18.09.2006 1M. Baldauf, DWD
Numerical contributions to the Priority Project ‘Runge-Kutta’
COSMO General Meeting, Working Group 2 (Numerics)
Bukarest, 18.09.2006
Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany
18.09.2006 2M. Baldauf, DWD
Outline
• Stability analysis of the p‘T‘-dynamics (PP ‚Runge-Kutta, Task 10)
• Vertical advection of 3. order (PP ‚Runge-Kutta‘, Task 8)
• New stability criteria for the small and the big timestep
• Tool for conservation properties of LM (PP ‚Runge-Kutta‘, Task 3)
18.09.2006 3M. Baldauf, DWD
Von-Neumann-stability analysis of a 2D (horizontal + vertical) system with advection + sound + buoyancy ( + smoothing, filtering)
constraints:• no bondaries (wave expansion in an extended medium)• assumptions about the base state: p0=const, T0=const
valid for atmospheric motions with about 2-3 km vertical extension• no orography (i.e. no metric terms)• only horizontal advection
PP ‚Runge-Kutta‘, Task 10
18.09.2006 4M. Baldauf, DWD
(p‘T‘-Dynamics)
18.09.2006 5M. Baldauf, DWD
Tool to inspect stability properties
von-Neumann-analysis for any combination of cs, U, dT0/dz, T, t, ... :
• calculate the amplification matrizes Q (4*4-matrix)• calculate the eigenvalues (use of LAPACK-EV-subroutines)• search of the biggest EV by scanning through kx x = -...+ , kz z = - ...+ .
‘verification’:
• analytically known stability limits (advection, sound, divergence damping) are calculated correctly
• physical stratification instability (i.e. N2<0) will be reproduced( combination ‘buoyancy + sound’ works)
18.09.2006 6M. Baldauf, DWD
Sound
Courant-numbers:
2 dxdampingstable for Cx<1forward-backw.+vertically Crank-Nic. (2,4,6>1/2)
2 dxneutralstable for Cx<1forward-backw.+vertically Crank-Nic. (2,4,6=1/2)
2 dx, 2dzneutralstable for Cx2+Cz
2<1forward-backward, staggered grid
4 dx, 4dzneutralstable for Cx2+Cz
2<2forward-backward (Mesinger, 1977), unstaggered grid
-uncond. unstablefully explicit
• temporal discret.:‘generalized’ Crank-Nicholson=1: implicit, =0: explicit
• spatial discret.: centered diff.
18.09.2006 7M. Baldauf, DWD
18.09.2006 8M. Baldauf, DWD
Choose CN-parameters for buoyancy in p‘T‘-dynamical core of the LMK=0.5 (‚pure‘ Crank-Nic.) =0.6 =0.7
=0.8 =0.9 =1.0 (purely implicit)
choose =0.7 as the best value
18.09.2006 9M. Baldauf, DWD
‚Verification‘ of the stability analysis tool:dependency from stratification
C=-0.35 C=-0.37 C=-0.38
C=-0.39 C=-0.395 C=-0.4
C=-0.41 C=-0.42 C=-0.45
critical value: C=-0.399 N=0
Tool works for ‚buoyancy + sound‘
18.09.2006 10M. Baldauf, DWD
xkd
18.09.2006 11M. Baldauf, DWD
Cdiv=0.025 Cdiv=0.05
Cdiv=0.075 Cdiv=0.1 Cdiv=0.15
Influence of Cdiv
Cdiv = xkd * (cs * t/ x)2
~0.35
18.09.2006 12M. Baldauf, DWD
without divergence filtering with 3D- divergence filtering with 2D- (only horizontal)divergence filtering
18.09.2006 13M. Baldauf, DWD
stability of the single waves for Cadv=1, Csnd=0.7
without divergence filtering with 3D- divergence filtering with 2D- (only horizontal)divergence filtering
18.09.2006 14M. Baldauf, DWD
summary
von-Neumann-stability analysis of a 2D (horizontal + vertical) system with advection + sound + buoyancy ( + smoothing, filtering)
• without divergence damping: weak instability of long waves remains• without orography: no relevant difference between 2D- and
(assumed better) 3D-Divergenzdämpfung.remark A. Gassmann: no difference in amplitude- but in phase-information
• experience from LMK: there exist cases which are unstable with 2D-divergence damping (orography?)there exist cases which can be stabilized by 2D-divergence damping implementation of 3D-div.-damping seems to be reasonable?
• LMK-Testsim. mit Bryan-Fritsch-Test: nur mit 3D-Divergenzdämpfung stabil
18.09.2006 15M. Baldauf, DWD
Improved vertical advection for the dynamical variables (u, v, w, T or T‘, p‘)
Motivation: explicitly resolved convection • vertical advection has increased importance use a scheme of higher order
(compare: horizontal adv. from 2. order to 5. order in RK-scheme)• greater w (~20 m/s) Courant-criterium is violated
implicit scheme or CNI-explicit scheme
up to now: implicit (Crank-Nicholson) advektion 2. order (centered differences)new: implicit (Crank-N.) advektion 3. order LES with a 5-banddiagonal-matrix
implicit Adv. 3. Ordn. in every RK-step: computation costs ~30% of total computation time!
planned: outside of the RK-scheme (splitting error?, stability with fast waves?)
PP ‚Runge-Kutta‘, Task 8
18.09.2006 16M. Baldauf, DWD
Implicit Vertical Advection for dynamic variables (u, v, w, T or T‘, p‘)
Generalized Crank-Nicholson-advection
(Dimensionless) Advection operator for centered differences 2. order (3-point-stencil):
Lin. eq. system with a tridiagonal matrix,needs ~3 N operations
Motivation for a better scheme:explicitly resolved convection, higher values of w
18.09.2006 17M. Baldauf, DWD
dim.less advection operator for upwind 3. order (4-point stencil)
=1/2: unconditionally stable, damping, truncation error 3. order>1/2: unconditionally stable, damping, trunc. error 1. order<1/2: unstable
Lin. eq. system with a 5-band diagonal matrixneeds ~14 N operations
LMK: Subr. complete_tendencies_uvwTpp_CN3
( Crowley 3. order, e.g. Tremback et al., 1987)case Cj>0
18.09.2006 18M. Baldauf, DWD
Idealized 1D advection testanalytic sol.implicit 2. orderimplicit 3. orderimplicit 4. order
C=1.580 timesteps
C=2.548 timesteps
18.09.2006 19M. Baldauf, DWD
Real case study: LMK (2.8 km resolution) ‚12.08.2004, 12UTC-run‘
implicit vertical adv. 2. order difference: 3. order - 2. order
18.09.2006 20M. Baldauf, DWD
Real case study: LMK (2.8 km resolution)‚ 25.06.2005, 00UTC-run‘
implicit vertical adv. 2. order difference: 3. order - 2. order
18.09.2006 21M. Baldauf, DWD
Calculation of small timestep
• Use correct gridlength: x = R cos (important for bigger areas)
• Use correct 2D criterion (dependency from x and y)
Subr. calc_small_timestep
To do:tcrit is calculated with T=303 K (?)• influence of buoyancy?
18.09.2006 22M. Baldauf, DWD
• 2D-advection in RK-schemes by a simple adding of tendencies(operator splitting (e.g. corner transport upstream (CTU) method) is not possible for upstream 3., 5., ... order )
• this is limited by
|Cx| + |Cy| < const.
this can be proven for RK2 + upwind 3. orderit holds empirically also for RK3 + upwind 5. order
• compare with the usual formulated 2-dimensional stability criterion:
Subr. check_cfl_horiz_advection
2-dim. horizontal Advection
18.09.2006 23M. Baldauf, DWD
Tool to inspect conservation properties of LM
• integral over a volume (arbitrary square-stone): ready• Subr. init_integral_3D: define square-stone (in the transformed grid!), domain decomp. • Function integral_3D_total: calc. volume integral• Function integral_3D_cond: calc. vol. integral over individual processor
• surface integral over the fluxes: work to do!
PP ‚Runge-Kutta‘, Task 3
balance equation for scalar :
temporal change
flux divergence
sources / sinks
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