Embed Size (px)
Citation preview
1
Frank Lunkeit
Meteorologische Modellierung
Introduction to Numerical Modeling
(of the Global Atmospheric Circulation)
0. Introduction: Where/what is the problem and basics
1. The (linear) evolution (decay) equation (discretization in time)
2. One-dimensional linear advection (discretization in time and space)
3. One-dimensional linear diffusion and one-dimensional linear transport equation
4. Nonlinear advection and 1d nonlinear transport equation (Burgers-equation)
5. More dimensions (grids)
6. Design of an atmospheric general circulation model (AGCM)
(7. An example: qg barotropic channel (weather prediction)) next semester
Outline
Frank Lunkeit
2
References
D. R. Durran: Numerical Methods for Wave Equations in Geophysical
Fluid Dynamic, Springer 1999, ISBN:0-387-98376-7.
G.J. Holtiner & R.T. Williams: Numerical Prediction and Dynamic
Meteorology, Second Edition, John Wiley & Sons 1980, ISBN: 0-471-
05971-4.
D. Randall: An Introduction to Atmospheric Modeling,
http://kiwi.atmos.colostate.edu/group/dave/at604.html
Frank Lunkeit
Introduction to Numerical Modeling
0. Introduction: The Problem and Basics
Frank Lunkeit
3
,..),,(1
TvuFx
Pfv
y
uv
x
uu
t
ux
Nonlinear (quasi-linear) coupled partial differential equations
(without a known analytic solution)
Example: Two dimensional flow
,...),,(1
TvuFy
Pfu
y
vv
x
vu
t
vy
y
v
x
u
yv
xu
t
,...),,( TvuFy
vx
ut
RTP
pcR
P
PT
0
Eq. of motion
Eq. of motion
Continuity eq.
First law of thermodynamics
Eq. of state
Pot. temperature
The Problem
Frank Lunkeit
Example: First law (1-d, p=const.,…)
t
T
x
Tu
2
2
x
TK
,...),( uTF
local
change
with time
advection diffusion other forcing
first
derivative
in time
first
derivative
in space
Second
derivative in
space
?
The Problem The Problem
Frank Lunkeit
4
From theory (the equations) to the numerics:
a) continuous functions -> discrete values
b) differential (integral) equations -> algebraic equations
Tasks:
a) choice of the discretization
b) calculation (representation) of the derivatives
Approaches (Algorithms):
- grid point methods (finite differences)
- series expansion (spectral method and finite elements)
Evaluation of a method:
- consistency, accuracy, convergence and stability
Numerical Solution of (Partial) Differential Equations
Frank Lunkeit
T
T(s)
s
s = t, x, y, z,…
T
s
{
∆s sj sj+1 sj-1
a) continuous function -> discrete values
∆s defines the resolution
Example: The Grid Point Method Example: The Grid Point Method
Frank Lunkeit
5
T
T(s)
s
s = t, x, y, z,…
T
s
{
∆s sj
sj+1 sj-1
b) derivatives -> finite differences (e.g. from Taylor-expansion)
...|)()()2
...|)()()1
sds
dTsTssT
sds
dTsTssT
s
s
...)()(
)1 from1
s
sTsT
ds
dT jj
...)()(
)2 from1
s
sTsT
ds
dT jj
Central differences
Forward differences
Backward differences
...2
)()()2)1 from
11
s
sTsT
ds
dT jj
Example: The Grid Point Method
Frank Lunkeit
derivatives -> analytical derivatives of the basis-functions
T*n differentiable orthogonal (Basis-) functions (Fourier-series;
Legendre Polynomials). The resolution is given by the number of
modes n
T
s
T*2(s)
T
T(s)
s
s = x, y, z,…(t)
+
T
s
T*1(s)
n
nn
ds
dTa
ds
dT *
)()( * sTasTn
nn
Example: Series Expansion -The Spectral Method
Frank Lunkeit
6
T
T(s)
s
s = x, y, z,…(t)
T
s +
example
F1,F2,… arbitrary function which is only locally non zero
Derivatives -> depending on the function
T
s
T
s
+
+ …. T(s)=aF1(s)+bF2(s)+….
F1(s)
F2(s)
F3(s)
Example: Series Expansion –Finite Elements
Frank Lunkeit
Evaluation: Consistency
ds
dT
s
T
s
0limThe discretization must converge to the differential:
Example: forward difference:
ds
dTs
ds
Td
ds
dT
s
sTssT
s
ds
Td
ds
dT
s
sTssT
s
ds
Tds
ds
dTsTssT
ss
...
2lim
)()(lim
...2
)()(
...2
)()()(
2
2
00
2
2
2
2
2
s
sTssT
s
T
)()(
Consistency from Taylor-expansion:
Frank Lunkeit
7
Frank Lunkeit
Evaluation: Accuracy
Accuracy: Smallest power of ∆s in the deviation from the truth
Example: forward differences
)(
...2
)()(
...2
)()()(
2
2
2
2
2
sOds
dT
s
T
s
ds
Td
ds
dT
s
sTssT
s
ds
Tds
ds
dTsTssT
s
sTssT
s
T
)()(
Taylor-expansion
=> forward differences are of first order accuracy
a: Accuracy (order of accuracy) of the discretization
b (mostly): Accuracy (order of accuracy) of the scheme
Considering the whole equation, i.e. including the ‚right hand side‘ (see, e.g.,
Section ‘Evolution (decay) equation’)
Evaluation: Convergence
Convergence: Convergence of the error to 0 for small ∆s:
0)()(lim0
tTtT As
Error: Deviation of the numerical solution (T) from the truth (analytical solution; TA)
Error measure (Norm)
a) Euclidean (quadratic) Norm:
b) Maximum Norm:
2/1
1
2
2
N
j
j
jNj
1max
Examples
Frank Lunkeit
8
Frank Lunkeit
or: A scheme is unstable if the numerical and the analytical solutions diverge with
time:
Stability: A scheme is stable if the difference between the numerical and the
analytical solution is bounded:
Evaluation: Stability
t
A tTtT )()(
Stability analyses: various methods (e.g.):
Direct (heuristic),
Energy Method and Von Neumann Method (see advection equation)
ttCtTtT A )()()(
(most important for practical use)
If TA(t) is bounded:
ttCtT )()(
t
tT )(
stable
unstable
Interrelation (Lax Equivalence Theorem):
‚For a consistent scheme stability is a necessary and sufficient condition for
convergence‘.
Stability may depend on ∆s
0. Introduction: The Problem and Basics
Summary:
• Reason to do numerics: No analytic solution of the problem
• Methods: Grid point method, spectral method, finite elements (,…)
• Finite Differences: forward, backward, central
• Evaluation: consistency, accuracy, convergence and stability
0. Introduction: The Problem and Basics
Frank Lunkeit
9
1. The (Linear) Evolution (Decay) Equation (Discretization in Time)
Introduction to Numerical Modeling
Frank Lunkeit
The (Linear) Evolution (Decay) Equation
FaTt
T
equation:
note: with a imaginary, i.e. a=ib, und complex T, T=TR+iTI an oscillation
equation results: e.g. inertial oscillation:
fut
vfv
t
u
)2 , )1
if
tivu
3) with
)2)3Im( );1)3Re( since
Frank Lunkeit
10
Frank Lunkeit
Analytical Solution
FaTt
T
a
FatTtT )exp()(
equation:
Describes the exponential decay of an
initial perturbation ∆T=T0-F/a (T0=initial
value) to the stationary solution F/a. time
scale: 1/a
solution:
example:
T0=0, F=1, a=0.05
initial value problem (T0 needed; eq. is first order in time)
Numerical Solution
FaTt
T
a) Discretization in time: t -> t0+n ∆t (∆t: timestep; n: integer)
T T(t)
t
T
t
{
∆t tn tn+1 tn-1
Tn Tn+1
Tn-1
timestep ∆t must be chosen adequately (physically meaningful)!
(continuous) (discrete)
Frank Lunkeit
11
Numerical Solution
FaTt
T
T
t
{
∆t tn tn+1 tn-1
Tn Tn+1
Tn-1
t
TT
t
T nn
1
t
TT
t
T nn
1
Centered differences
Forward differences
Backward differences
t
TT
t
T nn
2
11
Improper since T(t) known and T(t+∆t)
needed
Two level scheme (Einschritt-Verfahren)
Three level scheme (Zweischritt-Verfahren)
b) Derivatives -> finite Differences
Frank Lunkeit
Frank Lunkeit
Numerical Solution
FaTt
T
c) dealing with the ‚right hand side‘ (the tangent)
);(Tft
T
general: equation )(1 Tf
t
TT nn
discretization (two level)
Choice of T in f(T): )(1n
nn Tft
TT
Explicit (Euler)
)()( 11
nlinnnl
nn TfTft
TT
Implicit
Semi-implicit
)( 11
n
nn Tft
TT
T
t
{
∆t tn tn+1 tn-1
Tn Tn+1
Tn-1
(fnl=nonlinear part; flin=linear part)
12
Explicit (Euler)
)()( 11
nnnnnn TftTTTf
t
TT
Discretization:
Tangent (gradient) at tn , i.e. computed from Tn
T
t tn tn+1 tn-1
Tn Tn+1
Tn-1
evolution (decay) equation: )(1 FTatTT nnn
Frank Lunkeit
Implicit
)()( 1111
nnnn
nn TftTTTft
TTDiscretization:
Tangent (gradient) at tn+1 , i.e. computed from Tn+1 but used at tn.
T
t tn tn+1 tn-1
Tn Tn+1
Tn-1
evolution (decay) equation: at
FtTTFTa
t
TT nnn
nn
111
1
A re-arrangement of the equation is necessary to obtain Tn+1!
Frank Lunkeit
13
Two Level Scheme: Explicit (Euler)
)(11 FTatTTFaT
t
TTnnnn
nn
)lim(0 t
T
t
T
t
Consistency ?
t
Tt
t
T
t
T
t
TT
t
t
T
t
T
t
TTt
t
Tt
t
TtTttT
t
nn
t
nn
...2
limlim
...2
...2
)()(
2
2
0
1
0
2
2
1
2
2
2
Yes, from Taylor-expansion:
Accuracy (order) of the discretization: first order (Error goes with ∆t)
Frank Lunkeit
Two Level Scheme: Explicit (Euler)
)(11 FTatTTFaT
t
TTnnnn
nn
Accuracy (order) of the scheme?
ErrtTat
tTttTA
AA
)(
)()(Ansatz: insert the
analytic solution
Taylor-expansion: n
A
n
n
n
AAt
T
n
ttTttT
1 !
)()()(
Insert the Taylor-
expansion: )(
)(!
)()(
1 tTat
tTt
T
n
ttT
Err A
A
nn
nn
A
11
1
)!1(
)(
nn
nn
t
T
n
tErr => Err= O(1)
(here with F=0)
Since t
TtaT A
A
)(
Frank Lunkeit
14
Frank Lunkeit
Two Level Scheme: Explicit (Euler)
)(11 FTatTTFaT
t
TTnnnn
nn
Stability t
A tTtT )()( ?
))exp()()exp()( 0 tanTtntTatTtT AA (F=0)
ntaTtntTtaTttT )1()()1()( 00
t
A tTtTAta )()(12=> for (unstable)
=> choice of ∆t such that ∆t ≤ 1/a !
analytically:
numerically:
=> Amplitude increases/diminishes with factor ntaA )1(
(stable) for
)1( 21 Atabut for
bounded )()(12
t
A tTtTAta
+/- jumps
Convergence ? Yes (from Lax equivalence theorem) )0)()(lim(0
tTtT At
Note:
)( 0tTT
)()exp()1()1( 2tOtantanta n
Frank Lunkeit
Two Level Scheme: Implicit
Convergence? Yes
Consistency?
Accuracy (Order)? First order scheme (i.e. error grows with ∆t)
at
FtTTFTa
t
TT nnn
nn
111
1
yes, like explicit
Stability?
with F=0: nta
tTtntT
ta
tTttT
)1(
)()(
)1(
)()(
For all ∆t, since A < 1
but ∆t >1/a not appropriate to the problem!
=> Amplitude diminishes with factor ntaA
)1(
1
ttCtTtT A )()()(=>
=> implicit always stable
15
Two Level Scheme: Crank-Nicolson
Consistency?
Accuracy (Order)?
FTgTgat
TTnn
nn
))1(( 11
yes, like explicit und implicit
(g = 1 : explicit; g = 0 : implicit)
ErrttTgtTgat
tTttTAA
AA
)]()1()([
)()(Insert analytical solution:
Taylor-expansion: n
A
n
n
n
AAt
T
n
ttTttT
1 !
)()()(
Insert Taylor-expansion: )]!
)()()(1()([
)(!
)()(
1
1
n
n
nn
AA
A
nn
nn
A
t
T
n
ttTgtgTa
t
tTt
T
n
ttT
Err
)1
1(
!
)(
11
1
ng
t
T
n
tErr
nn
nn
=> for g= 1,0 (ex., im.): Err= O(1); for g=0.5: Err=O(2))
(with F=0)
=0 for n=1
and g=0.5 }
Frank Lunkeit
Two Level Scheme: Crank-Nicolson
FTgTgat
TTnn
nn
))1(( 11
Stability?
with F=0:
n
tag
tgatTtntT
tag
tgatTttT
))1(1(
)1()()(
))1(1(
)1()()(
For all ∆t, |A| < 1 (denominator > numerator)
but: ∆t >1/a not appropriate to the problem
=> Amplitude changes with factor
n
tag
tgaA
))1(1(
)1(
=>
=> always stable (for g <1)
))()1()(()()( ttTgtTgtatTttT
=> Convergence
ttCtTtT A )()()(
Frank Lunkeit
16
Frank Lunkeit
Two Level Scheme: 4th order Runge-Kutta
4321 226
1)()( KKKKtTttT
),( tTfdt
dTGeneral equation:
Runge-Kutta (4th order):
),)((
)2
,2
)((
)2
,2
)((
)),((with
34
23
12
1
ttKtTftK
tt
KtTftK
tt
KtTftK
ttTftK
Decay equation: on t)dependent explicitly(not )(),( TfFTatTf
Order: Err=O(∆t4)
Stability: Stable for 1|2462
1|443322
tatata
ta
Frank Lunkeit
Two level scheme: 4th order Runge-Kutta
4321 226
1)()( KKKKtTttT
),)(( ),2
,2
)((
)2
,2
)(( ),),((
342
3
121
ttKtTftKt
tK
tTftK
tt
KtTftKttTftK
Idea: Average of tangents for different T (T(t), T(t)+K1/2, T(t)+K2/2, T(t)+K3)
tn
T
t tn+1
T(t)
T(t)+K1/2 T(t)+K2/2
T(t)+K3 f(T(t))=K1/ ∆t
f(T(t)+K1/2)=K2/ ∆t
f(T(t)+K2/2)=K3/ ∆t
f(T(t)+K3)=K4/ ∆t
with ‚artificial‘ base points T(t)+Ki
Expensive since f(T) needs to be computed 4 times
17
Three Level Schemes
...6
|2
)()(
...6
|2
||)()(
...6
|2
||)()(
2
3
3
3
3
32
2
2
3
3
32
2
2
t
dt
Td
t
ttTttT
dt
dT
t
dt
Tdt
dt
Tdt
dt
dTtTttT
t
dt
Tdt
dt
Tdt
dt
dTtTttT
t
ttt
ttt
General: three level schemes do not only consider time t und t+∆t but also time t-∆t.
e.g. From Taylor-expansion:
=> Discretization is consistent, 2nd Order (Err=O(∆t2))
Decay equation: FtTat
ttTttT
)(
2
)()(Leap frog
Frank Lunkeit
Leap Frog
))((2)()()(2
)()(FtTatttTttTFtTa
t
ttTttT
i.e. evaluation of the gradient at t for step T(t-∆t)-> T(t+∆t)
t T(t-1) T(t)
Δt
T(t+1)
dT/dt = F - aT
t T(t-2)
Δt
T(t) T(t-1)
dT/dt = F - aT
T(t+1)
chronology:
Step n:
Step n+1:
Two time levels are needed (t-1 and t)!
Frank Lunkeit
18
Frank Lunkeit
Leap Frog
))((2)()()(2
)()(FtTatttTttTFtTa
t
ttTttT
Stability (Computational Mode)?
nnn TatTT 2 :)0F(with 11
121 nnn TTT Assumption: T changes each time
step by factor λ 1112 2 nnn TatTT
i.e. two solutions: 2/122
2,1 1 tata
For ∆t ->0 : λ1=1 (physical since correct); λ2=-1 (unphysical)
For ∆t > 0: physical mode 0 < λ1 < 1 (attenuation)
computational Mode λ2 < -1 i.e. oscillating instability!
Leap Frog not suited for damped (dissipative) systems!
Leap Frog: Unstable Computational Mode
T
t 1
ttttt ATTtaTTT 111 2
Leap-Frog initialization T0=0 ; T1<0
t=2 T2=T0-AT1= 0-AT1 > 0
t=3 T3=T1-AT2=T1-A(T0-AT1)=T1(1+4A2) < T1
t=4 T4=T2-AT3 > T2
2 3 4
Frank Lunkeit
19
Frank Lunkeit
Three Level Scheme: Adams-Bashforth(2)
FttTtTa
t
tTttT
))()(3(
2
)()(
gradient by averaging gradient from T(t) and gradient from ‚extrapolated‘ T‘(t+∆t)
(=T(t)+(T(t)-T(t-∆t)))
stability (computational mode)?
2)
2
31()3(
2 :)0F(with 111 at
Tat
TTTat
TT nnnnnn
121 nnn TTT
i.e. two solutions:
2/1
2
2,14
)2
31(
22
)2
31(
tata
ta
for ∆t ->0 : λ1=1 (physical, correct solution); λ2=0 (unphysical)
for ∆t > 0: physical mode 0 < λ1 < 1 (damped)
computational mode 0 < λ2 <1 (damped) => stability!
2)
2
31(2 atat
consistent and 2nd order (Err=O(∆t2))
1. The Decay Equation: Discretization in Time
Summary:
• Decay (linear evolution) equation: first order in time, initial value
• Schemes: implicit, explicit, semi-implicit, two-level, three-level
• Specific schemes: Euler, Crank-Nicolson, Runge-Kutta, Leapfrog, Adams-Bashford
• Computational mode
• Leapfrog unstable for dissipative systems!
The (Linear) Evolution (Decay) Equation
Frank Lunkeit
20
2. One-Dimensional Linear Advection (Discretization in Time and Space)
Introduction to Numerical Modeling
Frank Lunkeit
The Linear 1-d Advection Equation
x
Tu
t
T
equation: u=const.
analytical solution: )(),( utxftxT with any function f
example: Superposition of waves
with wave number k and
phase velocity u
k
k utxikTtxT ))(exp(),(
hyperbolic, first order in time and space: initial and boundary conditions needed
02
22
2
2
f
x
Te
t
Td
x
Tc
xt
Tb
t
Ta
hyperbolic: b2-4ac > 0
parabolic: b2-4ac = 0
elliptic: b2-4ac < 0
Frank Lunkeit
21
Numerical Solution: Spectral Method
x
Tu
t
T
Idea: transformation of T into new basis functions which are differentiable orthogonal
functions of x => derivations in space can be calculated analytically.
N
Nk
k ikxtTxtT )exp()(),(Example: Fourier-series
Tk(t) = time dependent (complex) Fourier coeff.; N = considered modes (resolution)
(N<< ∞ => T might not be perfectly represented)
)exp()()exp()(
ikxtTikuikxt
tTk
N
Nk
N
Nk
k
Insert into advection equation: =>
=> 2N+1 uncoupled ordinary differential equations kk iukTt
T
Frank Lunkeit
Spectral Method
Analytical solution:
kk iukTt
T
x
Tu
t
T
kk iukTt
T
Similar to the evolution (decay) equation but with imaginary a=iuk
and complex T (oscillation equation)
)exp(0 iuktTT kk
=> complete solution of the advection eq: ))(exp(0 utxikTT k
k
(as expected)
=> local: oscillation, global: non dispersive waves with phase velocity u
Frank Lunkeit
22
Frank Lunkeit
Spectral Method: Explicit
kk iukTt
T
x
Tu
t
T
Numerical solution: Analog to evolution (decay) eq. but: different characteristics
example: explicit (Euler) )()()(
tiukTt
tTttTiukT
t
T
Stability?
)( since 3
)(1error Phase b)
12
)(1)(1A with increases Amplitude a)
)3
)(1(
3
)()arctan( and
)(1)1(with
)exp()()1)(()(
)()()(
2
22
23
2
tuklyanalyticaltuk
tuktuk
tuktuk
tuktuktuk
tuktiukA
iAtTtiuktTttT
tiukTt
tTttT
A
A
N
always unstable
slower and k-dependent, i.e. numerical dispersion
(one wave k only,
index k omitted)
Frank Lunkeit
Spectral Method: Implicit
kk iukTt
T
x
Tu
t
T
example: implicit )()()(
ttiukTt
tTttTiukT
t
T
Stability?
)( since 3
)(1error Phase b)
decreases Amplitude a)
)3
)(1(
3
)()arctan( and
)(1
1with
)exp()()1(
)()(
)()()(
2
23
2
tuklyanalyticaltuk
tuktuk
tuktuktuk
tukA
iAtTtiuk
tTttT
ttiukTt
tTttT
A
A
N
always stable but k-dependent damping,
i.e. numerical diffusion
slower and k-dependent, i.e. numerical dispersion
23
t T(t-1) T(t)
Δt
T(t+1)
dT/dt=-iukT
t T(t-2)
Δt
T(t) T(t-1)
dT/dt=-iukT
T(t+1)
chronology:
Step n:
Step n+1:
Leap frog: n
nn iukTt
TTiukT
t
T
2 11
Recall:
Spectral Method: Leap Frog
Frank Lunkeit
Spectral Method: Leap Frog
Leap frog: n
nn iukTt
TTiukT
t
T
2 11
Stability?
)(2)()()(2
)()( ttTiukttTttTtiukT
t
ttTttT
22 )(1012)()(with tuktiuktiuktTttT
phase) ofout 180 mode (-) onal(computati
(faster) )6
)(1()
)(1arctan(
1 :error phase c)
neutral) modes(both 11)( with b)
mode) (-) nal'computatio' and )( (physical solutions Two a)
2
2
2
tuk
tuk
tuk
tuk
tuk
Leapfrog appropriate (but numerical dispersion); computational mode bothers
Frank Lunkeit
24
Leap Frog: Computational Mode
iukTt
T
equation: with k=0 (advection of the zonal mean) 0
t
T
1)(1 ) change (amplitude b)
)()(02
)()( (Leapfrog) a)
2
tuktiuk
ttTttTt
ttTttT
0
a)
...)6()4()2( TtTtTtT initial condition: T(t=0)=T0 even Δt correct
problem: T(Δt) needed but unknown (in most cases computed with Euler).
let T(Δt)=T0+E (E=small error) 2
)1()2
()( :solution complete 0
EETtnT n
2 :-1)( mode nalcomputatio b)
)2
(:1)( mode physical a) 0
E
ET
the computational mode is determined by the initialization error (E) only
Frank Lunkeit
Frank Lunkeit
Leap Frog: Robert-Asselin Filter
Modification of the leap frog scheme:
Computation of T(t) (or T(t-Δt)) by weighted averages of T(t+Δt), T(t) and T(t-Δt)
Calculation role:
)()( );()( 3.
.) ( ))()(2)(()()( 2.
))((2)()( 1.
**
**
*
tTttTttTtT
constfilterttTtTttTtTtT
tTftttTttT
step t
step t+1
t T*(t-1) T(t) Δt T(t+1)
dT/dt = f(T(t))
T*(t)
t T*(t-1) T(t) Δt T(t+1)
dT/dt = f(T(t))
T*(t)
25
Frank Lunkeit
Leap Frog with Robert-Asselin Filter
iukTt
T
)2( b)
2 a)
1
*
1
*
*
11
ttttt
ttt
TTTTT
tukTiTT
Stability?
1
*
*
11
2)221(
2
ttt
ttt
TtukiTT
tukTiTT
Tt+1 from a) insert into b)
(plus rearrange):
*
1
*
1
2221
12
t
t
t
t
T
T
tuki
tuki
T
T
old
Matrix-formulation:
new transition-matrix
XX AEigenvalue problem
02221
12
tuki
tuki
Eigenvalues λi from
2/122
2
2/122
1
)()1()(
)()1()(
tuktiuk
tuktiuk
=>
• For (uk∆t)2 <(1- γ)2 the scheme is always stable!
• For ∆t ->0 : λ1=1 (physical, neutral); λ2=2 γ-1 (computational, damped)
• For ∆t > 0 : |λ2|< |λ1|<1, i.e. both modes damped, stronger for computational
• Damping k dependent (larger k stronger damp.): Numerical diffusion
• Numerical dispersion (k dependent phase error)
Grid Point Method: Finite Differences
x
Tu
t
T
Discretization in space and time, i.e. space and time derivatives are approximated
by differences.
Two level schemes (explicit):
x
TTu
t
TT
x
Tu
t
Tn
j
n
j
n
j
n
j
2
11
1
centered differences
x
TTu
t
TT
x
Tu
t
Tn
j
n
j
n
j
n
j
1
1
downstream (u>0)
x
TTu
t
TT
x
Tu
t
Tn
j
n
j
n
j
n
j
1
1
upstream (u>0)
n = time; j = space
Frank Lunkeit
26
Finite Differences
Two level scheme (Einschritt-Verfahren): accuracy and order
)()()!
)()1(
!
)((
!
)(
2
!
)()1(
!
)(
!
)(
2
!
)()1(),(
!
)(),(),(
!
)(),(
2
),(),(),(),(
2
2
1
2
1
2
1
xOtOx
T
n
x
x
T
n
xu
t
T
n
t
x
Tu
t
TErr
x
x
T
n
x
x
T
n
x
ut
t
T
n
t
Err
Errx
x
T
n
xtxT
x
T
n
xtxT
ut
txTt
T
n
ttxT
Errx
txxTtxxTu
t
txTttxT
nn
nnn
nn
nn
nn
nn
n
nnn
n
nn
n
nn
n
nnn
n
nn
n
nn
a) centered differences:
2nd order in Δx
1st order in Δ t
Frank Lunkeit
)()(!
)()1(
!
)(
!
)()1(),(),(),(
!
)(),(
),(),(),(),(
2
1
2
1
xOtOx
T
n
xu
t
T
n
t
x
Tu
t
TErr
Errx
x
T
n
xtxTtxT
ut
txTt
T
n
ttxT
Errx
txxTtxTu
t
txTttxT
nn
nnn
nn
nn
n
nnn
n
nn
)()(!
)(
!
)(
),(!
)(),(),(
!
)(),(
),(),(),(),(
2
1
2
1
xOtOx
T
n
xu
t
T
n
t
x
Tu
t
TErr
Errx
txTx
T
n
xtxT
ut
txTt
T
n
ttxT
Errx
txTtxxTu
t
txTttxT
nn
nn
nn
nn
n
nn
n
nn
Finite Differences
Two level scheme: accuracy and order
b) Downsteam:
c) Upsteam:
1st order in Δx
and Δ t
1st order in Δx
and Δ t
Frank Lunkeit
27
Finite Differences: Stability
x
Tu
t
T
Two level scheme: Stability
x
TTu
t
TT n
j
n
j
n
j
n
j
2
11
1
a) centered differences: )(2
11
1 n
j
n
j
n
j
n
j TTx
tuTT
1. Energy Method
general: take a (scalar) quadratic norm for the total ‚energy‘ (E) in the space state of the
system and show that E remains bounded with time.
j
jTE 2)(Here:
(1)
square (1) and sum up:
j
n
j
n
j
n
j
j
n
j TTTT 2
11
21 ))(()( x
tu
2
j
n
j
n
j
n
j
n
j
n
j
n
j TTTTTT 2
11
2
11
2 )()(2)(
cyclic boundary conditions:
j
n
j
n
j
n
j
n
j
j
n
j TTTTT ))()((2)()( 11
22221
j
n
j
j
n
j
n
j TTT 22
11 )()(
Schwarz‘sche inequality:
j
n
j
j
n
j TT )()( 221=> unstable
Advantage: applicable also for non linear problems. Disadvantage: definition of E needed
Frank Lunkeit
Frank Lunkeit
Finite Differences: Stability
x
Tu
t
T
Two level scheme: Stability
x
TTu
t
TT n
j
n
j
n
j
n
j
2
11
1
a) centered differences: )(2
11
1 n
j
n
j
n
j
n
j TTx
tuTT
2. Von Neumann Method General: Linearize the problem. Replace dependence in space by Fourier expansion
(analytical). Investigate the ‚amplification factor‘ A of one time step (Tn+1=ATn). The scheme is
stable if |A|≤1.
xikj
k
kj etTtT )()(Here: already linear, Fourier-expansion:
(1)
Insert into (1): x
tu
Advantage: Relatively easy. Disadvantage: Linearization
linear -> only one k to consider
i
xikxik
eAA
xkieeA
)sin(1)(1
)( )1()1( xjikxjiknxikjnxikjn eeTeTeAT
21
22 ))(sin1( xkA ))sin(arctan( xk with
ikk etTAttT )()( Numerical solution (k):
1. Amplification factor |A| > 1 => unstable =>
tuk2. Phase error since (=analytical solution)
28
Frank Lunkeit
Finite Differences: Stability
x
Tu
t
T
Two level scheme: Stability analysis
x
TTu
t
TT n
j
n
j
n
j
n
j
1
1
b) Downstream: x
TTtuTT
n
j
n
jn
j
n
j
11
21
))1))(cos(1(21( xkA
))cos(1(1
)sin(arctan
xk
xk
Von Neumann method xikj
k
kj etTtT )()( Tn+1=ATn ) ; investigate (
Insert into (1): x
tu
i
xik
eAA
xkixkeA
)sin())cos(1(1)1(1
)( )1( xikjxjiknxikjnxikjn eeTeTeAT
with
ikk etTAttT )()( Numerical solution (k):
1. amplification factor |A| > 1 => unstable =>
tuk2. Phase error since (=analytical solution)
(1)
Frank Lunkeit
Finite Differences: Stability
x
Tu
t
T
Two level scheme: Stability analysis
x
TTu
t
TT n
j
n
j
n
j
n
j
1
1
c) Upstream: x
TTtuTT
n
j
n
jn
j
n
j
11
21
))1))(cos(1(21( xkA
))cos(1(1
)sin(arctan
xk
xk
Von Neumann method
Insert into (1): x
tu
i
xik
eAA
xkixkeA
)sin())cos(1(1)1(1
)( )1( xjikxikjnxikjnxikjn eeTeTeAT
with
(1)
Stable for 1A 1)1((211)1))(cos(1(21 xk 0))cos(1( since xk
10
x
tu=> Upstream stable for Courant-Friedrich-Levy criterion
100)1(2
x
tu
= Courant-number
Tn+1=ATn ) ; investigate ( xikj
k
kj etTtT )()(
29
Frank Lunkeit
Finite Differences: Upstream
One level scheme: Upstream x
TTtuTT
n
j
n
jn
j
n
j
11
21
))1))(cos(1(21( xkA
))cos(1(1
)sin(arctan
xk
xk
with
ikk etTAttT )()( Numerical solution (one wave k):
with u > 0
- |A|<1 => Numerical diffusion (for μ<1; stronger damping for larger k; O(μΔx2))
- Phase error k-dependent => Numerical dispersion (slower; O(Δx2))
x
tu
For small Δx:
...2
)1(12
)1(1
))1(1())1))(cos(1(21(
222
21
2221
txukxk
xkxkA
...36
1...36
1
))cos(1(1
)sin(arctan
2222222222 xk
x
tuxktuk
xkxkxk
xk
xk
Error
Frank Lunkeit
Finite Differences: Leap Frog
Three level scheme: Leapfrog x
TTu
t
TT
x
Tu
t
Tn
j
n
j
n
j
n
j
22
11
11
Stability:
1)sin(2
)(1
)(
)(
2
2
)1(1)1(1112
11
11
xkiAA
eeAA
eTeTAeTeTA
TTTT
xikxik
xjiknxjiknxikjnxikjn
n
j
n
j
n
j
n
j
• Stable (neutral; |A| = 1) for (ut/x)2 1 (Courant-Friedrich-Levy criterion)
• Phase error => Numerical dispersion
• Computational Mode (-) => Robert-Asselin Filter
...
6)(sin1
)sin(arctan and 333
33
22xk
xktuk
xk
xk
ieAxkxkiA )(sin1)sin( 22
1for 1with
2
2
x
tuA
x
tu
Von Neumann method Tn+1=ATn xikj
k
kj etTtT )()( ) ; investigate (
+ = physical mode
30
Frank Lunkeit
(Semi-) Lagrange Method
x
Tu
t
T
Idea: During advection the property (here T) of a particle/volume remains constant:
=> New T distribution from parcel re-distribution by advection
Explicit: New parcel positions from velocities
Implicit: Old parcel positions from velocities
0dt
dT
tuxttxxn )(1
tuxtxxn )(
T typically at fixed grid => interpolation (Semi-Lagrange) =>diffusion
u
x(t+∆t)
=x3
x(t)
T(x1) T(x2)
T T
One-Dimensional Linear Advection
• Numerical methods: Spectral, finite differences and semi-Lagrange
• Stability analysis: Energy method, Von Neumann method
• Numerical diffusion: Amplitude damping (wave number dependent)
• Numerical dispersion: Error in phase velocity (wave number dependent)
• Important parameter: Courant-number (ut/x)
• Courant-Friedrich-Levy criterion
Summary
Frank Lunkeit
31
3. One-Dimensional Linear Diffusion and One-Dimensional Linear Transport Equation
Introduction to Numerical Modeling
Frank Lunkeit
The (Linear) Diffusion Equation (Heat Equation)
2
2
x
TK
t
T
equation (one dimensional):
K=const.= Diffusion coeff.
an analytic solution:
)exp()exp(),( 2
0 tKkikxTtxT damped wave (wave number k):
Exponential decay of the
amplitude dependent on k2
(the shorter the faster)
(parabolic, first order in time,
second order in space)
Initial and boundary values needed
Frank Lunkeit
32
Numerical Solution: Spectral Method
2
2
x
TK
t
T
N
Nk
k ikxtTxtT )exp()(),(T as Fourier-series:
)exp()()exp()( 2 ikxtTkKikx
t
tTk
N
Nk
N
Nk
k
Insert =>
=> 2N uncoupled ordinary differential equations kk TKkt
T 2
Numerical solution similar to decay equation
Frank Lunkeit
Frank Lunkeit
Grid Point Method: Finite Differences
2
2
x
TK
t
T
Discretization for second derivative in space:
2
11
22
2 2)(2)()(
x
TTT
x
xTxxTxxT
x
T jjj
1iT iT 1iT
xx x xx
x
TT
x
T ii
1
x
TT
x
T ii
1
x
T
xx
T2
2
a) from discretization of x
T
)()(2)()(
...2
)()(
...2
)()(
2
22
2
2
22
2
22
xOx
xTxxTxxT
x
T
x
Tx
x
TxxTxxT
x
Tx
x
TxxTxxT
b) from Taylor expansion
2
2
x
T
33
Finite Differences
2
2
x
TK
t
T
Possible discretizations (two level schemes)
a) Explicit (forward in time centered in space; FTCS): 2
11
1 2
x
TTTK
t
TT n
j
n
j
n
j
n
j
n
j
b) Implicit: 2
11
1
1
1
1 2
x
TTTK
t
TT n
j
n
j
n
j
n
j
n
j
c) Crank-Nicolson: 2
11
2
11
1
1
1
1 2)1(
2
x
TTTKg
x
TTTgK
t
TT n
j
n
j
n
j
n
j
n
j
n
j
n
j
n
j
2
11
2
2 2
x
TTT
x
T jjj
with
Note: Leap frog is unstable
Frank Lunkeit
Frank Lunkeit
Finite Differences
Forward in Time Centered in Space; FTCS
),(!)1(
!!
),(2),(),(),(),(
2
2
2
xtOErrErrx
x
T
n
x
x
T
n
x
Kt
t
T
n
t
Errx
txTtxxTtxxTK
t
txTttxT
n
nnn
n
nn
n
nn
)2
(sin4
1)1)(cos(2
1))2(1( 2
222
xk
x
tKxk
x
tKAee
x
tKTAT xikxiknn
=> stable for 14
12
x
tKA
For small Δx: )1())2
(4
1( 22
2
1 tKkTxk
x
tKTT nnn
analytical: ...)1()exp( 221 tKkTtKkTT nnn
Accuracy/order:
Stability: Von Neumann method Tn+1=ATn xikj
k
kj etTtT )()( and (
convergence
)
34
The (Linear) Transport Equation (Advection and Diffusion)
2
2
x
TK
x
Tu
t
T
equation (one dimensional):
An analytical solution:
)exp())(exp(),( 2
0 tKkutxikTtxT
damped traveling wave
(2nd order parabolic)
Frank Lunkeit
Finite Differences
2
2
x
TK
x
Tu
t
T
Previous knowledge: FTCS improper for advection; Leap frog improper for diffusion
2
11
1
1
111
11 2
22 x
TTTK
x
TTu
t
TT n
j
n
j
n
j
n
j
n
j
n
j
n
j
=> mixed scheme:
Leap frog für Advektion FTCS für Diffusion
More general:
Time splitting method:
Partitioning of tendencies and different numerical treatments
Climate models (ECHAM, PlaSim): Adiabatic part (leap frog) und diabatic parts (Euler or impl.)
Frank Lunkeit
35
Frank Lunkeit
Finite Differences
2
11
1
1
111
11 2
22 x
TTTK
x
TTu
t
TT n
j
n
j
n
j
n
j
n
j
n
j
n
j
Time splitting (Leap frog und FTCS)
i.e. stable for 2
2
2
2
44
u
K
u
x
u
Kt
22
2
)()4(4 xuKK
xt
or
Stability (Von Neumann method) Tn+1=ATn xikj
k
kj etTtT )()( ) and (
(1)
i
xikxiknxikxiknnn
eAxkxkx
tKxkiA
xkx
tKxkAiA
eeTx
tKeeTTT
)(sin))cos(1(4
1)sin(
))cos(1(4
1)sin(2
22
22
2
2
2
1
2
11
x
tu
(+=physical)
1...1))cos(1(4
1 2
2/1
2
tKkxk
x
tKAwith (i.e. analytical +…)
2
22
22
2
61)(6
11
sin)cos(1(4
1
sinarctan
x
tKxktuk
xkxkx
tK
xk
(1)
dependent on K
1-d Linear Diffusion and 1-d Linear Transport Equation
Summary
a) Diffusion equation (heat equation)
• Spectral: like decay equation
• Grid point: FTCS
b) Transport equation
• Time spitting method
Frank Lunkeit
36
4. Nonlinear Advection and 1-d Nonlinear Transport Equation (Burgers Equation)
Introduction to Numerical Modeling
Frank Lunkeit
The Nonlinear Advection Equation (Inviscid Burgers Equation)
equation (one dimensional): 02
10
2
x
u
t
u
x
uu
t
u
dt
du
Solution: )(),( utxftxu with any function f
Like linear advection, but f depends on u itself (implicit equation)
Solvable (for practical use) in few special cases only.
A ‚formal‘ (geometrical) solution can be constructed by using the method of
characteristics
)sin()0,( xtxu Initial condition:
1
)sin(),(~),(k
s kxtkutxuSolution: kt
ktItku k
s
)(2),(~ with
Ik(kt) = first kind Bessel function
e.g. Platzmann 1964, Tellus:
Frank Lunkeit
37
The Method of Characteristics
0
x
uu
t
u
dt
du
00 )0,( xttxux
Lagrangian perspective: discussing the paths of individual parcels (points)
(paths = characteristics)
0dt
du=> velocity of individual parcel is constant in time
=> parcels are moving on a straight lines:
=> characteristics are straight lines with slope u(x0,t=0)
if u(x0,t=0) varies in space: characteristics cross at a certain time tc
=> b) after tc one parcel can have more than one location (unphysical!)
=> physical solutions only possible up to t=tc
from
=> a) discontinuities (shock waves) are formed
Frank Lunkeit
Frank Lunkeit After: Platzmann, G.W., 1964: An exact integral of complete spectral equations for unsteady
one-dimensional flow. Tellus, 16, 422-431.
Characteristics of Platzmanns Solution
Analytic solution characteristics
t=0
t=1
t=2
t=3
u
x
38
Frank Lunkeit
Non linear term after inserting sine-series:
1 2
21
1 2
21
2
2
1
1
]))sin[(]))(sin[(()(
)cos()sin()()(
)cos()()sin()(
21212
212
221
k k
kk
k k
kk
k
k
k
k
xkkxkktutuk
xkxktutuk
xktukxktux
uu
The non linear wave-wave interaction of the waves k1 and k2 forces waves with
wave number k1+k2 and k1-k2.
=> Energy/momentum is redistributed within the wave spectrum
0
x
uu
t
u
L
nk
2Example:
Sine-series (waves) with time dependent amplitudes
k
k kxtutxu )sin()(),(
The nonlinear term:
The Nonlinear Advection Equation
1 2 3 40j=
Problem: With a finite number of grid points (e.g. 0-M) waves with wave
numbers k > M/2 get erroneously interpreted (as M-k)
Example: M=4; k=3 => ‚false‘ wave with k‘=1
Grid Point Method: Aliasing
Frank Lunkeit
39
Frank Lunkeit
Aliasing: Consequence for Energy
20 cos)sin()0,( 00
x(kx) ku
x
ukxutxulet
=> kin. Energy:
2
0
2
022
00
2)(sin
2
udxkx
uE
Energy change:
2
0
2
0
2
0
2
2
1dx
x
uuudx
t
uudx
t
u
t
E
Example:
2
0
3
0
2
0
2
0
0)2sin()sin(2
)2sin(2
)cos()sin(with
dxkxkxku
t
E
kxuk
kxkxkux
uu
But With resolution M=3k
2
0
3
0
3
0sin
2)sin()sin(
2)sin()2sin(
kudxkxkx
ku
t
Ekxkx
gAlia
=> Energy increase (due to aliasing from limited resolution)
=> analytically the total energy is conserved
• At each time level short waves (k1+k2) may be forced which (potentially)
are not resolved and, therefore, erroneously interpreted.
• This leads to unrealistic increase of wave amplitudes (i.e. energy), in
particular in the short wave part of the spectrum, and finally to a ‘blow up’ of
the numerical solution.
• This mechanism is, in principal, independent of the time step length or the
grid size.
• Nonlinear instability
Nonlinear Instability
Frank Lunkeit
40
Solutions
• Artificial elimination of short waves
• Diffusive schemes (e.g. upstream)
• Explicit diffusion
• Appropriate discretization of non linear terms
Nonlinear Instability
Frank Lunkeit
Frank Lunkeit
Analytically:
i.e. total energy (~u2) is conserved
x
dxt
u
x
u
t
u
x
uu
t
uu
x
uu
t
u0
2
1
3
1
2
1 2322
If this would also be the case numerically => no non linear instability
Solution: Discretization of the non linear term
x
uuu
x
uu ii
i
i
2
111st try:
02
1
21
2
1
2112
j
jjjj
j
jj
j uuuuuu
udxt
uuudx
t
uu
t
E
02
12
2
31
2
31
2
23
2
23
2
12
2
1 uuuuuuuuuuuu
example: 3 Points, cyclic boundary conditions:
Not appropriate
Discretization of the Nonlinear Term
41
j
jjjjj
jjjjjjjjjjjjj
j
uuuuuxt
E
x
uuu
x
uuu
x
uuu
x
uuuu
x
uu
06
1
2
)(
2
)(
2
)(
3
1
6
2121
21211121212nd try:
But: Still transfere of energy into ‚wrong‘ waves (unphysical)
06
1123213312132231321 uuuuuuuuuuuuuuuuuu
example: 3 points, cyclic boundary
conditions
appropriate
j
jjjjjjj
jjjjjj
jjjjj
j
uuuuuuuxt
E
uuuuuuxx
uuuuu
x
uu
06
1
6
1
23
2
111
2
1
2
111
2
1
11113rd try:
appropriate
Discretization of the Nonlinear Term
Frank Lunkeit
x
uuuu
t
uu
x
uu
t
unnnnn
i
n
i iiii
6
2121
1
options
Two level Euler:
x
uuuu
t
uu
x
uu
t
unnnnn
i
n
i iiii
62
2121
11
Three level Leap frog:
Caution: a) slightly unstable due to t-discretization
b) linear instability may occur => mind the CFL criterion
Finite Differences
Frank Lunkeit
42
Frank Lunkeit
=> right hand side:
N
Nk
k
k k
kk
k
k
k
k
ikxF
xkkiuuik
xikuikxikux
uu
2
2
212
221
)exp(
))(exp(
)exp()exp(
1 2
21
2
2
1
1
0
x
uu
t
u
N
Nk
k ikxtutxu )exp()(),(
Ansatz: transformation of u into new basis functions which are differentiable
orthogonal functions of x, e.g. Fourier-series:
N
Nk
k ikxt
u
t
u)exp(=> left hand side:
Since contributions to
wavenumbers -2N to 2N are
obtained
with k=k1+k2 and
2
1
)(L
Ll
lklk uulkiF (interaction coefficients)
L1=max(-N,k-N); L2=min(N,k+N), i.e. L1=k-N, L2=N for k ≥ 0,
and L1=-N, L2=k+N for k < 0
Spectral Method
Frank Lunkeit
with
N
Nk
k
N
Nk
k ikxFikxt
u
x
uu
t
u 2
2
)exp()exp( with
2
1
)(L
Ll
lklk uulkiF
L1=max(-N,k-N); L2=min(N,k+N)
NkN
k
N
Nk
k
N
Nk
k ikxFikxFikxF2
2
2
)exp()exp()exp(
resolved
waves
unresolved
waves
Neglecting the unresolved waves leads to 2N coupled ordinary differential
equations (ODE‘s):
2
1
)(L
Ll
lklkk uulkiFt
u L1=max(-N,k-N); L2=min(N,k+N)
-N ≤ k ≤ N
Cut off: no aliasing but not conserving moments higher than u2
To be solved using common methods
Spectral Method
43
Frank Lunkeit
spectral method versus finite differences:
spectral: exact spatial derivatives but numerical effort goes with N2
finite differences: numerical effort goes with N but spatial derivatives not exact
Improvement by combination: The spectral transform method
Idea: compute derivatives (and linear terms) in spectral space and non linear term
on corresponding grid (≥ 3N+1 grid points to avoid aliasing). The effort goes with N
ln(N) (using a fast fourier transform (fft)).
Non linear advection:
Step 1: transform u into grid point domain (inverse Fourier transformation (via fft))
Step 2: compute u2 on grid points
Step 3: transform u2 into spectral space (via fft)
Step 4: compute the x-derivative of u2 in spectral space, and do the time stepping
Step 1: ….
02
10
2
x
u
t
u
x
uu
t
u
The Spectral Transform Method
Frank Lunkeit
The (Viscid) Burgers Equation: Advection and Diffusion
equation (one dimension): 2
2
x
uK
x
uu
t
u
Various applications to describe processes in science and technology in a ‚simple‘
way (e.g. road traffic) and important test bed for numerical schemes.
Numerical Solution: Spectral or finite differences with the known schemes.
In most cases by using time splitting (separating advection and diffusion)
(formal) analytic solution using Cole-Hopf transformation: x
zK
x
z
z
Ku
ln2
2
2
2
2
22
2
2
2
22
ln2
ln2
ln2
2
1
x
zK
t
z
x
z
xK
x
z
xK
t
z
xK
x
uK
x
u
t
u
dK
txG
dK
txG
t
x
u
)2
),;(exp(
)2
),;(exp(
=>
t
xdutxG
2
)()(),;(
2
0
0
Linear heat equation
44
Numerical Solution of Burgers Equation using different K
Frank Lunkeit
Nonlinear Advection and 1-d Nonlinear Transport Equation (Burgers Equation)
Summary
• Viscid and inviscid Burgers equation
• Aliasing
• Non linear instability
• Energy conserving discretization
• Spectral transform method
Frank Lunkeit
45
5. More Dimensions (Grids)
Introduction to Numerical Modeling
Frank Lunkeit
Problem: not only one dimension and variable but three (x,y,z) dimensions and
various (coupled) variables (scalars and vectors).
typically: Scalars (temperature, height, vorticity, divergence, etc.) and 3d flow (u,v,w)
Numerics:
Grid point models: discretization using staggered grids (e.g. scalars shifted
compared to vectors).
(Semi-) Spectral models: (non linear terms) and parameterizations on corresponding
(Gaussian) grids with staggering in the vertical.
Three Dimensions with Scalar and Vector Fields
Frank Lunkeit
46
Frank Lunkeit
u u
v
v
T
u u
v
v
T u u
v
v
T
u u
v
v
T u u
v
v
T
E = two shifted C-grids
Grids
Horizontal: ‚classical‘ grids according to Arakawa (T=scalar, (u,v)=flow)
w vertically staggered to all other
variables (u,v,T), horizontally at T
or u or v grid points
z
w
u,v,T
u,v,T
w
w
Grids
Vertical: staggered
Frank Lunkeit
47
Frank Lunkeit
Grids
Global grids
‚classical‘: lon-lat grid
New: Icosahedron
ui ui-1
v
v
Ti
Staggered grid: Discretization using appropriate averaging:
x
TTu
x
TTu
x
Tu ii
iii
i
i
111
2
1
z
wk+1
Tk
wk
Example: Advection (x-direction) on C-grid:
Vertical advection:
z
TTw
z
TTw
z
Tw kk
kkk
k
k
11
1
2
1
x
TTuu
x
Tu ii
ii
i
2)(
2
1 111or (better)
∆x
z
TTww
z
Tw kk
kk
k
2)(
2
1 111or
Grids: Discretizations
Frank Lunkeit
48
More Dimensions (Grids)
Summary
• Grids after Arakawa (A,B,C,D,E)
• Staggered grid
• Lon-lat- und icosahedral-grids
Frank Lunkeit
6. Design of an Atmospheric General Circulation Model (AGCM)
Introduction to Numerical Modeling
Frank Lunkeit
49
Frank Lunkeit
An Atmospheric General Circulation Model (AGCM) ECHAM
Variables:
prognostic (dynamic) variables
boundary conditions (partially
prognostic)
Processes:
Adiabatic: moist atmospheric fluid
dynamics
Diabatic: forcing (sources) and
dissipation (sinks)
Representation (equations):
Adiabatic: resolved scales; based
on fundamental laws
(approximated; e.g.
primitive eq.)
Diabatic: unresolved scales;
parametrerized, i.e.
linked to the prognostic
variables (resolved
scales)
Numerics
Representation and discretization:
Horizontal: spectral transform method (Legendre polynomials) or grid point (finite
diff.); semi-Lagrangian tracer transport (q)
Vertical: staggered grid (z-, p-, σ- and/or θ-system (mostly mixed: p (top) and σ
(ground))
Time: time splitting method; adiabatic: mostly leapfrog with Asselin filter; semi-
implicit (divergence eq.); diabatic: explicit or implicit (e.g. diffusion)
Resolution:
Horizontal: about 10-500km
Vertical: about 20-100 levels; 0-30/100km
Time: some minutes
An Atmospheric General Circulation Model (AGCM)
Frank Lunkeit
50
Numerics
Input: Initial fields (or restart files) and boundary conditions
Output: Prognostic variables, deduced parameters, history (restart files)
Work flow:
1 Initialization: read boundary conditions (z0, albedo, SST, ice, etc.)
Set initial conditions (e.g. normal mode initialization for NWP,
start date for climate) or read history (restart files)
2 Time stepping: a) update boundary conditions (e.g. radiation, SST)
b) compute adiabatic and diabatic tendencies
c) update prognostic variables
d) write output
a) …
3 Termination: write history (restart files) for continuation
An Atmospheric General Circulation Model (AGCM)
Frank Lunkeit
Design of an Atmospheric General Circulation Model (AGCM)
Summary
• Time splitting method (adiabatic/diabatic)
• Resolved and unresolved processes -> parameterizations
• Prognostic and diagnostic variables
• Boundary an initial conditions
Frank Lunkeit
