the spectral method
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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
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