ecmwf governing equations 3 slide 1 numerical methods iv (time stepping) by nils wedi (room 007;...

ECMWFGoverning Equations 3 Slide 1

Numerical methods IV(time stepping)

by Nils Wedi (room 007; ext. 2657)

In part based on previous material by Mariano Hortal and Agathe Untch


What is the basis for a stable numerical implementation ?

A: Removal of fast - supposedly insignificant - external and/or

internal acoustic modes (relaxed or eliminated), making use of

infinite sound speed (cs) and/or the hydrostatic approximation

from the governing equations BEFORE numerics is introduced.

B: Use of the full equations WITH a semi-implicit numerical framework, reducing the propagation speed (cs 0) of fast acoustic and buoyancy disturbances, retaining the slow convective-advective component (ideally) undistorted.

C: Split-explicit integration of the full equations, since explicit NOT practical (~100 times slower)

Determines the choice of the numerical scheme


Choices for numerical implementation

Avoiding the solution of an elliptic equation

fractional step methods (eg. split-explicit); Skamrock and Klemp (1992); Durran (1999)

Solving an elliptic equation

Projection method; Durran (1999)

Semi-implicit Durran (1999); Cullen et. al.(1994); Benard et al. (2004); Benard (2004); Benard et al. (2005)

Preconditioned conjugate-residual solvers (eg. GMRES) or multigrid methods for solving the resulting Poisson or Helmholtz equations; Skamarock et. al. (1997); Saad (2003)

Direct Methods; Martinsson (2009)


Split-explicit integrationSkamarock and Klemp (1992); Durran (1999);Doms and Schättler (1999);

‘Slow’ part of solution‘Fast’ part of solution

e.g. implemented in popular limited-area models: Deutschland Modell, WRF model


Semi-implicit schemes

(i) coefficients constant in time and horizontally (hydrostatic models Robert et al. (1972), Benard et al. (2004), Benard (2004) ECMWF/Arpege/Aladin NH)

(ii) coefficients constant in time Thomas (1998); Qian, (1998); see references in Bénard (2004)

(iii) non-constant coefficients Skamarock et. al. (1997), (UK Met Office NH model, EULAG model)

non-linear term, treated explicit

linearised term, treated implicit


Design of semi-implicit methods

Treat all terms involving the fastest propagation speeds implicitly (acoustic waves, gravity waves).

Assume that the energy in those components is negligible.Consider the solvability of the resulting implicit system,

which is typically an elliptic equation.


Example: Shallow water equations

0

0

0

0

u u hU g

t x xh h u

U Ht x x

Linear analytic solution:

ikxti eeutxu 0),(

Phase speed: gHUk

c

0

Linearized:

H denotes here a mean state depth.


0

0

( ) 0

u u uu v fv

t x y x

v v vu v fu

t x y y

u vu v

t x y x y

Shallow water equations

: advection

In the linear version:2

0202 VU

st

2

st

2 20 02

st

U V

: gravity-wave (or sometimes called ‘adjustment’) term

combinedadvection adjustment

In the atmosphere

5

300 /

10

gH m s

s m

in synoptic-scale models ==> Δt≤ 236 sec ~ 4 min

2 20 0U V


Explicit time-stepping

• Leap-frog explicit scheme

1 1

1 1

1 1

n n n n nj j j j x j

n n n n nj j j j y j

n n n n nj j j j j

tu u tV u

st

v v tV vs

ttV V

s

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&

( , )

( , )

j j j

x y

x j j x j x

y j j y j y

gH

x y s

V u v

A A A

A A A

&&&&&&&&&&&&&&

&&&&&&&&&&&&&&

x x x

x x x

x x xj-Δx j+Δx

j+Δy

j-Δy

j

Stability:

If we treat implicitly the advection terms we do not get a Helmholtz equation

2

st


Increasing the allowed timestep

• Forward-backward scheme

0

0

x

u

t

xt

u

)(2

)(2

11

11

1

111

nj

nj

nj

nj

nj

nj

nj

nj

x

tuu

uux

tforward

backward

von Neumann gives 2)sin(

xkx

tdoubles the leapfrogtimestep

• Pressure averaging

)]}()[())(21{(2

1

211

11

11

1111

11

nj

nj

nj

nj

nj

nj

nj

nj

xt

uu

if ε=0 ------> leapfrog

if ε=1/4 we get 2)sin(

xkx

t doubles the leapfrogtimestep


Split-explicit time-stepping

1

1

1

s n n nj j s j j

s n n nj j s j j

s n n nj j s j j

u u t V u

v v t V v

t V

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

Stability as beforebut M times a simplerproblem.

Note: The fast solution may be computed implicitly.

2f

st

1

1

1

fn s nj j x j

fn s nj j y j

fn s nj j j

tu u

st

v vs

tV

s

&&&&&&&&&&&&&&

slowfast

2 20 02

s

st

U V

s ft M t

Potential drawbacks: splitting errors, conservation. However recent advances for NH NWP suggested in (Klemp et. al. 2007)


Semi-implicit time-stepping

1 1 1 1

1 1 1 1

1 1 1 1

( )2

( )2

( )2

n n n n n nj j j j x j j

n n n n n nj j j j y j j

n n n n n nj j j j j j

tu u tV u

st

v v tV vs

ttV V V

s

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&& &&&&&&&&&&&&&&

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&

( , )

( , )

j j j

x y

x j j x j x

y j j y j y

x y s

V u v

A A A

A A A

&&&&&&&&&&&&&&

&&&&&&&&&&&&&&

x x x

x x x

x x xj-Δx j+Δx

j+Δy

j-Δy

j

Solve:2

2 1 1 , 12

4( )

( )n n n nj j

sF

t

Helmholtz equation !

Stability: now only limited by the advection termsNote: if we also treat the advection terms implicitly we do not get a Helmholtz equation!


Compressible Euler equations

Davies et al. (2003)


Compressible Euler equations


A semi-Lagrangian semi-implicit solution procedure

Davies et al. (1998,2005)

(not as implemented, Davies et al. (2005) for details)



Non-constant-coefficient approach!

Helmholtz equation (solutions see e.g. Skamarock et al. 1997, Smolarkiewicz et al. 2000)


Semi-implicit time integration in IFS

Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment.

In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation)and long internal gravity waves. => implicit treatment of the adjustment terms.

L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at “+” that can be solved almost analytically in a spectral model.


Two-time-level semi-Lagrangian semi-implicit time integration in the hydrostatic IFS

xRHSDt

DX

0 0 1/2 1/2 *0.5 0.5 xX tL X tL tRHS tL X

Notations:X : advected variableRHS: right-hand side of the equationL: part of RHS treated implicitlySuperscripts: “0” indicates value at dep. point (t)

“1/2” indicates value at mid-point (t+0.5Δt) “+” indicates value at arrival point (t+Δt)

0 1/20.5 ( )tt L L L L For compact notation define:

“semi-implicit correction term”

L=RHS => implicit schemeL= part of RHS => semi-implicit (β=1)L=0 => explicit (β=0)

1/2x tt

DXRHS L

Dt =>



1

0

h

TT

vv

)v(1

)(ln

)1(1

lnvv

dp

pp

t

KPpq

T

Dt

DT

KPpTRkfDt

D

ss

v

vdhh

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

semi-implicitequations

semi-implicit corrections


1

0

1

0

γ

1

d r

r

d r r

pd r

r

sr

R X dpX d

p d

R T dpX X d

c p d

dpX X d

p d


semi-implicitequations

Where:

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

DRHSpt

DRHSDt

DT

pTRTRHSDt

D

ttps

ttT

srdtth

)(ln

lnγv

v

Reference state for linearization:

rT ref. temperature

srp ref. surf. pressure

=> lin. geopotential for X=T

=> lin. energy conv. term for X=D


Linear system to be solved

2 *

*

*

0.5 ( log( ))

0.5

log( ) 0.5

d r s

s

D t T R T p D

T t D T

p t D P

2 *

*

*

0.5 ( log( ))

0.5

log( ) 0.5

d r s

s

D t T R T p D

T t D T

p t D P

Eliminate all variables to find also aHelmholtz equation for D+ :

2 2 2 * 2 * *(1 0.25 ( ) ) 0.5 ( )d r d rt R T D D t T R T P 2 2 2 * 2 * *(1 0.25 ( ) ) 0.5 ( )d r d rt R T D D t T R T P

DD~

I 2

rdTR γ operator acting only on the verticalI = unity operator



DD~

I 2

rdTR γ

Vertically coupled set of Helmholtz equations. Coupling through

Uncouple by transforming to the eigenspace of this matrix gamma(i.e. diagonalise gamma). Unity matrix “I” stays diagonal. =>

DDi

~1 2

One equation for each LevNi 1

In spectral space (spherical harmonics space):

mn

mni DD

a

nn ~)1(1

2

22

( 1)m mn n

n nY Y

a

because

Once D+ has been computed, it is easy to compute the other variables at “+”.

ecmwf governing equations 3 slide 1 numerical methods iv (time stepping) by nils wedi (room 007;...

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