Stabilitätsanalyse des Runge-Kutta- Zeitintegrationsschemas für das konvektionserlaubende Modell COSMO-DE

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1 Stabilitätsanalyse des Runge-Kutta- Zeitintegrationsschemas für das konvektionserlaubende Modell COSMO-DE DACH-Tagung , Bonn Michael Baldauf GB FE, Deutscher Wetterdienst, Offenbach M. Baldauf (DWD)

2 COSMO-DE (LMK) Basic time integration scheme for dynamical core: 3-stage Runge-Kutta (Wicker, Skamarock, 2002, MWR) Efficiency achieved by: time-splitting: slow process with large time step ΔT = 25 sec. fast process with small time step Δt = 4.6 sec Implicit vertical discretization non-hydrostatic resolved convection Δx = 2.8 km 42 * 46 * 50 GP Δt = 25 sec., T = 2 h (since 6. April 2007) Questions: Which stability statements can be made? Stability analysis in time and space Temporal order of the integration scheme? possible improvements/alternatives? M. Baldauf (DWD)

3 The current 3-stage RK-scheme in the COSMO-model Wicker, Skamarock (2002) MWR Vertical advection Horizontal advection pressure gradient, divergence terms, Coriolis force, physics tendencies, In the following: no vertical advection considered α=0 M. Baldauf (DWD)

4 Temporal order conditions for the time-split RK3 slow processes (e.g. advection): fast processes (e.g. sound, gravity wave expansion): Insertion into the time-split RK3WS integration scheme: n s := ΔT / Δt compare with exp [ ΔT (P s + P f ) ] only 2nd order for: slow process = Euler forward and n s ; fast process can be of 2nd order M. Baldauf (DWD)

5 Linear advection equation (-dim.) discretized: Spatial discretisations of the advection operator (order... 6) (u>0 assumed) upwind st order centered diff. 2nd order iadv_order= iadv_order=2 iadv_order=3 iadv_order=4 iadv_order=5 iadv_order=6 from the Theorem and Lemma below: all Nth order (LC-) Runge-Kutta-schemes have the same linear stability properties for u=const.! Hundsdorfer et al. (995) JCP Wicker, Skamarock (2002) MWR M. Baldauf (DWD)

6 Stable Courant-numbers for advection schemes up cd2 up3 cd4 up5 cd6 LC-RK (Euler) LC-RK LC-RK LC-RK LC-RK LC-RK LC-RK M. Baldauf (DWD)

7 Stability limit for the effective Courant-number for advection schemes C eff := C / s, s = stage of RK-scheme up cd2 up3 cd4 up5 cd6 Euler LC-RK LC-RK LC-RK LC-RK LC-RK LC-RK Baldauf (2008): Stability analysis for linear discretizations of the advection equation with Runge-Kutta time integration, J. Comput. Phys. General theory for Linear Case (LC)-Runge-Kutta-schemes, applied to the Dahlquist test problem M. Baldauf (DWD)

8 Stability analysis of a 2D (horizontal + vertical) system with Sound + Buoyancy + Advection ( + Smoothing, Filtering ) Advektion Schall Auftrieb p=p 0 +p T=T 0 +T Divergence damping: Von Neumann stability analysis Restrictions: no boundaries (wave expansion in extended medium) base state: p 0 =const, T 0 =const (for the coefficients) (but stratification dt 0 /dz 0 possible) application to structures with a vertical extension << 0 km no orography (i.e. no metric terms) only horizontal advection M. Baldauf (DWD)

9 Stabilität einzelner Wellen für Advektion + Schall (partial operator splitting mit RK3, C adv =.0, C snd,x =0.6) (ohne Div.-dämpfung) Sound term discretization: β s =0.5 Spatial: centered differences Temporal: Forward-backward (Mesinger, 977) Vertically implicit (β s ½) k z Δz stable for C snd,x <, C snd,z arbitrary β s =0.7 k x Δx M. Baldauf (DWD)

10 Divergence damping leads to a diffusion of divergence damping of sound waves Isotropic div. damping recommended: Gassmann, Herzog (2007) MWR Courant-numbers: 2D, explicit, stagg. grid: stable for C div,x + C div,z < ½ 2D, hor. expl.-vert. impl., stagg. grid: stable for C div,x < ½, C div,z arbitrary C div,x = 0. für COSMO-DE α div =60000 m²/s Skamarock, Klemp (992) MWR, Wicker, Skamarock (2002) MWR M. Baldauf (DWD)

11 Stabilität einzelner Wellen für Advektion + Schall (partial operator splitting mit RK3, C adv =.0, C snd,x =0.6) ohne Div.-dämpfung mit Div.-dämpfung β s =0.5 k z Δz β s =0.7 k x Δx M. Baldauf (DWD)

12 M. Baldauf (DWD)

13 Stabilität einzelner Wellen für Advektion + Schall + Auftrieb (partial operator splitting mit RK3, C adv =.0, C snd,x =0.6) ohne Div.-dämpfung mit Div.-dämpfung β s =0.5 k z Δz β s =0.7 k x Δx M. Baldauf (DWD)

14 Choice of CN-parameters for buoyancy in the p T -Dynamic of COSMO-DE β=0.5 ( purely Crank-Nic.) β=0.6 β=0.7 β=0.8 β=0.9 β=.0 (purely implicit) C adv = u * ΔT / Δx choose β=0.7 as the best value C snd = c s * Δt / Δx M. Baldauf (DWD)

15 Simplest'-LC-RK4 (4-stage, 2nd order) Are 4-stage RK-schemes competitive? 0 ¼ ¼ /3 0 /3 ½ 0 0 ½ M. Baldauf (DWD)

16 S-LC-RK3 (3-stage, 2nd order) + up5 ΔT/Δt = 6 no smoothing S-LC-RK4 (4-stage, 2nd order) + cd4 ΔT/Δt = 2 no smoothing + 4th order diffusion instab. by long waves instab. by short waves instab. by long waves M. Baldauf (DWD)

17 Simplest'-LC-RK4 (4-stage, 2nd order) Are 4-stage RK-schemes competitive? 0 ¼ ¼ /3 0 /3 ½ 0 0 ½ RK-SSP(4,3) (4-stage, 3rd order) (Ruuth, Spiteri, 2004) 0 /2 /2 /2 /2 /2 /6 /6 /6 /6 /6 /6 /2 'classical' RK4 (4-stage, 4th order) (Numerical recipies) 0 ½ ½ ½ 0 ½ 0 0 /6 /3 /3 /6 (negative coeff. arises if written in 'substepping-form') M. Baldauf (DWD)

18 Other RK4 + cd4 + add. smoothing... S-LC-RK4 (4-stage, 2nd order) SSP(4,3)-RK 4-stage, 3rd order (Ruuth, Spiteri, 2004) is not a LC-RK-scheme! 'classical' RK4 (4-stage, 4th order) strong stability preserving schemes do not automatically work together with timesplitting completely unstable (!?) due to negative coefficients in the substepping-form? M. Baldauf (DWD)

19 Summary RK3 time-splitting: only of 2nd order in time for Euler-forward advection and n s ; fast processes (sound, gravity waves) in 2nd order helps for finite n s von Neumann stability-analysis of 2D Euler equations (Sound-Buoyancy- Advection) in time and space: without metric terms: no off-centering for sound needed (β s = ½) (with metric: some off-centering recommendable) C div ~ 0. is recommended buoyancy needs off-centering (β B = 0.7, and even then is slightly unstable) RK4 + cd4 + weak add. smoothing could be a more efficient alternative to RK3 + up5; with better advection properties than RK3 + up3 Baldauf, 200: Linear stability analysis of Runge-Kutta based partial time-splitting schemes for the Euler equations, MWR M. Baldauf (DWD)