1

The Spectral Method

2

Definition

mmm xettx )()(),( where (em,en)=δm,n

en= basis of a Hilbert space(.,.): scalar product in this space

L

dxeffe0

*),(In L2 space where f*: complex conjugate of f

Discretization: limit the sum to a finite number of terms

M

mmm xettx )()(),(

(consistent if the em’s are appropriately ordered)

3

The Galerkin procedure• Linear case

• If em’s are eigenfunctions of H: Hem=λmem

)(),(

H

t

txH: linear space operator

RxetHxett

M

mmm

M

mmm

)()()()( R: discretization error

we assume R to be a function of the omitted em’s onlytherefore R is orthogonal to em (m ≤ M) (alternatively we minimize ||R||2)

M

m

M

mnnmmnm

m eReHeeedt

d),(),(),(

||δm,n dxHee mn

*|| ||

0

tnnn

n nedt

d

0

analytical solution (no need to discretize in t)

compute once and store

4

One-dimensional linear advection equation

0

t

)()0,( f

periodic boundary conditions ),2(),( tnt

• Analytical solution

)(),( tft phase speed: γ=cte (rad/s)

• Basis functions: Fourier functions eimλ (eigenfunctions of ∂/∂λ)

2

0,2 nm

inim dee

M

Mm

imm ett )(),(

ω being a real field ==> ω-m(t)= ωm*(t); we need to solve only for 0≤m ≤M

• Galerkin procedure

Mmindt

tdn

n

0;0)(

this is a system of 2M+1 equations (decoupled) for the(complex) ωn coefficients

5

One-dimensional linear advection equation (2)

• Exact solutiontin

nn et )0()(

ωn(0)

nγ/2π

- in physical space

M

m

timm

M

m

imtimm tfeeet )()0()0(),( )(

the same form as the analytical solutionno dispersion due to the space discretizationbecause the derivatives are computed analytically

6

Calculation of the initial conditions• computation of ωm(0) given ω(λ,0)

- Direct Fourier transform

2

0

)0,()0( deA immm

where Am: normalization factor

- Inverse Fourier transform

m

immm etBt )(),(

Bm: normalization factors

• Discrete Fourier transforms

- Direct :

K

i

imimm

ieA1

' )()0(

- Inverse : M

m

immmi

ietBt )(),( '

Transformations are exact if K2M+1

Procedure: Fast Fourier Transform (FFT) algorithm

Products of two functions have no aliassing if K3M+1

7

The linear gridUnfitted function

Fitted withquadraticgrid

Fitted withlineargrid

3M+1 points in λ ensure noaliassing in computations ofquadratic terms (case ofEulerian advection) Quadratic grid

2M+1 points in λ ensureexact transforms of linearterms to grid-point and backLinear grid

8

Stability analysism

m imdt

d

• Leapfrog scheme

no need to discretizein time if we do nothave other terms in theequation

nm

nm

nm im

t

2

11

imnnm eWTry 0

Substituting and dividing by W(n-1)

1)(012 2222 tmtimWWtimW

11 tmW 1 W conditionally stable and neutral

- Comparison with finite differencesU0=RγM~N/2Δx=2πR/N

x

tUtM 0

1

0 x

tU

if using the quadratic grid M~N/3 ---->

2

30 x

tU

9

Graphical representation

tt t

m

2

)( ttm )( ttm

)(tmtt

m

tt

tt

m

tt

10

Non-linear advection equation

t

M

j

M

n

nnjjmm

M

m d

deeet

dt

d)(

M

m

immeF

2

= there are more wavenumbers on the r.h.s. than in the original function

Galerkin procedure

kk Ftdt

d )( k=0 …. M

therefore Fm m>M are not usedbut no aliassing produced because of misrepresentation

11

Non-linear advection equation (cont)Calculation of Fk

• Interaction coefficients

M

j

M

nknjnjk eeeinF ),(

Ijnk ---> interaction coeff. matrixI is not a sparse matrix

• Transform method

M

nnn

M

n

nn eind

de

I. FFTf(λl); l=1, … L

M

jjje

g(λl); l=1, … LI. FFT

F(λl) = - f(λl)•g(λl) M

kkkeF

D. FFT

can be shown to have no aliassing if L 3M+1

12

One-dimensional gravity-wave equations

conditionsperiodicboundaryinitial

Ht

h

hg

t

)(&

0

0

M

m

imm ett )(),(

M

m

imm ethth )(),(

• Galerkin procedure

kk

kk

Hkidt

dh

hgkidt

d

tgHikkk

k

tgHikkk

k

ehhgHhkdt

hd

egHkdt

d

02

2

02

2

2

no need to discretize in time

13

One-dimensional gravity-wave equations (cont)

• Explicit time stepping (leapfrog)

nk

nk

nk

nk

nk

nk

tHikhh

tghik

2

211

11no need to transformto grid-point space

• Stability and dispersion (von Neumann method)

tnink

tnink

ehh

e

0

0assume

substituting: gHtkt 222 )()(sin

gHMtgHtMreal

11)( 22

gH

x

~smaller thanwith finitedifferences

gHtm

gHtm

mv f

)asin( dispersion due solely

to the time discretization

14

One-dimensional gravity-wave equations (cont)

• Implicit time stepping

)(2

12

)(2

12

1111

1111

nm

nm

nm

nm

nm

nm

nm

nm

tHimhh

hhtgim 1nm substituting

)])(1(2[)(1

1 221122

1 gHtmhtHimgHtm

h nm

nm

nm

Decoupled set of equations because the basis functions eimλ are eigenfunctions of the space operator ∂/ ∂λ with eigenvalues im

• Stability and dispersion

using von Neumann we get

tanyforrealgHtmt 222 )()(tan stable

gHtm

gHtm

mv f

)atan( dispersion larger than

in leapfrog scheme

15

Shallow water equations

conditionsperiodicboundaryandinitial

y

v

x

u

yv

xu

t

yfu

y

vv

x

vu

t

v

xfv

y

uv

x

uu

t

u

)(

0)(

0

0

Linearize about a basic state U0, V0, Φ0 and assume f=cte=f0 (f-plane approx)

),,('

),,('

),,('

0

0

0

tyx

tyxvVv

tyxuUu

substitute and neglect productsof perturbations

)''

('''

''

'''

''

'''

000

000

000

y

v

x

u

yV

xU

t

yuf

y

vV

x

vU

t

v

xvf

y

uV

x

uU

t

u

16

Leapfrog (explicit) time schemeStability according to von Neumann

ilyimxtni

ilyimxtni

ilyimxtni

eee

eeevv

eeeuu

0

0

0

'

'

'

assume and substitute

0)(2

02

02

00000000

00000000

00000000

ilvimuilVimUt

ee

ilufilvVimvUt

eev

imvfiluVimuUt

eeu

titi

titi

titi

17

Leapfrog (explicit) time scheme stability (cont)

0)sin(1

0)sin(1

0)sin(1

000000000

00000000

00000000

lvmulVmUtt

luiflvVmvUvtt

mvifluVmuUutt

or, calling ),,(~

000 vuZ

0~~~~)sin(

00

ZHZlVmUt

t

where

0

0

0~~

00

0

0

lm

lif

mif

H

18

Leapfrog (explicit) time scheme stability (cont)

calling

lVmUt

t00

)sin(

0~

)~~~~

( ZIH

this system has non-trivial solutions if 0~~~~ IH

0)( 20

220

3 flm)(

0

220

20 lmf

for α to have a real solution

)sin( t )( 00 lVmUt 1

The most restrictive case is when )( 220

20 lmf which gives

)(

122

02

000 LMfLVMUt

19

)(cos)(

0

2220

20 tlmf

Semi-implicit time scheme stabilityFollowing the same steps as in the explicit scheme we arrive at:

0)cos()cos(

)cos(0

)cos(0~~

00

0

0

tltm

tlif

tmif

H

therefore

])(cos)()[()sin( 2220

2000 tlmflVmUtt

if 00022

0 )()( flVmUlm

)()tan( 220 lmtt

LVMUt

00

1

20

Spherical harmonicsOrthogonal basis for spherical geometry

immn

mn ePY )(),( m: zonal wavenumber

n: total wavenumber

λ= longitudeμ= sin(θ) θ: latitudePn

m: Associated Legendre functions of the first kind

)()(

0;)1()1(!2

1

)!(

)!()12()( 22/2

mn

mn

nmn

mnm

nmn

PP

md

d

nmn

mnnP

1

1,)()(

2

1sn

ms

mn dPP

2

0,2 lm

ilim dee

21

Spectral representation

M

Mm

N

mn

mn

mn YtXtX ),(),(),,,(

n-|m|: effective meridional wavenumber

1

1

2

0

)(),,,(4

1),( ddePtXtX imm

nmn

spectral transform

Since X is a real field, Xn-m=(Xn

m)*

• Fourier coefficients

2

0

),,,(2

1),,( detXtX im

m direct Fourier transform

N

mn

mn

mnm PtXtX )(),(),,(

orinverse Legendretransform

22

Some spherical harmonics (n=5)

23

Spherical harmonics (cont)• Properties of the spherical harmonics

mn

mn imY

Y

eigenfunctions of the operator ∂/ ∂λ

mn

mn Y

a

nnY

22 )1(

eigenfunctions of the laplacian operatorsemi-implicit method leads to a decoupledset of equations

2/1

2

22

1112

14;)1()1(

n

mnYnYn

Y mn

mn

mn

mn

mn

mn

latitudinal derivatives easy to compute although the spherical harmonics are not eigenfunctions of the latitudinal derivative operator

24

Spherical harmonics (cont.)• Usual truncations

N=min(|m|+J, K) pentagonal truncationM=J=K triangularK=J+M rhomboidalK=J>M trapezoidal

m

n

Mm

n

M

m

n

Mm

n

M

pentagonal triangular

rhomboidal trapezoidal

K=J=MK

K

K=J

J

J

25

Gaussian gridUse of the transform method for non-linear terms

• Integrals with respect to λ ---> 3M+1 points equally spaced in λ

• Integrals with respect to μ computed exactly by means of Gaussian quadrature using the values at the points where

0)(0 GN

P Gaussian latitudes

In triangular truncation NG (3N+1)/2

Gaussian latitudes are approximately equally spacedsame spacing as for λ

26

The reduced Gaussian grid

Full grid Reduced grid

• Triangular truncation is isotropic• Associated Legendre functions are very small when m is large and |μ| near 1

27

The linear Gaussian gridQuadratic

gridLineargrid

Exact transformsto g.p. and back

Yes Yes

Alias-freecomputation ofquadratic terms

Yes No

Number ofdegrees offreedom

at least(3N+1)2/2

at least(2N+1)2/2

Ratio of degreesof freedomg.p. / spec

4.472at T213

2.at TL319

Gibbsphenomena

Yes Yesbut to a lesser

extent

28

Two resolutions using the same Gaussian grid

T213 orography TL319 orography

29

Diffusion0;2

KAKt

A

mn

mn A

a

nnK

t

A2

)1(

very simple to apply

2

24 )1(

a

nn

- Leapfrog

•Stability

01)1(

22

2

a

nntK

physical solution 0≤λ≤1 stablecomputational solution λ≤-1 unstable

- Forward

)1(

2

nKn

at

- Backward (implicit)

)()1()()(

2ttA

a

nnK

t

tAttA mn

mn

mn

decoupled system

of equations

2

)1(1

1

a

nntK

0≤λ≤1 Stable