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were so smooth “ I walked around in a fool™s paradise for weeks. Actually, I was solving
a Sturm-Liouville problem of the Fourth Kind with a nasty singularity right on the expan-
sion interval. Enlightment came when I graphed the second derivative of the eigenmodes,
and found it resembled the ¬ngerpainting of a crazed pre-schooler. I recovered from this
¬asco by applying a change-of-coordinate as described earlier. Plotting the reciprocal of
differences in eigenvalues for two different numerical resolutions should protect one from
a similar blunder. Well, most of the time.
Boyd (1996b) solved the so-called “equatorial beta-plane” version of the tidal equations,
and obtained two modes whose eigenvalues, equal in magnitude but opposite in sign,
changed almost negligibly with N . One was the so-called “Kelvin” wave, which is known
to have the simplest structure of all tidal modes and therefore should indeed be resolved
most accurately by any numerical solution. The other, however, was a sort of “anti-Kelvin”
wave that has no counterpart in reality.
The proof lies in the graph: Fig. 7.13 shows the zonal velocity for the two modes. The
wild oscillations of the lower mode, which become only wilder with increasing N , show
that it is a numerical artifact. Yet its eigenvalue is unchanged through the ¬rst twelve
nonzero digits when N is increased from 36 to 50!
Obviously, the prudent arithmurgist will compare the eigenmodes, and not merely the
eigenvalues, for different N , at least occasionally. As pilots say, even when ¬‚ying on in-
struments, it is a good idea to occasionally look out the window.
“Negligence” is a shorthand for missing important modes. The QR/QZ algorithm com-
putes all the eigenvalues of a given matrix, so with this method for the algebraic eigenvalue
problem, failure is possible for a given mode only by choosing N too small. When QR is

True Kelvin


-1 -0.5 0 0.5 1


-1 -0.5 0 0.5 1

Figure 7.13: Example Two (In¬nite Interval): Absolute errors in the eigenvalues for 16-
point (solid disks) and 32-point (open circles) T Bn discretization.

replaced by an iterative, ¬nd-one-mode-at-a-time algorithm, however, it is very easy to
overlook eigenmodes even if N is large enough to resolve them.
Boyd (1985a) records an embarrassing example. By combining the QR algorithm with
a Chebyshev pseudospectral method and a detour into the complex plane, I found that
my earlier calculations (Boyd, 1981a) had missed the third and seventh eigenmodes. Oops!
The earlier work had combined a local iterative method with continuation in the parameter.
However, these modes have very small imaginary part, and the older numerical method,
which did not use a complex-plane mapping, was suf¬ciently inaccurate to miss them.
It really is a good idea to use the QR/QZ method whenever possible. If one must iterate,
one must worry over the workstation like a parent watching over a toddler asleep with a
high fever.
Coding errors have also sent eigenvalue computations into the Black Swamp of Pub-
lished Errors. Of course, the misplaced “if” statement or the ever-dangerous sign error
can imperil any kind of calculation. Galerkin calculations, however, seem to be especially
prone to slips.
A good program-checking strategy is to apply the code to a simple problem with a
known solution. Unfortunately for Galerkin algorithms, solvable test problems usually
have diagonal Galerkin discretization matrices, and therefore do not test the off-diagonal
elements at all.
In ye olden days when computers were young and full of vacuum tubes, it was common
to derive the Galerkin matrix elements by hand calculations using recurrence relations.
Fox & Parker (1968) and Fox, Hayes and Mayer (1973) are good exemplars of this old
style; these recurrences are used in Orszag™s (1971b) classic article and furnish a couple
of pages of appendix in Gottlieb and Orszag (1977). It is so beautiful and elegant, revealing
whatever sparsity is present in the Galerkin matrix, that it is still widely used.
Unfortunately, these recurrences are risky. Each change of problem, even from the tar-
get differential equation to a test equation, requires redoing much of the algebra. Conse-
quently, a successful test is no guarantee that the algebra “ much of it different “ is correct
for the target problem, too.
O™Connor (1995) applied recurrences to Laplace™s Tidal Equation. This seemed partic-
ularly appropriate because the exact Galerkin matrix is sparse. However, the published
article contains one spurious mode and slightly incorrect eigenvalues for all the others as
corrected in O™Connor (1996). I know of other examples where the published work even-
tually fell under suspicion, but cannot name names because the authors never found their
Galerkin-by-recurrence-relation has been used successfully, but one should be very cau-
tious. Exact quadrature is no excuse for taking risks.

Table 7.3: A Selected Bibliography of Spectral Calculations for Eigenproblems

References Comments
Longuet-Higgins(1968) Laplace™s Tidal Equation; spherical harmonics;
special continued fraction algorithm
for tridiagonal Galerkin matrix
Birkhoff&Fix(1970) Fourier and Hermite function Galerkin methods
Orszag (1971b) Chebyshev; 4th order Orr-Sommerfeld stability eqn.
Noted usefulness of spectral combined with QR/QZ
Fox&Hayes Double eigenvalue problem: 2d order ODE with
&Mayers(1973) 3 boundary conditions and 2 eigenparameters
Boyd (1978a) Chebyshev domain truncation for quantum quartic oscillator
Chebyshev curve-¬tting for analytical approximations
to eigenvalue
Boyd (1978c) more remarks on Chebyshev/QR connection
Boyd (1978b) Chebyshev, Fourier & spherical harmonic bases on sphere
Banerjee (1978) Eigenvalue problem: quantum anharmonic operator
Banerjee et al. (1978) Hermite func. ψn (±y) with variable ±
Boyd (1981a) Eigenproblem with interior singularity
Boyd (1982b, c) Atmospheric waves in shear ¬‚ow
Liu&Ortiz(1982) Singular perturbations; tau method
Boyd (1983d) Analytic solutions for continuous spectrum (¬‚uids)
Lund&Riley(1984) sinc basis with mapping for radial Schroedinger equation
Liu&Ortiz&Pun(1984) Steklov PDE eigenproblem
Boyd (1985a) Change-of-coordinate to detour around around interior
singularity into the complex plane
Brenier&Roux Comparison of Chebyshev tau and Galerkin
&Bontoux(1986) for convection
Liu&Ortiz(1986) PDE eigenproblems; tau method
Liu&Ortiz(1987a) complex plane; Orr-Sommerfeld equation
Liu&Ortiz(1987b) powers of eigenparameter in ODE
T Bn basis: x ∈ [’∞, ∞]
Boyd (1987a)
T Ln basis: x ∈ [0, ∞], Charney stability problem
Boyd (1987b)
Eggert&Jarratt sinc basis;¬nite and semi-in¬nite intervals, too, through
&Lund(1987) map singularities at end of ¬nite interval
Zebib (1987b) Removal of spurious eigenvalues
Navarra(1987) Very large meteorological problem (up to 13,000 unknowns)
via Arnoldi™s algorithm
Tensor product of T Ln (z) — T Bm (y)
Lin & Pierrehumbert (1988)
for two-dimensional baroclinic instability
Gardner&Trogdon Modi¬ed tau scheme to remove “spurious” eigenvalues
Boyd (1990d) Chebyshev computation of quantum scattering
(continuous spectrum)
Malik(1990) Hypersonic boundary layer stability; spectral multidomain
Jarratt&Lund&Bowers(1990) sinc basis; endpoint singularities, ¬nite interval
McFadden&Murray Elimination of spurious eigenvalues, tau method

Table 7.3: Bibliography of Spectral Calculations for Eigenproblems[continued]

References Comments
Boyd (1992a) Arctan/tan mapping for periodic eigenproblems
with internal fronts
Falques & Iranzo (1992) T Ln and Laguerre, edge waves in shear
Su&Khomami (1992) Two-layer non-Newtonian ¬‚uids
Mayer&Powell(1992) Instabilities of vortex trailing behind aircraft
One-dimensional in r in cylindrical coordinates
Integration along arc in complex plane for near-neutral modes
Khorrami&Malik (1993) Spatial eigenvalues in hydrodynamic instability
Chen (1993) T Ln (x) basis for semi-in¬nite interval; nonparallel ¬‚ow
Boyd (1993) Symbolic solutions in Maple & REDUCE
Huang&Sloan(1994b) Pseudospectral method; preconditioning
Boyd (1996b) Legendre, quantum and tidal equations;
traps and snares in eigencalculations
Gill& Sneddon Complex-plane maps (revisited) for eigenfunctions
(1995,1996) singular on or near interior of (real) computational domain
Dawkins&Dunbar Prove tau method, for uxxxx = »uxx , always has 2 spurious
eigenvalues larger than N 4
&Douglass (1998)
O™Connor(1995,1996) Laplace™s Tidal Equation in meridian-bounded basin
Sneddon (1996) Complex-plane mappings for a semi-in¬nite interval
Straughan&Walker(1996) Porous convection; compound matrix & Chebyshev tau
Dongarra&Straughan Chebyshev tau/QZ for hydrodynamic stability
&Walker(1996) Comparisons: 4th order vs. lower order systems
Boomkamp&Boersma Pseudospectral/QZ algorithm for eigenvalue problem;
&Miesen&Beijnon(1997) stability of two-phase ¬‚ow; 3 subdomains
Chapter 8

Symmetry & Parity

“That hexagonal and quincuncial symmetry . . . that doth neatly declare how nature Ge-
ometrizeth and observeth order in all things”
” Sir Thomas Brown in The Garden of Cyrus (1658)

8.1 Introduction
If the solution to a differential equation possesses some kind of symmetry, then one can
compute it using a reduced basis set that omits all basis functions or combinations of basis
functions that lack this symmetry. The branch of mathematics known as “group theory” is
a systematic tool for looking for such symmetries. Group theory is a mandatory graduate
school topic for solid-state physicists, physical chemists, and inorganic chemists, but it is
also important in ¬‚uid mechanics and many other ¬elds of engineering.
However, in ¬‚uid mechanics, the formal machinery of group theory is usually not
worth the bother. Most of the observed symmetries can be found by inspection or sim-
ple tests.

8.2 Parity
The simplest symmetry is known as “parity”.
De¬nition 21 (PARITY) A function f (x) is said to be SYMMETRIC about the origin or to pos-
sess “EVEN PARITY” if for all x,
f (x) = f (’x)
A function is said to be ANTISYMMETRIC with respect to the origin or to possess “ODD PAR-
ITY” if for all x,
f (x) = ’f (’x) (8.2)
A function which possesses one or the other of these properties is said to be of DEFINITE PARITY.
The word “PARITY” is used as a catch-all to describe either of these symmetries.
Note that trivial function f (x) ≡ 0 is of BOTH EVEN & ODD parity ” the only function
with this property.


Parity is important because most of the standard basis sets ” the sines and cosines of
a Fourier series, and also Chebyshev, Legendre and Hermite polynomials ” have de¬nite
parity. (The exceptions are the basis sets for the semi-in¬nite interval, i. e., the Laguerre
functions and the rational Chebyshev functions T Ln (y).) If we can determine in advance
that the solution of a differential equation has de¬nite parity, we can HALVE the basis set
by using only basis functions of the SAME PARITY.
The terms of a Fourier series not only possess de¬nite parity with respect to the origin,
but also with respect to x = π/2. Consequently, it is sometimes possible to reduce the
basis set by a factor of four for Fourier series if the differential equation has solutions that
also have this property of double parity. The most studied example is Mathieu™s equation,
whose eigenfunctions fall into the same four classes that the sines and cosines do.


(i) All cosines, {1, cos(nx)}, are SYMMETRIC about the origin.
All sines {sin(nx)} are ANTISYMMETRIC about x = 0.

(ii) The EVEN degree cosines {1, cos(2x), cos(4x), . . . } and the ODD sines {sin(x),
sin(3x), sin(5x), . . . } are SYMMETRIC about x = ±π/2.
The cosines of ODD degree {cos(x), cos(3x), cos(5x), . . . } and the sines of EVEN
degree {sin(2x), sin(4x), . . . } are ANTISYMMETRIC about x = ±π/2.

(iii) All orthogonal polynomials of EVEN degree (except Laguerre) are SYMMETRIC:
{T2n (x), P2n (x), C2n (x), and H2n (x)}.
All orthogonal polynomials of ODD degree (except Ln (y)) are

(iv) The rational Chebyshev functions on the in¬nite interval, T Bn (y), have the same
symmetry properties as most orthogonal polynomials (EVEN parity for EVEN sub-
script, ODD parity for ODD degree), but the rational Chebyshev functions on the
semi-in¬nite interval, y ∈ [0, ∞], T Ln (y), have no parity.

The double parity of the sines and cosines is summarized in Table 8.1 and illustrated in
Fig. 8.1.
PROOF: (i) and (ii) are obvious. (iii) is a consequence of the fact that all the polynomials
of even degree (except Laguerre) are sums only of even powers of x, and all those of odd
degree are sums of {x, x3 , x5 , . . . }. [This can be rigorously proved by induction by using
the three-term recurrences of Appendix A.] Eq. (iv) may be proved along the same lines as
(iii); each rational function T Bn (y) has a denominator which is symmetric about the origin
and a numerator which is a polynomial of only even or only odd powers of y. Theorem 22


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