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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-

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