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The system of ODEs has been replaced by a so-called “Differential-Algebraic” (DAE) set of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