Deutscher Wetterdienst

Transcription

1 Stability Analysis of the Runge-Kutta Time Integration Schemes for the Convection resolving Model COSMO-DE (LMK) COSMO User Seminar, Langen March 2008 Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

2 The operational Model Chain of DWD: GME, COSMO-EU and -DE (since 6. April 2007) Deutscher Wetterdienst GME COSMO-EU (LME) COSMO-DE (LMK) hydrostatic parameterised convection Δx 40 km * 40 GP Δt = 33 sec., T = 7 days used: Leapfrog + cd2 (Klemp, Wilhelmson, 978) planned: RK3 + up3 better solution possible? non-hydrostatic parameterised convection Δx = 7 km 665 * 657 * 40 GP Δt = 40 sec., T= 78 h used: RK3 + up5 (Wicker, Skamarock, 2002) non-hydrostatic resolved convection Δx = 2.8 km 42 * 46 * 50 GP Δt = 25 sec., T = 8 h 2

3 Runge-Kutta-Time-Integration autonomous ODE-system explicit N-stage Runge-Kutta-method (to integrate from t n to t n+ ): (other representations possible, e.g. substepping-form ) Butcher-Tableau: 3

4 Consistency condition at least st order accuracy: example: 4 conditions for RK 3rd order: stage # of β ij # of cond. for # of cond. for N N th order RK N th order LC-RK (?)

5 Standard-Testproblem linear, homogeneous ODE-system (M equations) with time-independent M*M Matrix P Theorem: for the Standard-Testproblem, an N-stage RK-method is of order N, if the N conditions hold. Such a scheme is called here Linear Case-Runge-Kutta (LC-RK) of order N. (h N+ (l) are polynomials in the β ij, for which an explicit expression can be given) example: the 3rd order RK method used for WRF (Wicker, Skamarock, 2002) is a 3-stage, 2nd order RK, but a 3rd order LC-RK Lemma: all LC-RK methods of order N behave similar for the standard testproblem. Lemma: All stage=order RK methods of order N (i.e. N-stage and Nth order also for nonlinear ODE's) are a subset of the LC-RK methods of order N all stage=order RK methods of the same order have the same stability properties for the standard testproblem! 5

6 Linear advection equation (-dim.) Spatial discretisations of the advection operator (order... 6) upwind st order centered diff. 2nd order upwind 3rd order centered diff. 4th order upwind 5th order cent. diff. 6th order from the Theorem and Lemma above: all Nth order (LC-) Runge-Kutta-schemes have the same linear stability properties for u=const.! 6

7 Stability limit for the effective Courant-number for advection schemes von Neumann stability analysis 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), submitted to JCP 7

8 Time splitting (or sub-cycling): distinguish between slow (adv.) and fast (sound, buoyancy) processes Matrix P in the standard test problem becomes time-dependent! general stability statements from above not applicable nevertheless LC-RK is not only of academic interest... Stability analysis of a 2D (horizontal + vertical) system with Sound + Buoyancy + Advection ( + Smoothing, Filtering ) 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 of about 2-3 km no orography (i.e. no metric terms) only horizontal advection 8

9 (p T -Dynamics) 9

10 0

11 Choice of CN-parameters for buoyancy in the p T -Dynamics of COSMO-DE β=0.5 ( purely Crank-Nic.) β=0.6 β=0.7 β=0.8 max. amplific. factor 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

12 Comparison between 3-stage RK-schemes Motivation: a model crash during the storm Kyrill ( 8. Jan ) Simplest'-LC-RK3 (3-stage, 2nd order) (Wicker, Skamarock, 2002) 0 /3 /3 /2 0 /2 0 0 TVD-RK3 (3-stage, 3rd order) (Fehlberg, 970; Shu, Osher, 988; Hundsdorfer et al., 995) 0 /2 /4 /4 /6 /6 2/3 Configuration of the following tests: Buoyancy: ~ standard atmosphere, off-centering β=0.7 Sound: off-centering β=0.7 aspect ratio: Δx/Δz=0 with divergence damping (C div = 0.) 2

13 S-LC-RK3 (3-stage, 2nd order) + up5 (Wicker, Skamarock, 2002) TVD-RK3 (Shu, Osher, 988) + up5 different small timesteps ΔT/Δt = 6 possible 2 2 3

14 Simplest'-LC-RK4 (4-stage, 2nd order) Are 4-stage RK-schemes competitive? 0 ¼ ¼ /3 0 /3 ½ 0 0 ½

15 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 5

16 Are 4-stage RK-schemes competitive? Simplest'-LC-RK4 (4-stage, 2nd order) 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') 6

17 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?) 7

18 Summary rather general RK-theory for the Standard-Testproblem developed application to the stability problem of D advection equation (partly analytical results for critical Courant numbers) linear stability of 2D Sound-Buoyancy-Advection-system determination of Crank-Nicolson-coeff. for sound and buoyancy discretization TVD-RK3 reduces spurious instabilities but also reduces the possible range for C adv RK4 + cd4 + add. smoothing could be a more efficient alternative to RK3 + up5; perhaps with better advection properties than RK3 + up3 careful choice of the 4-stage RK is needed. Is a better RK4 possible for timesplitting? 8

19 9

20 Stable Courant-numbers for advection schemes up cd2 up3 cd4 up5 cd6 Leapfrog LC-RK (Euler) LC-RK LC-RK LC-RK LC-RK LC-RK LC-RK Wicker, Skamarock, 2002: Leapfrog, RK2, RK3 20

21 2

22 Stabilitätsanalyse Tool von-neumann-analyse für je eine Komb. von c s, U, dt 0 /dz,... : bestimme die Stabilitätsmatrix Q (4*4-Matrizen) berechne deren Eigenwerte (Verwendung von LAPACK-EW-Routinen) Suche den maximalen Eigenwert durch Abscannen im Bereich kx Δx = -π...+ π, kz Δz = - π...+ π ab. Verifikation : analytisch bekannte Stabilitätsgrenzen (Advektion, Schall, Divergenzdämpfung) werden korrekt berechnet physikalische Schichtungsinstabilität (d.h. N 2 <0) wird wiedergegeben ( auch Kombination Auftrieb + Schall funktioniert) 22

23 Sound temporal discret.: generalized Crank-Nicholson β=: implicit, β=0: explicit spatial discret.: centered diff. Courant-numbers: fully explicit uncond. unstable - forward-backward (Mesinger, 977), unstaggered grid stable for C x2 +C z2 <2 neutral 4 dx, 4dz forward-backward, staggered grid stable for C x2 +C z2 < neutral 2 dx, 2dz forward-backw.+vertically Crank-Nic. (β 2,4,6 =/2) stable for C x < neutral 2 dx forward-backw.+vertically Crank-Nic. (β 2,4,6 >/2) stable for C x < damping 2 dx 23

24 xkd 24

25 Need of a divergence damping: without divergence damping with 3D-(isotropic)- divergence damping with quasi-3d (non-isotropic) divergence damping 25

26 Stability of single waves in C adv =, C snd =0.7 without divergence damping with 3D-(isotropic)- divergence damping with quasi-3d (non-isotropic) divergence damping 26

27 4-stage, 2nd order RK (MS-LC-RK4) + cd4 no smoothing + 4th order diffusion instab by short waves 27