2. “Algebraic maps”

Lx

y=√ x ∈ [’1, 1] (17.34)

,

1 ’ x2

3. “Exponential maps”

t ∈ [’π, π] (17.35)

y = sinh(Lt),

where y ∈ [’∞, ∞]. The names for these families of maps are chosen by how rapidly y

increases with x ’ ±1.

Grosch & Orszag(1977) and Boyd (1982a) were big boosters of algebraic mappings.

These lead to the rational Chebyshev functions, described in the next seven sections. Their

strength is that they are “minimalist” mappings, creating a not-very-violent change from

one coordinate to the other, and they have optimal properties in the limit N ’ ∞ for ¬xed

L.

Nevertheless, the other two families of mappings are useful, too. The logarithmic map-

pings have the disadvantage that well-behaved functions on the in¬nite interval are turned

into nasty functions on x ∈ [’1, 1]. For example, the mapping y = arctanh(x) transforms

sech± (y) ’ (1 ’ x2 )± (17.36)

The Hermite and rational Chebyshev series for the hyperbolic secant function converge

exponentially fast for any positive ±, even non-integral ±. However, the function (1 ’ x2 )±

has branch points at both endpoints unless ± is an integer.

The arctanh mapping is useful almost entirely in the reverse direction, transforming

functions with endpoint singularities at the ends of a ¬nite interval into well-behaved func-

tions on the in¬nite interval. One can then apply a standard in¬nite interval basis to obtain

very accurate solutions to singular problems (Chap. 16, Secs. 5).

There is one exception: in the theory of solitary waves and of shocks, there are many

perturbation series which involve only powers of the hyperbolic secant and/or tangent.

The arctanh mapping (with L = 1) transforms these polynomials in hyperbolic functions

of y into ordinary polynomials in x, making it easy to compute these expansions to high

order (Boyd, 1995a, 95j).

The exponential maps go so far to the opposite extreme that they are inferior to the

algebraic maps in the limit N ’ ∞. Cloot and Weideman(1990, 1992), Weideman and

Cloot(1990) and Boyd (1994b) have shown that the Weideman and Cloot map (17.35) is

CHAPTER 17. METHODS FOR UNBOUNDED INTERVALS

356

nevertheless extremely ef¬cient in real world situations because the asymptotic limit is

approached very, very slowly with N . We will illustrate the effectiveness of the Weideman-

Cloot change-of-coordinate later in the chapter. First, though, we shall turn to algebraic

mappings where there has been much wider experience.

17.7 Algebraically Mapped Chebyshev Polynomials: T Bn (y)

These basis functions are de¬ned on the interval y ∈ [’∞, ∞] by

T Bn (y) ≡ Tn (x) ≡ cos(nt) (17.37)

where the coordinates are related via

Lx y

√ (17.38a)

y= ; x=

1 ’ x2 L2 + y 2

= arccot(y/L) (17.38b)

y= L cot(t) ; t

The ¬rst 11 T Bn (y) are listed in Table 17.5. As one can see from the table or from the

right member of (17.38a), the T B2n (y) are rational functions which are symmetric about y =

0. The odd degree T B2n+1 (y) are antisymmetric and are in the form of a rational function

divided by a left-over factor of L2 + y 2 . We shall refer to all these basis functions as

the “rational Chebyshev functions on an in¬nite interval” even though the odd degree

members of the basis are not rational. Fig. 17.4 illustrates the ¬rst four odd T Bn (y). Note

that the map parameter merely changes the y-scale of the wave (the oscillations become

narrower when L increases) without altering the shape, so these graphs for L = 1 apply for

all L if we replace y by (y/L).

Figure 17.4: Graphs of the ¬rst four rational Chebyshev functions on x ∈ [’∞, ∞]. Because

the odd degree T Bn (y) are all antisymmetric about y = 0, they are illustrated only for

positive y. All asymptote to 1 as y ’ ∞.

17.7. RATIONAL CHEBYSHEV FUNCTIONS: T BN 357

Table 17.4: A Selected Bibliography of Rational Chebyshev Functions

References Comments

Algebraic map: in¬nite interval in y to x ∈ [’1, 1]

Grosch&Orszag(1977)

Tn (x) basis, equivalent to rational Chebyshev in y

Boyd (1982a) Steepest descent asymptotics for spectral coef¬cients

Christov (1982) Rational functions related to T Bn ;

application to rational solution of Burgers-like model

Cain&Ferziger&Reynolds(1984) Map in¬nite interval to ¬nite through y = Lcotan(x)

and apply Fourier in x; equivalent to T Bn (y)

Boyd (1985a) Change-of-coordinate to detour around interior

singularity into the complex plane

Boyd (1987a) Theory and asymptotics for T Bn

Boyd (1987b) Theory & examples for semi-in¬nite interval: T Ln

Boyd (1987c) Quadrature on in¬nite and semi-in¬nite intervals

Tensor product of T Ln (z) — T Bm (y)

Lin & Pierrehumbert (1988)

for two-dimensional baroclinic instability

Boyd (1990a) Theory; relation of T Bn to other rational basis sets

Christov&Bekyarov(1990), Nonlinear eigenvalue (soliton); basis is cousin of T Bn

Bekyarov&Christov(1991)

Boyd (1990b,1991a,1991d, T Bn /radiation function basis for nonlocal solitons

1991e,1995b,1995j)

Boyd (1990d) T Bn /radiation function basis for quantum scattering

(continuous spectrum)

Boyd (1991d,1995a) T Bn for nonlinear eigenvalue (soliton)

Weideman&Cloot(1990) Comparisons with the Weideman-Cloot sinh-mapping

Cloot (1991) Comparisons with solution-adaptive mapping

Cloot&Weideman (1992) Comparisons with the Weideman-Cloot sinh-mapping;

adaptive algorithm to vary the map parameter with time

Liu&Liu&Tang(1992,1994) T Bn to solve nonlinear boundary value problems for

heteroclinic (shock-like) & homoclinic (soliton-like) solutions

Falqu´ s&Iranzo(1992)

e Edge waves in the ocean with T Ln

Boyd (1993) Algebraic manipulation language computations

Chen (1993) Eigenvalues (hydrodynamic stability) using T Ln

Weideman(1994a,1994b,1995a) Error function series, good for complex z, using

basis {(L + iz)/(L ’ iz)}n , n = ’∞, ∞

Hilbert Transform computation via {(L + iz)/(L ’ iz)}n

Weideman(1995b)

Compares well with alternative Fourier algorithm.

Boyd (1996c) Legendre, quantum and tidal equations;

traps and snares in eigencalculations

Gill&Sneddon(1995,1996a,1996b) Complex-plane maps (revisited) for eigenfunctions

singular on or near interior of (real) computational domain

Sneddon (1996) Complex-plane mappings for a semi-in¬nite interval

Yang&Akylas(1996) T Bn /radiation function basis for nonlocal solitons

Matsushima&Marcus(1997) Polar coordinates: radial basis is rational functions

which are images of associated Legendre functions

to defeat the “pole problem”

CHAPTER 17. METHODS FOR UNBOUNDED INTERVALS

358

Table 17.5: Rational Chebyshev functions for the in¬nite interval: T Bn (y).

(For map parameter L = 1).

n T Bn (y)

[Symmetric about y = 0]

0 1

(y 2 ’ 1)/(y 2 + 1)

2

(y 4 ’ 6y 2 + 1)/(y 2 + 1)2

4

(y 6 ’ 15y 4 + 15y 2 ’ 1)/(y 2 + 1)3

6

(y 8 ’ 28y 6 + 70y 4 ’ 28y 2 + 1)/(y 2 + 1)4

8

(y 10 ’ 45y 8 + 210y 6 ’ 210y 4 + 45y 2 + 1)/(y 2 + 1)5

10

[Antisymmetric about y = 0]

y/(y 2 + 1)1/2

1

y(y 2 ’ 3)/(y 2 + 1)3/2

3

y(y 4 ’ 10y 2 + 5)/(y 2 + 1)5/2

5

y(y 6 ’ 21y 4 + 35y 2 ’ 7)/(y 2 + 1)7/2

7

y(y 8 ’ 36y 6 + 126y 4 ’ 84y 2 + 9)/(y 2 + 1)9/2

9

17.7. RATIONAL CHEBYSHEV FUNCTIONS: T BN 359

Figure 17.5: Schematic showing the mapping relationships between the rational functions

T Bn (y), the Chebyshev polynomials Tn (x), and the Fourier functions cos(nt). The large

arrows indicate the mappings that transform one set of basis functions into another. Each

series converges within the cross-hatched region; the mappings transform the boundaries

of one region into those of another.

Fig. 17.5 shows the mappings between y, x, and t and the relationships between the

basis functions in each coordinate and their domains of convergence. The series converges

within the largest shaded area that does not contain a singularity of the function being

expanded.

The rational Chebyshev functions are orthogonal on [’∞, ∞] with the weight function

(for L = 1) of 1/(1 + y 2 ). However, the easiest way to program is to use the trigonometric

representation. The necessary derivative conversion formulas are given in Appendix E.

The earliest use of these functions (without a special notation) is Boyd (1982a). How-

ever, Grosch & Orszag (1977) implicitly used the same basis by mapping to the interval

x ∈ [’1, 1] and applying ordinary Chebyshev polynomials in x. The most complete treat-

ment is Boyd (1987a).

These orthogonal rational functions have many useful properties. For example, if the

differential equation has polynomial or rational coef¬cients, one can show [by converting the

equation from y to the trigonometric argument t and using trigonometric identities] that

the corresponding Galerkin matrix will be banded. A fuller discussion is given in Boyd

(1987a).

The rate of convergence of the T Bn (y) series is normally exponential but subgeomet-

ric. In the same way that the convergence domain of both sinc and Hermite expansions

is a symmetric strip parallel to the real y-axis, the domain of convergence for the rational

Chebyshev functions is the exterior of a bipolar coordinate surface in the complex y-plane

with foci at ±iL as shown in the left panel of Fig. 17.5. In words, this means that if f (y) is

rational with no singularity at ∞, then the T Bn (y) expansion of f (y) will converge geomet-

rically. The closer the poles of f (y) are to those of the mapping at ±iL, the more rapid the

convergence.

In practice, geometric convergence is rare because the solutions of most differential

CHAPTER 17. METHODS FOR UNBOUNDED INTERVALS

360

Figure 17.6: Graphical construction, devised by Mark Storz, of the interpolation points

for the rational Chebyshev functions which are orthogonal on y ∈ [’∞, ∞]. The evenly

spaced grid points on the semicircle of unit radius (dots) become the roots of the T Bn (y)

(—™s) on the line labelled “y”. L is the map parameter

equations are singular at in¬nity. This limits the convergence domain of the T Bn (y) to the

real y-axis and reduces the convergence rate from geometric to subgeometric. Nonetheless,

the rational Chebyshev functions are still an extremely useful basic set, easy to program

and just as ef¬cient as the Hermite and sinc expansions in almost all cases.

Fig. 17.6 is Storz™ graphical construction of the interpolation grid. Note that ” in con-

trast to Chebyshev polynomials ” the smallest distance between adjacent grid points is

O(1/N ). This implies that the explicit time-stepping restrictions will be no more severe for

these orthogonal rational functions than when ¬nite differences replace them as proved by

Weideman (1992).

He shows that rational Chebyshev differentiation matrices have good properties: like

those of Hermite functions, the Galerkin matrices are banded and the ¬rst derivative is

skew-symmetric while the second derivative is symmetric. The corresponding collocation

matrices are full matrices without symmetry. However, Weideman argues that the pseu-

dospectral differentiation matrices are “asymptotically normal” in the sense that determin-

ing timestep limits from the eigenvalues of the differentiation matrices is legitimate.

In contrast, the Legendre polynomial differentiation matrices are highly non-normal.

The concept of “pseudoeigenvalues” is needed to realistically estimate timesteps (Trefethen

and Trummer, 1987, Trefethen, 1988, Reddy and Trefethen, 1990, and Trefethen et al., 1993).

Weideman (1992, 1994a, 1994b, 1995a, 1995b) is but one of several authors who ¬nd it

convenient to work with complex-valued forms of the rational Chebyshev functions. Chris-

tov (1982) and Christov and Bekyarov (1990), for example, use linear combinations of the

functions

1 (iy ’ 1)n

ρn ≡ √ n = 0, ±1, ±2, . . . (17.39)

,

π (iy + 1)n+1

These functions were introduced by Norbert Weiner as the Fourier transforms of Laguerre

functions (Higgins, 1977). Boyd (1990a) shows that with the change-of-coordinate y =

cot(t), these complex-valued functions are simply the terms of a Fourier series in t, or

the difference of two such functions, and thus are mathematically equivalent to the T Bn

de¬ned here or the SBn which will be de¬ned in the section after next. One may use

whatever form of the rational Chebyshev functions is convenient; we prefer the T Bn basis

de¬ned above.

Matsushima and Marcus (1997) have introduced an interesting variant: rational func-

tions which are the images of associated Legendre functions rather than Chebyshev poly-

17.8. BEHAVIORAL VERSUS NUMERICAL BOUNDARY CONDITIONS 361

nomials. Although this sacri¬ces the applicability of the Fast Fourier Transform, the ra-

tional associated Legendre functions, when used as radial basis functions in polar coor-

dinates, have no “pole problem”. This means that when using an explicit time-marching

algorithm, one can use a time step an order of magnitude larger than with a rational Cheby-

shev basis. For boundary value or eigenproblems, or when the time-marching method is

implicit, this advantage disappears, so the rational associated Legendre functions are use-

ful for one restricted but important niche: explicit time-integrations of unbounded domains

in polar coordinates.

17.8 Behavioral versus Numerical Boundary Conditions

In Chap. 6, Sec. 3, we brie¬‚y discussed two kinds of boundary conditions: “behavioral”

and “numerical”. When a problem is periodic, the boundary conditions are behavioral,

that is to say, we require that the solution has the behavior of spatial periodicity. We can-