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

2

velocity

1.5

1

0.5

0

-1 -0.5 0 0.5 1

Anti-Kelvin

1

velocity:scaled

0.5

0

-0.5

-1

-1 -0.5 0 0.5 1

cos(colatitude)

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.

CHAPTER 7. LINEAR EIGENVALUE PROBLEMS

156

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

errors.

Galerkin-by-recurrence-relation has been used successfully, but one should be very cau-

tious. Exact quadrature is no excuse for taking risks.

7.12. COMMON ERRORS 157

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

&Douglas(1989)

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

&Boisvert(1990)

CHAPTER 7. LINEAR EIGENVALUE PROBLEMS

158

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,

(8.1)

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.

159

CHAPTER 8. SYMMETRY & PARITY

160

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.

Theorem 22 (PARITY OF BASIS FUNCTIONS)

(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:

(m)

{T2n (x), P2n (x), C2n (x), and H2n (x)}.

All orthogonal polynomials of ODD degree (except Ln (y)) are

ANTISYMMETRIC.

(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