a Galerkin method. However, as noted
CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY
386

by Gottlieb and Orszag(1977), the Bessel series usually converges only at an algebraic rate.
(An analysis of the precise rate of convergence is given in Boyd and Flyer(1999).)
Bouaoudia and Marcus(1991) developed a fast transform for the polar “Robert” func-
tions, which mimic the spherical basis invented by Robert(1966):

cos(mθ)
φm,n (r, θ) = rm Tn (r) (18.10)
sin(mθ)

where n is restricted to have the same parity (i. e., even or odd) as m. Unfortunately,
Merilees(1973a) showed that the analogous spherical basis is horribly ill-conditioned and
therefore useless except at very low resolution. The problem is that for large m and small
n, the basis functions are zero over almost all the interval because of the rm factor; note that
rm = exp(m log(r)) ≈ exp(’m(1 ’ r)) for r ≈ 1, and thus decays exponentially fast away
from the boundary at r = 1. In the narrow region around r = 1 where the small n functions
are non-zero, T0 , T2 and other low degree basis functions vary so slowly that φm,0 ≈ φm,2 ,
destroying the linear independence of the functions in the basis. Marcus abandoned the
polar Robert functions in his later work, and instead went to One-sided Jacobi polynomials
as explained below.
The third bad option is to expand fm (r) as a series of Shifted-Chebyshev polynomials
with a linear argument:

Tj— (r) ≡ Tj (2r ’ 1), r ∈ [0, 1] (18.11)

where the asterisk is the usual notation for the Shifted-Chebyshev polynomials. The reason
that this option is bad is that the Shifted-Chebyshev grid has points clustered near both
r = 0 and r = 1. However, the disk bounded by r = ρ has an area which is only the
fraction ρ2 of the area of the unit disk. Near the origin, points are separated by O(1/N 2 ).
It follows that the high density of points near the origin is giving high resolution of only a
tiny, O(1/N 4 )-in-area portion of the disk. Unless the physics imposes near-singularities or
high gradients near the origin, clustering grid points at r = 0 is a bad idea. And even in this
special situation of small-scale features near the origin, it is conceptually better to regard
the use of Shifted-Chebyshev polynomials as “mapping to resolve large gradients near the
origin” rather than “Shifted-Chebyshev polynomials are a good general representation for
radial dependence”.
Fig. 18.2 compares the Shifted-Chebyshev series (dashed) with two other options. The
shallow slope of the coef¬cients of the Shifted-Chebyshev series shows that one needs
roughly twice as many terms to obtain a given degree of accuracy with the Shifted-Chebyshev
polynomials as with the alternatives.
The Shifted-Chebyshev polynomials of quadratic argument and the Parity-Restricted
Chebyshev polynomials, which are also shown in the ¬gure, both work well, but generate
only a single curve on the graph. The reason is that these two options, although seemingly
very different, are in fact the same because of the identity

Tj (2r2 ’ 1) ≡ T2j (r), ∀j, r (18.12)
parity
The grid with parity uses rj = cos(tj /2) whereas the shifted grid with quadratic argu-
ment has rj = {(1 + cos(tj ))/2}1/2 where the tj are an evenly spaced grid on the interval
t ∈ [0, π] for Lobatto, Chebyshev-roots, or Radau grids. (The Radau grid includes r = 1 but
not r = 0.) The trigonometric identity

t 1 + cos(t)
(18.13)
cos =
2 2
18.5. RADIAL BASIS SETS AND RADIAL GRIDS 387

shows that
parity
∀j (18.14)
rj = rj ,

Thus, both the basis functions and the interpolation grid are identical for these two meth-
ods. The distance between adjacent points close to origin is O(1/N ).
Nevertheless, when there is a large number M of points in the polar angle θ and N
points in radius, the distance between adjacent gridpoints on the circle of radius r1 , the
smallest radial gridpoint, will be O(1/(M N )). Since the explicit time-stepping limit is
roughly the amount of time required for advection or diffusion from one grid point to
the next, it follows that the Parity-Restricted Chebyshev polynomials will give a time step
limit which is O(1/(M N )) for advective stability and O(M ’2 N ’2 ) for diffusion. Ouch! On
the sphere, this motivates the choice of spherical harmonics as the basis, which eliminates
this “pole problem” of a very short time-step. A similarly-motivated basis for the disk is
described in the next section.

18.5.1 One-Sided Jacobi Basis for the Radial Coordinate
Matsushima and Marcus(1995) and Verkley(1997a,1997b) independently proposed One-
Sided Jacobi polynomials, multiplied by rm :

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. 79
( 136 .)



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