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N
dj f j
1
uslow (T ; ) ∼ ’ (12.60)
dT j
» »
j=0


This in¬nite series is noteworthy for a couple of reasons. First, the asymptotic equality
sign is used because this is an asymptotic series that diverges except for special cases.
However, when /» ∼ O(1/10) as in meteorology, one can use six to ten terms of the series
to obtain an extremely accurate approximation, and the fact that the error would grow with
additional terms has no practical relevance.
The second is that the asymptotic series does not contain an arbitrary constant of inte-
gration. The general solution to this ¬rst order ODE is

(12.61)
u(t) = A exp(»/ t) + uslow ( t; )

where A is an arbitrary constant determined by the initial condition. If we pick any old
u(0), then u(t) will be a mixture of fast and slow motion. Initialization onto the slow man-
ifold means setting u(0) = uslow (0; ) so that A = 0.
The method is almost the same for the coupled system of fast and slow modes, Eq.(12.54),
except for two differences. First, only the time derivative of the “fast” components is ne-
glected so that the “Machenhauer” approximation is

St + i ω S . — S = fS
i ωF . — F (12.62)
= fF
12.12. NONLINEAR GALERKIN METHODS 243

The system of ODEs has been replaced by a so-called “Differential-Algebraic” (DAE) set of
equations.
The second difference is that fS and fF are not pre-speci¬ed, known functions of time
but rather depend nonlinearly on the unknowns. Since the fast component is typically
small compared to the slow modes “ if it wasn™t one would hardly be integrating near
the slow manifold ” one can build the dependence on F into the perturbation theory.
However, the coupling between slow modes is often so strongly nonlinear than it cannot
be approximated. The DAE must usually be solved by numerical integration of the ODE
part plus algebraic relations which become increasingly complicated as one goes to higher
order.
Indeed, the dif¬culties of separating various orders in the perturbation theory esca-
late so rapidly with order that Tribbia(1984) suggested replacing perturbation by itera-
tion, which is easier numerically. With this simpli¬cation, the Baer-Tribbia method and its
cousins are in widespread operational use in numerical weather prediction to adjust raw
observational data onto the slow manifold. (See the reviews in Daley, 1991.) Errico (1984,
1989a) has shown that the Machenhauer approximation [lowest order multiple scales] is a
good description of the relative magnitudes of terms in general circulation models, which
include such complications as latent heat release, chemistry and radiative transfer, pro-
vided that the label of “fast” mode is restricted to frequencies of no more than a few hours.


12.12 Nonlinear Galerkin Methods
Although the Baer-Tribbia series and related algorithms have been used almost exclusively
for initialization, the split into fast and slow modes applies for all time and is the basis for
the following class of algorithms.
De¬nition 29 (NONLINEAR GALERKIN) A numerical algorithm for discretizing a partial
differential equation is called a “Nonlinear Galerkin” scheme if the system of ODEs in time that
results from a conventional spatial discretization is modi¬ed to a Differential-Algebraic Equation
(DAE) system by replacing the ODEs for the time derivatives of the fast modes by the method of
multiple scales (in time).
Nonlinear Galerkin methods can be classi¬ed by the order m of the multiple scales approxima-
tion which is used for the algebraic relations. NG(0) is the complete omission of the fast modes
from the spectral basis. NG(1) is the lowest non-trivial approximation, known in meteorogy as the
“Machenhauer approximation”. For the single linear forced ODE, du/dt = »u + f (t), the ¬rst
three approximations are

≡0 NG(0)
u
∼ ’ f /» NG(1)
u
∼ ’ f /» ’ df /dt/»2 NG(2) (12.63)
u

Kasahara (1977, 1978) experimented with forecasting models in which the usual spher-
ical harmonic basis was replaced by a basis of the Hough functions, which are the normal
modes of the linear terms in the model. This was a great improvement over the old quasi-
geostrophic models, which ¬ltered gravity modes, because the Hough basis could include
some low frequency gravitational modes. However, it was still only an N G(0) approxima-
tion: high frequency Hough modes were simply discarded.
Daley (1980) carried out the ¬rst experiments with NG(1) for a multi-level weather
prediction model. He found that the algorithm compared favorably with stanard semi-
implicit algorithms in both ef¬ciency and accuracy. Unfortunately, the Nonlinear Galerkin
method was no better than semi-implicit algorithms either. Haugen and Machenhauer
CHAPTER 12. IMPLICIT SCHEMES & THE SLOW MANIFOLD
244




Table 12.2: A Selected Bibliography of Nonlinear Galerkin Methods
Note: KS is an abbreviation for “Kuramoto-Sivashinsky” , FD for “¬nite difference”, “CN” for the Crank-
Nicholson scheme and “BDF” for Backward Differentiation time-marching schemes. Most articles with non-
spectral space discretizations are omitted.

Reference Area Comments
Kasahara (1977, 1978) Meteorology NG(0)
Daley (1980) Meteorology NG(1)
Daley (1981) Meteorology Review: normal mode
initialization
Foias et al.(1988) Reaction-diffusion Independent rediscovery;
& KS call it
“Euler-Galerkin” method
Marion & T´ mam(1989)
e Dissipative systems Independent rediscovery of NG
T´ mam (1989)
e Navier-Stokes Recursive calculation
high order NG series
Jauberteau, Rosier Fluid Dynamics
& T´ mam(1990a,b)
e
Jolly, Kevrekidis KS Eq. Computation of inertial manifolds;
& Titi(1990, 1991) Fourier
Dubois, Jauberteau Fluids Fourier, 2D, 3D
& T´ mam (1990)
e number of fast modes
adaptively varied
Foias et al.(1991) KS, forced Burgers Blow-up
T´ mam (1991b)
e Numerical analysis Stability analysis
T´ mam (1991a)
e Fluid Mechanics Review
Promislow Ginzburg-Landau Eq. numerical experiments;
& T´ mam(1991)
e higher order N G
T´ mam (1992)
e Review
Devulder & Marion(1992) Numerical Analysis
Margolin & Jones(1992) Burgers NG with spatial FD
Pascal & Basdevant(1992) 2D turbulence Improvements & ef¬ciency tests
Gottlieb & T´ mam(1993)
e Burgers Fourier pseudospectral
Foias et al.(1994) KS Blow-up
Garc´ ia-Archilla Numerical analysis Comparisons of NG
& de Frutos(1995) with other time-marching schemes
Garc´ia-Archilla(1995) Numerical analysis Comparisons of NG with other schemes
Boyd (1994e) Numerical Analysis Theory for inferiority
of NG to CN & BDF
Vermer&van Loon(1994) chemical NG(1) popular as PSSA:
kinetics “Pseudo-steady-state approx.”
Debussche&Dubois Homogeneous
&T´ mam(1995)
e turbulence
Jolly&Xiong(1995) 2D turbulence
Jones&Margolin Comparison with Galerkin;
&Titi(1995) forced Burgers & KS equations
Dubois&Jauberteau&Te´ mam(1998)
e Comprehensive review
12.13. WEAKNESSES OF THE NONLINEAR GALERKIN METHOD 245



1
NG(1)
0.8

0.6
BE
0.4

0.2
CN
0
0 0.2 0.4 0.6 0.8 1
µ„
Figure 12.7: Relative errors in approximating the time derivative of exp(i t) as a function
of and the time step „ . The relative error in the Nonlinear Galerkin method, which sim-
ply discards the time derivative, is one independent of the time step (horizontal line). The
errors in the Backwards Euler (BE) and Crank-Nicholson (CN) implicit methods are func-
tions of the product of with „ and decrease linearly (circles) and quadratically (asterisks)
as „ ’ 0. However, even for „ = 1 ” only 2π grid points in time per temporal period ”
the errors are still much smaller than N G(1).


(1993) also tried an NG scheme in combination with a semi-Lagrangian (SL) treatment of
advection. For long time steps (sixty minutes), the NG scheme was inaccurate unless two
Machenhauer-Tribbia iterations were used. This made the NG/SL combination a little less
ef¬cient than the semi-implicit SL algorithm which has now become very popular for both
global and regional models.
Nonlinear Galerkin methods were independently rediscovered by Roger T´ mam and e
his collaborators (Marion and T´ mam, 1989, Jauberteau, Rosier, and T´ mam, 1990) and also
e e
by Foias et al.(1988). (Parenthetically, note that similar approximations under the name of
the “quasi-static” approximation or “pseudo-steady-state approximation” have also been
independently developed in chemical kinetics where the ratio of time scales often exceeds a
million (Vermer and van Loon, 1994)!). Table 12.2 describes a large number of applications.
The practical usefulness of Nonlinear Galerkin time-marching is controversial. Daley,
Haugen and Machenhauer (1993), Boyd (1994e), Garc´ ia-Archilla(1995) and Garc´
ia-Archilla
and de Frutos (1995) are negative Pascal and Basdevant (1992) conclude that NG is useful
in reducing costs for turbulence modelling; most of the other authors in Table 12.2 are also
optimistic about the merits of the Nonlinear Galerkin method.


12.13 Weaknesses of the Nonlinear Galerkin Method
In comparing the Nonlinear Galerkin method with implicit and semi-implicit schemes,
there are two key themes.
CHAPTER 12. IMPLICIT SCHEMES & THE SLOW MANIFOLD
246

1. All temporal ¬nite difference approximations have an accuracy which is proportional
to N G(1) multiplied by a constant times „ r where r is the order of the ¬nite difference
method and „ is the timestep.
2. The computational complexity and expense of Nonlinear Galerkin methods, N G(m),
grows very rapidly with the order m.
Demonstrating the second assertion would take us too far a¬eld, but it is really only com-
mon sense: N G schemes are derived by perturbation theory, and the complexity of almost
all perturbation schemes escalates like an avalanche. T´ mam (1989) gives recursive for-
e
mulas for calculating high order N G formulas; Promislow and T´ mam(1991) is unusual
e
in performing numerical experiments with high order NG schemes, that is, N G(m) with
m > 1. The special dif¬culty which is explained in these articles is that all NG schemes
higher than Machenhauer™s [N G(1)] approximation require the time derivative of the non-
linear terms which force the fast modes. There is no simple way to compute the time
derivative of order (m ’ 1) as needed by N G(m), although complicated and expensive
iterations can be implemented.
The ¬rst point is more fundamental. The sole error of Backwards Euler, Crank-Nicholson
and other time-marching schemes is that they approximate the time derivatives. Since
N G(1) approximates the time derivative by completely ignoring it, it follows that the error
in N G(1) is proportional to the accuracy of this assertion that the time derivative is small.
However, the relative smallness of the time derivative is an advantage for implicit time-
marching schemes, too, because their error in approximating the derivative is multiplied
by the smallness of the term which is being approximated.
For example, consider the simple problem
(12.64)
ut + iu = exp(i t)
for which the exact slow manifold (Boyd, 1994e, H.-O. Kreiss, 1991) is
1
uslow (t; ) ≡ ’ i (12.65)
exp(i t)
1+
The exact derivative of the slow solution is i uslow . The relative errors in approximating
this, in other words, [ut (exact)’ut (approximate)]/ut (exact), are graphed as Fig. 12.7. Note
that because the complex exponential is an eigenfunction of the difference operators, the
slow manifold for all the approximation schemes (including N G(m)) is proportional to
exp(i t). This factor cancels so that the relative error is independent of time.
The most remarkable thing about the graph is the fact that even when the temporal
resolution is poor (no more than 6 points per period of the slow manifold), the humble
BE method has only half the error of N G(1) whereas the Crank-Nicholson scheme has
only 1/12 the error of N G(1). This seems little less than amazing because the BE and
CN methods are general time-stepping schemes that in theory can be applied to any time-
dependent problem. In contrast, the Nonlinear Galerkin method is specialized to track the
slow manifold only; it has no meaning for time-dependent problems with but a single time
scale. Why is the specialized method so bad compared to the general methods?
The answer is the obvious one that an approximation, even a bad one, is better than
no approximation at all. N G(1) is a non-approximation of the time derivative. In contrast,
implicit schemes try to accurately approximately ut as is also done by higher order (m > 1)
NG methods.
Of course, with a really long time step, the ¬nite differencing schemes would give O(1)
errors, but this is insane. In practical applications, it is necessary to integrate the fast and
slow modes as a coupled system. A time step which is long enough to make Crank-
Nicholson poorer than N G(1) for the fast modes will also do a terrible job of advancing
12.13. WEAKNESSES OF THE NONLINEAR GALERKIN METHOD 247


F F F F F F F F F F1
NG(0) ["truncation"]

NG(1) ["diagnostic"]
0.1
B
B
Backwards-Euler B B B J
BB J
JJ
B J
NG(2)
H 0.01

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. 52
( 136 .)



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