Post on 03-Jan-2016
description
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
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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
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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
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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
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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
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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
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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
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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
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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