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spectrum
of under-
spectrum
resolved
of energy-
fronts
containing
eddies




π /h
0 wavenumber k
Figure 11.11: Solid: spectrum of energy-containing eddies (“synoptic eddies” in Phillips™
simulation). Circles: Fourier coef¬cients of the unresolved fronts.


will spontaneously develop fronts too thin to be numerically resolved (if the viscosity is
very weak) and (ii) the energy spectrum will fall off as O(k ’3 ) for wavenumbers smaller
than kdiss , the beginning of the dissipation range, where k is the total wavenumber. For
the Fourier coef¬cients (as opposed to the energy, whose spectrum is proportional to the
absolute value squared of the Fourier coef¬cients), this implies |ak | ∼ O(1/k 3/2 ). The crucial
point is that the Fourier coef¬cients of a jump discontinuity (like a front or the “sawtooth”
function) decrease more slowly as O(1/k). The implication is that the fronts have rather
small energy, or otherwise the energy spectrum would decay as O(1/k 2 ).
Thus, it is possible for the underresolved small features to contain little energy com-
pared to large, well-resolved features. Thus, spectral blocking in Phillips™ model could be
described (conjecturily) by Fig. 11.11.
On the other hand, Brown and Minion(1995), Minion and Brown(1997) and the stud-
ies reviewed by Aoyagi (1995) have identi¬ed some speci¬c computational instabilities.
These seem to occur only for marginally-resolved or poorly-resolved ¬‚ow, thank good-
ness. When sideband resonance is the villain, the bland statement “too little resolution”
seems insuf¬cient. Perhaps with a little tinkering of the algorithm, it might be stabilized
against sideband resonace (or whatever) so that the integration could be extended without
a shift to a ¬ner spatial grid.
The theory of aliasing instability and blow-up is on the whole in a pretty sorry state. It
matters because many important ¬‚ows can only be marginally resolved, forcing the use of
energy-conserving algorithms, dealiasing and so on. We could ¬ght the instability much
better if we only understood what we were ¬ghting!


11.9 Summary
Spectral blocking, 2h-waves and blow-up are as much a problem today as in the time of
Phillips™ ¬rst GCM experiment. The bad news is that despite Phillips™ own demonstration
11.9. SUMMARY 219


s


log ak
me
;
r h Sche
lle
ma ptive
Sa
Ad

l
/Artificia
Filtering y
Viscosit
t

metric
w-sym
Ske ction
adve
log ak




Spectral Energy-conserving
Blocking
t
Front-tracking: PPM
FCT,etc.



Blow-Up
Dealiasing
; 2/3 Rule




Figure 11.12: Schematic of the time evolution of a model from a smooth initial condition
(top left) to a ¬‚ow that has lost some accuracy to spectral blocking (middle left) before
blowing up (bottom left) and six types of remedies for spectral blocking and blow-up (right
arrows).


of the wrong-way energy cascade created by aliasing, the causes of blow-up are still not
well understood. The good news is that we have a quiver full of arrows to shoot at the
problem (Fig. 11.12).
The ¬rst arrow is to increase the resolution. Phillips found that this didn™t help very
much, buying only a few hours of additional time. However, this may only re¬‚ect the fact
that frontogenesis happens very rapidly once the eddies have grown to a reasonable size. It
is dangerous to always blame blow-up on a numerical villain, a computational Bogeyman
named Aliasing Instability. It may just be that the ¬‚ow is physically generating features
whose width is 1/1000 of the size of the computational domain, and a model with N = 100
fails simply because it can™t resolve it.
Adaptive schemes are very useful in this context because often a ¬‚ow can be resolved
with rather small N over much of its lifetime before the front develops. With a variable
resolution, that is, a ¬ne grid around the front only, the narrow features may be resolved
with only moderate N . Cloot, Herbst, and Weideman (1990), Bayliss, Gottlieb, Matkowsky
and Minkoff (1989), Bayliss and Turkel(1992), Bayliss, Garbey and Matkowsky (1995) and
CHAPTER 11. ALIASING, SPECTRAL BLOCKING, & BLOW-UP
220

Augenbaum (1989, 1990) are illustrations.
The second arrow is ¬ltering or the use of an arti¬cial, computational viscosity (Chapter
18, Sec. 20).
The third arrow is skew-symmetric advection. This forces the numerical discretization
of advection to provide pure translation without adding a little bit of spurious numerical
growth or decay. It seems to be helpful with marginally-resolved ¬‚ows.
The fourth arrow is an energy-conserving algorithm. Canuto et al. (1988, Sec. 4.6) give
an example of a magnetohydrodynamic ¬‚ow where this is helpful. Energy-conserving
schemes are likely to be useful for climate modelling, too, where the ¬‚ow is integrated
over a very long period of time. Symplectic integration schemes for Hamiltonian systems
conserve not only energy but the Hamiltonian structure.
The ¬fth arrow is to use a front-resolving scheme like PPM or MUSCL or Flux-Corrected
Transport (FCT) or the like. Spectral generalizations of most of these algorithms are now
available.
The sixth arrow is the Two-Thirds Rule. This is really just a special case of ¬ltering, but
we list it separately because it is often discussed in the literature as something separate
with a different motive from conventional ¬lters: to dealias a quadratically nonlinear com-
putation. However, Orszag™s procedure discards 1/3 of the numerical degrees of freedom
in one dimension, 5/9 in two dimensions and 19/27 in three dimensions. One pays a very
heavy price just to eliminate aliasing, so this is probably a last resort.
Note that the Two-Thirds Rule is an “all-or-nothing” ¬lter: the coef¬cients are unmod-
i¬ed or completely discarded. To resolve fronts and other narrow features, ¬lters which
vary smoothly and continuously with wavenumber have become quite popular and success-
ful. Experience and some theory suggests that a smooth ¬lter is better than a step function
(in wavenumber) like the Two-Thirds Rule. Since the higher wavenumbers are clobbered
by the ¬lter anyway, one is likely to gain most of the bene¬ts of dealiasing.
There are two important things to remember before applying these remedies. The ¬rst
is that for a well-resolved ¬‚ow, none of them are needed! A spectral code will do a very
good job of preserving the energy over a fairly long time simply because of its inherent ex-
ponential accuracy. It is only when the ¬‚ow develops fronts, shocks and other pathologies
that one ever needs to think about tinkering with the algorithm to enforce conservation of
the discrete energy. Otherwise, one is incurring extra cost, both your time and the com-
puter™s, for no particular gain.
The second point is that when these remedies succeed, they succeed dangerously. That
is to say, these interpolation-with-physics-constraints force the computed ¬‚ow to be smooth
and the energy to stay constant even when the numerical errors are large. If the eddies have
a characteristic scale of 1000 km, it is clear that a very coarse grid with h=5000 km cannot
possibly furnish an accurate solution. However, with an Arakwa-type energy-conserving
scheme, the numerical solution won™t blow up. Indeed, the dreadfully-underresolved so-
lution may look quite smooth, even though it™s nonsense.
For his thesis, a colleague ported the energy-conserving Arakawa-Mintz model from
the atmosphere to the ocean. Unfortunately, because ocean eddies have a diameter of O(100
km),which is very small compared to the width of an ocean basin, it was not possible to
explicitly resolve the eddies in 2 B. C. [Before Cray2 ]. However, it was possible to make a
short run at higher resolution. The horror! The horror! With smaller h, everything changed!
Nevertheless, the production run was published. Although it certainly had major er-
rors, my friend was unable to convince his advisor of this because the Arakawa scheme
(and viscosity) made the computed solutions look smooth even though the ¬‚ows were actu-
ally rubbish.
2 The Cray-1, the ¬rst supercomputer, was introduced in 1976.
11.9. SUMMARY 221

In contrast, a code which is resolving a ¬‚ow ought to approximately conserve energy
for a long time, even if this property is not explicitly built into the discretization. If the code
isn™t resolving the ¬‚ow, well, perhaps it ought to blow up.
Smoothness and energy conservation are like the shining glitter of gold. Perhaps it
really is gold, or perhaps it is only iron pyrites, “fool™s gold”, which gleams as brightly
but is worth almost nothing. The number-cruncher must be as suspicious of glitter as the
mineralogist.
Chapter 12

Implicit Time-Marching, the Slow
Manifold and Nonlinear Galerkin
Methods


“In the terminology which you graciously ascribe to me, we might say that
the atmosphere is a musical instrument on which one can play many tunes.
High notes are sound waves, low notes are long inertial [Rossby] waves, and
nature is a musician more of the Beethoven than of Chopin type. He much
prefers the low notes and only occasionally plays arpeggios in the treble and
then only with a light hand. The oceans and the continents are the elephants in
Saint-Saens™ animal suite, marching in a slow cumbrous rhythm, one step every
day or so. Of course there are overtones: sound waves, billow clouds (gravity
waves), inertial oscillations, etc., but these are unimportant and are heard only
at N. Y. U. and M. I. T.”

“Jule Charney (1917-1982) (from a 1947 letter to Philip Thompson)




12.1 Introduction
When the partial differential equation is linear and the “method of lines” is applied to
discretize the spatial dependence, the system of ordinary differential equations in time is
linear, too, and can be written
du
(12.1)
= Λu + f (t)
dt

where Λ is a large square matrix of dimension N where N is the number of elements in the
column vector u and where f (t) is the discretization of the inhomogeneous terms in the
differential equation. (We assume that the boundary conditions have been dealt with by
basis recombination or whatever; the vector f may contain terms proportional to inhomo-
geneous boundary conditions in some or all of its elements.)

222
12.1. INTRODUCTION 223

The “con¬guration space” of the model is just the N -dimensional space of the vector
u, the spectral coef¬cients or grid point values. The instantaneous state of the model is
a point in con¬guration space; its evolution through time is a one-dimensional curve in
con¬guration space, parameterized by the single coordinate t.
Eq.(12.1) is completely general and applies whether the spatial discretization is spectral,
¬nite difference or ¬nite element and also whether the unknowns are grid point values or
Chebyshev polynomial coef¬cients. In general, the elements of the matrix Λ are functions
of time. In most of this chapter, however, we shall make the further assumption that Λ is
independent of time.
In a world which is highly nonlinear and time-dependent, these assumptions would
seem drastic. Indeed, such approximations cannot capture the mysteries of solitary waves,
chaos, aliasing instability and a wide variety of other phenomenon. Nevertheless, such a
simpli¬ed model as (12.1) can teach us much about time-marching algorithms. The reason
is that accurate time-stepping requires a time step which is short compared to the time
scales on which the coef¬cients of the differential equation and u itself are changing. This
implies that the matrix Λ will not change much between then and time t = t0 + „ where „
is the timestep; otherwise, we should shorten the timestep.
The conceptual advantage of the linear-with-time-independent coef¬cient approxima-
tion is that the linear, constant coef¬cient ODE system (12.1) has the general solution
t
(12.2)
u = exp(Λt)u0 ) + exp(Λt) exp(’Λs) f (s) ds

where u0 ≡ u(t = 0). The matrix exponential is most easily interpreted by expanding u in
terms of the eigenvectors of the square matrix Λ. If

(12.3)
Λej = »j ej

and
N
(12.4)
u= bj ej
j=1


then in terms of the eigenvector basis, Λ becomes a diagonal matrix, with its eigenvalues
»j as the diagonal elements, and the ODE system collapses to the uncoupled set of ordinary
differential equations

dbj
(12.5)
= »j bj + gj (t)
dt

where gj (t) is the j-th coef¬cient of the eigenexpansion of the forcing function f (t).
It follows that we can learn almost everything about time-marching schemes “ to the
extent that neglecting nonlinearity and other time-dependence of the coef¬cients of the
PDE is legitimate “ by understanding how different algorithms integrate

du
(12.6)
= »u + f (t)
dt
where all the unknowns are now scalars, not vectors. For a simple wave equation like

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. 47
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