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often used in a narrower sense to label elements which are discontinuous at inter-
element boundaries.
IMBRICATE SERIES: An in¬nite series for a spatially periodic function which is the su-
perposition of an in¬nite number of evenly spaced identical copies of a “pattern”
function A(x). It may be shown that all periodic functions have imbricate series in
addition to their Fourier expansions, and often the imbricate series converge faster.
Imbricate series may be generalized to an arbitrary number of dimensions.
IMPLICIT (TIME-MARCHING): A time-integration scheme in which it is necessary to
solve a boundary-value problem to compute the solution at the new time level. The
Crank-Nicholson method is an example. (Chap. 12.)
IMPLICITLY-IMPLICIT: A label for a time-dependent partial differential equation in which
the time derivative is multiplied by a differential operator L so that it is necessary to
solve a boundary value problem at every time step even to apply an explicit time-
marching scheme. (Chap. 9.)
INFINITE ORDER CONVERGENCE: A spectral series possesses the property of “in¬-
nite order convergence” if the error decreases faster than any ¬nite inverse power
of N as N , the number of terms in the truncated series, increases. A synonym for
“exponential convergence”.
INTERPOLANT: An approximation PN ’1 (x) whose free parameters or coef¬cients are
chosen by the requirement that
f (xi ) = PN ’1 (xi ) i = 1, . . . , N
at a set of N grid points. The process of computing such an approximation is INTER-

ISOLA: For a nonlinear equation that depends upon two parameters, ± and „ , an “isola”
is a one-parameter family of solutions, parameterized by some quantity s, such that
(±(s), „ (s)) is a simple closed curve in the ±’„ plane. (From Latin “insula”, “island”,
through Italian “isolato”, “isolate”.)

JURY PROBLEM: Numerical slang for a boundary value problem in one or more dimen-
sions. The reason for the name is that when discretized by any reasonable difference
or spectral scheme, the solution must be computed on the entire grid simultaneously.
All the boundary and interior values render a “verdict” on the solution at each point.

LANCZOS-TAU METHOD: Technique for solving differential equations which computes
an exact solution to an approximate differential equation. (The differential equation is
perturbed in such a way that an exact solution of the perturbed problem is possi-
ble.) This philosophy is opposite to the usual strategy of computing an approximate
solution to the original, exact differential equation. (Chap. 21.)

LIMIT POINT: A point where a solution u(±) of a nonlinear equation curves back so that
there are two solutions for ± on one side of ± = ± limit and no solutions on the other
side of the limit point. As the limit point is approached, du/d± ’ ∞. Special methods
[“pseudoarclength continuation” or “globally convergent homotopy”] are needed to
“turn the corner” and march from the lower branch through the limit point onto the
upper branch or vice versa. Synonyms are “FOLD POINT”, “TURNING POINT”,

MARCHING PROBLEM: Arithmurgical slang for a time-dependent problem. It is not
necessary to compute the solution through the entire space-time plane at once. In-
stead, one can march from one time level to the next, computing over all space at
¬xed t independently of the unknown future. See also JURY PROBLEM.

COLLOCATION. (Now rare.)

MMT: MATRIX MULTIPLICATION TRANSFORM: This is N -point interpolation using
N -point Gaussian quadrature or summation of an N -point spectral series at each of
the N grid points. In either direction, the transform is computed by multiplying
an N -dimensional column vector by an N -dimensional square matrix. (PMMT is the
same except that the transform is split into two multiplications by (N/2)-dimensional
square matrices to exploit parity. (Chap. 10.)

NATURAL BOUNDARY CONDITION: A boundary condition is “natural” if it need not
be explicitly imposed, but rather is implicitly satis¬ed by the choice of basis functions.
Examples include periodicity (implicit in Fourier basis) and analyticity at an endpoint
where the coef¬cients of the differential equation are singular (implicit in Chebyshev

NON-INTERPOLATING: (i) Adjective used to label that family of spectral methods which
do not employ an auxiliary interpolation grid to compute the series coef¬cients.
Galerkin™s method and the Lanczos-tau method are examples (ii) Label for a semi-
Lagrangian time-marching scheme invented by Ritchie (Chap. 14.)

NUMERICAL BOUNDARY CONDITION: A constraint such as u(’1) = ’0.5 which in-
volves a number. It is always necessary to modify either the basis set or the Galerkin
or pseudospectral matrix to enforce such conditions.
Glossary 583

ORTHOGONAL COLLOCATION: A collocation method in which the grid points are the
roots or extrema of the basis functions. Synonym for PSEUDOSPECTRAL.
PARITY: A symmetry of functions such that either f (x) = f (’x) for all x (“SYMMETRIC
with respect to the origin”) or f (x) = ’f (’x) for all x (“ANTISYMMETRIC with
respect to the origin”. A symmetric function is said to be of “even parity” while an
antisymmetric function is said to be of “odd parity”.
PERIODIC: A function f (x) is “periodic” with period L if and only if
f (x + L) = f (x)
for all x.
PMMT: PARITY MATRIX MULTIPLICATION: Same as the MMT except parity is ex-
ploited by taking sums and differences of the input vector to compute its symmetric
and antisymmetric parts and each is transformed separately by a matrix-vector mul-
tiplication. The two transforms are combined to give the transform of the original
unsymmetric series or set of grid point values. (Chap. 10.)
PRECONDITIONING: A technique for accelerating the convergence of an iteration for
solving Λx ’ f by changing the iteration matrix to H ’1 Λ. The matrix H is the “pre-
conditioning matrix” and is chosen to be an approximation to Λ (in the sense of hav-
ing approximately the same eigenvalues), but is much less expensive to invert than Λ.
(Chap. 15.)
PSEUDOSPECTRAL: an algorithm which uses an interpolation grid to determine the co-
ef¬cients of a spectral series. Synonyms are ORTHOGONAL COLLOCATION &
RESIDUAL FUNCTION: When an approximate solution uN is substituted into a differ-
ential, integral, or matrix equation, the result is the RESIDUAL function, usually de-
noted R(x; a0 , a1 , . . . , aN ). The residual function would be identically zero if the
approximate solution were exact. (Chap. 1.)
RULE OF THREE NAMES: Every term in this glossary has at least two synonyms.
SEMI-IMPLICIT (TIME-MARCHING): a time-integration method that treats some terms
implicitly and others explicitly. Such algorithms are very common in hydrodynam-
ics where diffusion is treated implicitly but the nonlinear advection is explicit; this
avoids inverting large systems of nonlinear algebraic equations at each time step.
SEMI-LAGRANGIAN: A family of time-marching algorithms for ¬‚uid mechanics which
use the method of characteristics to integrate the advection terms while employing
conventional strategies for the rest. Excellent shock resolution; now widely employed
in numerical weather forecasting. (Chap. 14.)
SEPARABLE: An adjective applied to a partial differential (PDE) equation which denotes
that the PDE can be split into a sequence of ordinary differential equations including
Sturm-Liouville eigenproblems. The procedure for reducing a PDE to ODEs is the

SKEW-SYMMETRIC: A matrix A is SKEW-SYMMETRIC if and only if its tranpose is its
negative, that is, A = ’A. All eigenvalues of a skew-symmetric matrix are pure
imaginary or zero. Similarly, an operator is skew-symmetric if its adjoint is its nega-
tive; it, too, has pure imaginary eigenvalues.

SKYLINE SOLVER: An algorithm for the LU FACTORIZATION of a SPARSE matrix which
omits nonzero operations. It is a generalization of a BANDED matrix solver because
the “skyline” allows for different numbers of nonzero elements in different rows (Ap-
pendix B).

SPARSE MATRIX: A matrix whose elements are mostly zero. The opposite of a DENSE

SPECTRAL: (i) A catch-all term for all methods (including pseudospectral techiques) which
expand the unknown as a series of global, in¬nitely differentiable expansion func-
tions. (ii) In a narrower sense, it denotes those algorithms that use only the expansion
functions and their integrals and recurrence relations (as opposed to pseudospectral
methods, which need the values of the functions on a grid of interpolation points).
[De¬nition (ii) is not used in this book.]

SPECTRAL BLOCKING: The (spurious) accumulation of energy at or near the smallest
resolved scales. Common in nonlinear hydrodynamics; also arises when the CFL
limit is violated only for the shortest resolvable scales. The signature of spectral
blocking is that the log |an | rises instead of falls with n near n = N , the truncation
limit. The usual remedy is to add a little dissipation which is highly scale-selective,
damping only wavenumbers or coef¬cients close to the truncation limit. (Chap. 11.)

SPECTRAL ELEMENT METHOD: A technique in which the computational domain is
subdivided into several subintervals or blocks and then separate spectral series are
employed on each interval. It differs from the SUBDOMAIN method only by em-
ploying a variational principle and integration-by-parts to more closely parallel ¬nite
element algorithms.

SPECTRAL RINGING: Spurious, small scale oscillations which appear when a function
with a discontinuity or discontinuous slope is approximated by a truncated spectral
series; synonym for GIBBS™ OSCILLATIONS.

SPONGE LAYER: To absorb waves near the ends of a ¬nite interval y ∈ [’L, L] when
domain truncation is used to simulate an in¬nite spatial interval, a large arti¬cial
viscosity is often added in the neighborhood of y = ±L. The regions of large damping
are called “sponge layers” because they absorb the waves so that the center of the
domain is uncorrupted by spurious waves re¬‚ected from y = ±L.

STATIC CONDENSATION: A direct method for solving the matrix equations which re-
sult from ¬nite element or spectral element discretizations. As a ¬rst step, the degrees
of freedom internal to each element are eliminated in favor of the grid point values
on subdomain walls. This generates a dense matrix of much smaller size than the
original sparse matrix, which is then solved by standard LU factorization.

SUBDOMAIN SPECTRAL METHOD: Instead of solving the problem by using a single
series which is valid over the whole domain, the computational region is split into
several smaller regions or “subdomains”. A separate Chebyshev or Legendre expan-
sion is then de¬ned on each subdomain, and the expansions are matched together at
the subdomain boundaries. There are many variants of this strategy, depending upon
whether a variational principle is or is not used and also upon how the algorithm en-
forces continuity from one subdomain to the next. See also SPECTRAL ELEMENTS.
(Chap. 22.)
Glossary 585

SUBGEOMETRIC CONVERGENCE: There are three equivalent de¬nitions. (i) The se-
ries converges exponentially fast with n, but too slowly for the coef¬cients to be
bounded in magnitude by c exp(’pn) for any positive constants c and p. (ii) The
index of exponential convergence r < 1. [r = 1 is “geometric convergence”.]

lim log(|an |)/n = 0

Subgeometrically-converging series are the expansions of functions which are singu-
lar but in¬nitely differentiable at some point (or points) on the expansion interval
(often at in¬nity). In mathematical jargon, such weakly singular functions “are in
C ∞ , but not C „¦ .”
SUPERGEOMETRIC CONVERGENCE: If the coef¬cients can be bounded by

|an | ¤ O{s exp[’(n/j) log n]}

for suf¬ciently large n and some constants s and j, then the convergence rate is “su-
pergeometric”. This is possible only when the function being expanded is an “entire”
function, that is, one which has no singularities anywhere in the complex plane ex-
cept at ∞.
SUPERIOR LIMIT: For a sequence {an }, the superior limit or supremum limit is writ-
ten lim sup{an } and denotes the lower bound of the almost upper bounds of the se-
quence. (A number is an almost upper bound for a sequence if only a ¬nite number
of members of the sequence exceed the “almost upper bound” in value.) Strictly
speaking, de¬nitions of convergence rates should be expressed in terms of superior
limits, rather than ordinary limits, to allow for oscillations and zeros in the sequence
as n ’ ∞. A synonym is “supremum limit”.
SYMMETRIC: (i) [Of a matrix]: Aij = Aji . (Appendix B.) The longer but more precise
term “CENTROSYMMETRIC” is sometimes used as a synonym. (ii) [Of a function
f (x)]: f (x) = f (’x) for all x. (Chap. 8.)
TAU-METHOD: A mean weighted residual method for which the weighting functions are
the Chebyshev polynomials. Also called the LANCZOS TAU-METHOD. (Chapt. 21.)
TENSOR PRODUCT BASIS: A multi-dimensional basis whose elements are the products
of one-dimensional basis elements. In two dimensions,

¦mn (x, y) ≡ φm (x) ψn (y)

TENSOR PRODUCT GRID: A multi-dimensional grid whose M N points are formed from
the corresponding one-dimensional grids:

xij ≡ (xi , yj ) i = 1, . . . , M & j = 1, . . . , N

TRUNCATION ERROR: The error made by neglecting all coef¬cients an in the spectral
series such that n > N for some “truncation” N .
WEAKLY NONLOCAL SOLITARY WAVE: A steadily-translating, ¬nite amplitude wave
that decays to a small amplitude oscillation (rather to zero) as one moves away from
the core of the disturbance.

adaptive coordinate transformations, see bifurcation point, 544
coordinate transformations,adaptive boundary conditions, 10
algebraic convergence absorbing, 341
calculation of the index for ODE so- basis recombination, 64, 112“115
lution, 43 behavioral versus numerical, 75, 109“
examples 111, 361
half-wave recti¬er function, 22 boundary-bordering, 111“112
sawtooth function, 21 coordinate transformation of bound-
index of, de¬ned, 25 ary conditions on derivatives, 575“
algebraic manipulation language, see sym- 576
bolic manipulation language and homogenization of boundary condi-
spectral methods tions, 112“114
aliasing polar coordinates, at origin, 383
de¬ned, 204 boundary layer, 57
Resolution Rule-of-Thumb, 59
equality on the grid (=), 205
symbolic manipulation example, 465
two-h waves, 206
aliasing instability
cardinal functions
dealiasing, 211“216
general orthogonal polynomials, 104
energy-conserving schemes and, 213“
polynomial, de¬ned, 83
pseudospectral Chebyshev ODE-solving,
history, 210
skew-symmetric advection and, 213
Whittaker, see sinc function
theory, 216“218
Chebyshev polynomials
Two-Thirds Rule for Dealiasing, 211“
asymptotic coef¬cients, calculation of,


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