7 9 11 13 15 17 19 21 23 25

3

Fourier degree n

Figure 19.1: Absolute values of the Fourier coef¬cients of the Mathieu function, ce15 (x),

when the parameter q = 10. This is the motive for the “sideband truncation”: only ¬ve of

the coef¬cients are non-negligible, but they are not the coef¬cients of the lowest ¬ve Fourier

terms.

{1, cos(2x), cos(4x) . . . }

Even-Even symmetry . . .

ce2n (x) ...

{cos(x), cos(3x), cos(5x) . . . }

Even-Odd symmetry

ce2n+1 (x) ... ...

{sin(2x), sin(4x), sin(6x) . . . }

Odd-Odd symmetry

se2n (x) ... ...

{sin(x), sin(3x), sin(5x) . . . }

Odd-Even symmetry

se2n+1 (x) . . . ...

For small q, one can show via perturbation theory that an±2 is O(q), an±4 is O(q 2 ), an±6 is

O(q 3 ) and so on. The coef¬cient an±4 is only O(q 2 ) because (19.3) shows that the pertur-

bation cannot couple these components directly to cos(nx), but only to the much smaller

components, cos[(n ± 2)x].

Fig. 19.1 illustrates these remarks by plotting the absolute value of the Fourier cosine

coef¬cients for ce15 (x) for q = 10 (Lowan et al., 1951). We do not need 1 or cos(2x) or cos(4x)

in our basis set (for this eigenvalue and this value of q) because their coef¬cients are very,

very small. The important basis functions are those which are oscillating on nearly the

same scale as the dominant basis function, cos(nx).

De¬nition 42 (SIDEBAND TRUNCATION)

When the basis set is restricted to basis functions of the form φn±m (x) where m n, this is

said to be a “SIDEBAND TRUNCATION”.

The basis function φn (x), normally the function with the largest coef¬cient, is the “fundamen-

tal” and the other basis functions in the truncation are the “SIDEBANDS”.

When we apply Galerkin™s method with a sideband truncation to ¬ve basis functions,

the resulting matrix problem is of the form

19.2. SIDEBAND TRUNCATION 445

[» ’ (n ’ 4)2 ] ’q 0 0 0 an’4

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

. .

. [» ’ (n ’ 2)2 ] .

’q

. .

0 0 0 an’2

. .

. .

. .

. .

. .

’q [» ’ n ] ’q

2 =0

. .

0 0 an

. .

. .

. .

. .

. 2.

’q [» ’ (n + 2) ] . ’q

.

0 0 an+2

. .

. .

. .

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

’q [» ’ (n + 4)2 ]

0 0 0 an+4

(19.4)

Eq. (19.4) applies to all four classes of basis functions provided that n ≥ 7. The condition

for a nontrivial solution is that the determinant of the 5 — 5 matrix in (19.4) should equal

0, which gives a quintic equation to determine ». If we truncate to just the fundamental

plus two sidebands, then the “secular determinant” is that of the inner 3 — 3 block in (19.4)

which is enclosed by dots.

The polynomial equations that determine » are

P3 (»; q, n) ≡ »3 ’ (3 n2 + 8)»2 + (3 n4 ’ 2 q 2 + 16)» (19.5)

+ 8 q 2 ’ 16 n2 + 2 n2 q 2 + 8 n4 ’ n6

plus a similar equation for the 5 — 5 determinant (not displayed). Since we are interested

in the mode whose unperturbed eigenvalue is n2 and since the eigenvalue is changed only

a little bit by the perturbation provided that n2 q, we do not need to apply a general

polynomial equation solver to P3 (») and P5 (»). Instead, we can apply Newton™s iteration

once to obtain an accurate analytical solution in the form

(19.6)

» = »0 + δ

where

»0 ≡ ’n2 (19.7)

The 3 — 3 truncation gives

P3 (»0 )

=’ (19.8)

δ3

dP3 (»0 )/d»

4 q2

(19.9)

=

q 2 + 8 n2 ’ 8

and similarly1

32 q 4 + 512 q 2 n2 ’ 1024 q 2

(19.10)

δ5 =

3 q 4 ’ 896 q 2 + 64 n2 q 2 + 1024 n4 ’ 5120 n2 + 4096

Fig. 19.2 compares δ3 (q) and δ5 (q) with the exact correction to the eigenvalue for n = 15.

Although we have included no cosines lower than cos(11x), the 5 — 5 sideband truncation

is almost indistinguishable from the exact eigenvalue correction for q ¤ 25.

When q/n2 is large, the perturbation is so strong that the non-negligible sidebands ex-

tend all the way to the lowest mode, thus implicitly reverting to a normal spectral expan-

sion. “Sideband truncation” has only a limited range of applicability.

1 The

determinants and the Newton™s iteration were evaluated using the algebraic manipulation language

REDUCE

CHAPTER 19. SPECIAL TRICKS

446

1.5

δ5

Eigenvalue correction

1

δ3

0.5

0

0 5 10 15 20 25

q

Figure 19.2: A comparison of the exact eigenvalue correction, δ(q), for the mode ce15 with

the three-basis-function and ¬ve-basis-function approximations computed in the text. The

¬ve-term approximation, δ5 (q), is almost indistinguishable from the exact δ(q).

Nevertheless, there are other physical problems where this trick is useful. An exam-

ple would be the direct (as opposed to perturbative) calculation of “envelope solitons”

(Boyd, 1983a, b, c). There are many interesting phenomena which can be modelled by a

fast “carrier” wave modulated by a slowly-varying “envelope” or amplitude; a sideband

truncation is appropriate for all of them.

19.3 Special Basis Functions, I: Corner Singularities

When the computational domain has sharp angles, the PDE solution is usually weakly

singular at such angles or corners. As noted in Chap. 16, mapping methods are often very

effective in dealing with corner singularities because the solution is bounded even at the

corners. However, mappings rob resolution from the interior of the domain to squeeze

more grid points near the walls. A better option, at least in theory, is to subtract the branch

points from the solution so that the Chebyshev series is only required to represent the non-

singular part of u(x, y).

Schultz, Lee, and Boyd (1989) describe an illuminating example: the wall-driven in-

compressible ¬‚ow in a two-dimensional rectangular cavity. The velocity is discontinuous

at the upper corners where the moving wall meets two stationary boundaries. The result is

that if one de¬nes a local polar coordinate system (r1 , θ1 ) centered on the upper left-hand

corner, the ¬‚ow is singular there with the strongest branch point taking the form

π2

r1 θ1 (cos θ1 ’ sin θ1 ) ’

π

sin θ1

2 2

(19.11)

ψs1 = π2

’1

4

plus a similar form for the other upper corner.

When the streamfunction is partitioned as

(19.12)

ψ(x, y) = ψa (x, y) + ψs (x, y)

where ψs (x, y) includes the two singular terms of the form of (19.11) and ψa (x, y) is ex-

panded as a 30—30 two-dimensional Chebyshev series, the streamfunction for zero Reynolds™

19.3. SPECIAL BASIS FUNCTIONS, I: CORNER SINGULARITIES 447

number is accurate to at least 1 part in 10,000,000,000, relative to its maximum. The singularity-

subtraction has added at least six decimal places of accuracy for this truncation.

The remarkable thing about this accuracy is that the driven cavity ¬‚ow has weaker

singularities at the lower corners, too. Moffatt (1964) showed that as the corners are ap-

proached, the ¬‚ow becomes more and more linear because the velocities tend to zero (rel-

ative to the walls). This near-corner linearity allowed him to show that the ¬‚ow consists

of an in¬nite sequence of eddies of rapidly diminishing size and intensity “ the so-called

“Moffatt eddies”. Because the singular term (19.11) is driven directly by the wall, it is com-

pletely known. However, the “Moffatt eddies” are driven by the large eddy in the center

of the cavity, so the “Moffatt eddies” are known only to within multiplicative constants

which must be determined by matching to the interior ¬‚ow.

This problem illustrates two general themes. The ¬rst is that one can generally deter-

mine the form of the singular basis functions, as in (19.11), by a linear asymptotic analysis

even when the problem as a whole is strongly nonlinear.

The second theme is that weak singularities may often be ignored. The “Moffatt eddies”

are described by functions with branch points, but neither ψ nor its ¬rst couple of deriva-

tives are unbounded. The result is that one can obtain ten decimal place accuracy without

applying special tricks to these corner eddies.

Which corner branch points can be ignored and which should be represented by singu-

lar basis functions depends on (i) the target accuracy and (ii) the strength of the nonlinear-

ity.

If the goal is thirty decimal place accuracy, for example, one could greatly improve

ef¬ciency by supplementing the Chebyshev basis by special functions in the form of Mof-

fatt eddies of undetermined strength. The greater the target accuracy, the more singular

functions that should be included in the spectral basis set.

The strength of the nonlinearity is important, too. Boyd (1986) solved another problem,

the two-dimensional Bratu equation, which has corner singularities. He showed that as the

amplitude becomes larger, the solution series converges more and more slowly. The corner

singularities become less and less important in this same limit.

Schultz, Lee and Boyd (1989) show that the same is true of the driven cavity ¬‚ow. As

the Reynolds number increases, the ¬‚ow develops interior fronts. These regions of steep

gradients are smoothed by viscosity so that the ¬‚ow is analytic everywhere in these frontal

zones ” in contrast to the corner eddies, which are genuinely pathological at the corners.

Nevertheless, the Chebyshev series converges more and more slowly as the internal fronts

become steeper. Unless the target accuracy is greater than ten decimal places, these internal

fronts are a greater numerical challenge than resolving the Moffatt eddies in the corners.

For many problems, the spectral coef¬cients are the sum of two or more indepen-

dent contributions. The corner singularities, because their contributions decay only alge-

braically with n, must dominate in the limit n ’ ∞. However, if the corner branch points

are weak and the internal fronts are strong, the exponentially decaying contributions from

the complex poles or branch points associated with the fronts may be larger than the contri-

butions of the corner singularities for all n < ncross-over . For example, consider a Chebyshev

coef¬cient which depends on n as

n 0.000001

an ∼ 10 exp ’ (19.13)

+

n5

3

A pure mathematician would be content with the statement that an ∼ 0.000001/n5 in the

limit n ’ ∞. The numerical analyst is called to a higher standard: the exponentially-

decaying term in (19.13) is larger than the second, algebraically decaying term for all n <

120.

CHAPTER 19. SPECIAL TRICKS

448

Boyd (1987e, 1988d) gives other vivid examples of this competition between competing

numerical in¬‚uences. Each problem exhibits a “cross-over truncation” where the slope

of the error as a function of N dramatically changes. Boyd (1988d) is a one-dimensional

application of singular basis functions to resolve logarithmic endpoint singularities.

The driven cavity ¬‚ow and the Bratu equation both exhibit a similar tug-of-war be-

tween branch points of different strengths and different nearness-to-the-square. When the

corner singularities win, singular basis functions are extremely helpful.

19.4 Special Basis Functions, II: Wave Scattering

The scattering of waves incident upon a re¬‚ector is another problem where special radia-

tion functions are very useful (Boyd, 1990d). To illustrate this idea, we solve the quantum

mechanics problem of the scattering of a sine wave by a potential well in one dimension.

¨

The physical background is described in Morse and Feshbach (1953). The Schrodinger

2