Deutscher Wetterdienst
COSMO Priority Project:
Further developments of the Runge-Kutta
Time Integration Scheme
report ‘Oct. 2007 – Sept. 2008’ / final report
COSMO General Meeting, Cracow
15.-19.09.2008
Michael Baldauf
Deutscher Wetterdienst, Offenbach, Germany
FE 13 – 22.03.2016
1
Deutscher Wetterdienst
Tasks of the Priority Project ‚Runge-Kutta‘:
Repair detected model deficiencies:
1. Looking at pressure bias
4. Advection of moisture quantities in conservation form
6. Deep valleys
7. Different filter options for orography
14. DFI for RK
New developments:
8. Higher order discretization in the vertical for Runge Kutta scheme
9. Physics coupling scheme
10. Testing of alternative fast wave scheme
13. Divergence damping in a truely 3D-version
Tool development:
2. Continue RK case studies
3. Conservation tool
5. Investigation of convergence
11. Development of a more conservative dynamics (planned)
12. Development of an efficient semi-implicit solver in combination with RK time integration scheme (planned)
FE 13 – 22.03.2016
2
Deutscher Wetterdienst
Task 1: Looking at pressure bias
(Torrisi, Zängl)
Talk by Lucio Torrisi
Goals:
verifications of LM 7 km runs showed a higher positive pressure bias for the RK core
than for the Leapfrog core, whereas other variables show comparable behaviour.
Reasons and solutions?
starting
point of
the task:
RK
Leapfrog
FE 13 – 22.03.2016
3
Deutscher Wetterdienst
Task 2: Continue RK case studies
(deMorsier,Torrisi)
Identify problems of the RK scheme
Several unstable cases found in previous winter periods (e.g. ‚13. Jan. 2004‘)
most of them could be simulated with Semi-Lagrange Adv. for moisture variables
Winter storms Kyrill ('18.01.2007') and Lothar ('26.12.1999') simulated with
MeteoCH new pre-operational model chain (2.2 km and 6.6 km):
 new configuration:
1.) WRF-like RK3 used (instead of TVD-RK3)
(as found at DWD for the Kyrill case)
2.) Semi-Lagrange-Adv. for moisture (instead of Bott-scheme)
3.) new level distribution especially in boundary layer (cures problems with TKE scheme)
FE 13 – 22.03.2016
5
Deutscher Wetterdienst
Vertical level distribution
 Test-chains in July 2007 using operational (L60.2) and a new (L60.1)
vertical level distributions
 Three test cases
 12.7.2006 (convection)
 23.12.2006 (fog)
 18.1.2007 (Kyrill)
TP
BL
6
Deutscher Wetterdienst
TKE Instability
L60 v2
23.12.2006 (fog)
gridpoint (60,180)
L60 v1
• Strong checkerboard instability in TKE-field
• Same effect also 12.7.2006 (convection) and 18.1.2007 (Kyrill)
7
Deutscher Wetterdienst
Stability of TKE-diffusion
 If equation is solved explicitly, stability constraints apply
 In COSMO, the diffusion constant is limited
 Default value for securi = 0.85 is wrong!!!
Alternative: vertically implicit (Crank-Nicholson) scheme was implemented
this cures the most problems; some artefacts remain (stability functions?)
8
O. Fuhrer
Deutscher Wetterdienst
Task 3: Conservation
(Baldauf)
Tool for inspection of conservation properties will be developed.
balance equation for scalar :
temporal
change
flux
divergence
sources
/ sinks
integration area = arbitrarily
chosen cuboid (in the transformed
grid, i.e. terrain-following)
Status: available in LM 3.23:
• Subr. init_integral_3D: define cuboid (in the transformed grid!), prepare domain decomp.
• Function integral_3D_total: calc. volume integral
V ijk Vijk
• Subr. surface_integral_total: calc. surface integrals
V jijk * Aijk
• prelimineary idealised tests were carried out
• report finished; will be published in the next COSMO-Newsletter Nr. 7 (2007)
Task is finished
(Study of conservation properties will be continued in collaboration with MPI-Hamburg, see WG2 Task 2.10.1)
FE 13 – 22.03.2016
9
Deutscher Wetterdienst
Task 4: Advection of moisture quantities in conservation form
(Förstner, Baldauf)
Status: two schemes available
implementation of the Bott (1989)-scheme into the Courant-number independent
advection algorithm for the moisture densities with mass consistency (Easter, 1993,
Skamarock, 2004, 2006)
Task was finished in Sept. 2006 because implemented
schemes (Bott-2, Bott-4) behaved well
But in the meanwhile: stability problems occured in some cases (steep orography!)
revival of the task necessary!
Semi-Lagrange-scheme (backtraj. 2nd order, tri-cubic interpolation)
multiplicative filling algorithm for global conservation
FE 13 – 22.03.2016
10
Deutscher Wetterdienst
Task 5: investigation of convergence
(Ceci, Vitagliano)
Goals: determination of the spatial and temporal order of convergence of the RKscheme in combination with advection schemes of higher order.
Test cases:
•
•
•
•
•
linear, 2D, hydrostatic mountain flow (h=10 m, a=10 km)
linear, 2D, non-hydrostatic mountain flow (h=10 m, a=500 m)
nonlinear, 2D mountain flows (dry case) (h=500 m, a=10 km)
linear, 3D mountain flow
nonlinear mountain flows with precipitation
FE 13 – 22.03.2016
11
Deutscher Wetterdienst
Deliver a program to calculate linear analytic solutions
(e.g. for convergence tests)
(M. Baldauf)
Starting point: compressible Euler equations
Preconditions:
• no friction
• only adiabatic processes (in particular no phase changes)
• ideal gas law
• cp=const., cV=const., R=const.
• no Coriolis force
• all movements take place on a plane (no earth curvature)
these preconditions can easily be fulfilled by a dynamical core (‚switches‘)
Only 2 approximations will be made:
1. linearisation (1/Fr<<<1  very flat mountains; not too small U)
2. the assumption that kz=const (see below; not absolutely necessary)
confidence into the accuracy of the linear solution for comparison with
numerical models
FE 13 – 22.03.2016
12
Deutscher Wetterdienst
Stationary case (=0)
From perturbation equations: express u', v', ' and p' by w'
 equation 2nd order for w'(kx, ky, z):
with coefficient functions:
The only approximation so far is linearisation!
FE 13 – 22.03.2016
13
Deutscher Wetterdienst
Example 2: 2D-test case from Schaer et al (2002)
w [m/s]
FE 13 – 22.03.2016
14
Deutscher Wetterdienst
w [m/s]
Example 3: 3D Gaussian Hill
FE 13 – 22.03.2016
15
Deutscher Wetterdienst
Initialization of the perturbation pressure field



G. Zängl
The present initialization of the perturbation pressure field
(executed in src_artifdata for idealized simulations; otherwise in
int2LM) is not exactly consistent with the discretized buoyancy
term in the vertical momentum equation
The error is too small to be noticeable in real-case applications;
however, it becomes evident in idealized simulations with
constant flow and a very low mountain (or no mountain at all)
To fix the problem, a new initialization procedure has been
developed by solving the discretized vertical wind equation (for
dw/dt = 0) for p‘; ideally, this would ensure strict absence of
buoyancy at the lateral model boundaries
FE 13 – 22.03.2016
16
Deutscher Wetterdienst
Simulation with flat surface, u = 10m/s, and fixed relaxation b.c.‘s, t = 12 h
Fields: θ (contour interval 2 K), w (colours)
Old p‘ initialization
Error amplitude: 1 mm/s
FE 13 – 22.03.2016
New p‘ initialization
Error amplitude: 10-4 mm/s
17
Deutscher Wetterdienst
Gaussian mountain height=750 m size=10 km
P. L. Vitagliano, G. Ceci
Horizontal resolution 4 km
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100m
Solution at Y=0 symmetry plane - RK3 UP5
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100m
Solution at Y=0 symmetry plane - RK3 UP3
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
3th order upwind
5th order upwind
3D TEST CASES: HYDROSTATIC FLOW
Krakow - September, 15th
2008
18
Deutscher Wetterdienst
Gaussian mountain height=750 m size=10 km
Horizontal resolution 8 km
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100m
Solution at Y=0 symmetry plane - RK3 UP3
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100m
Solution at Y=0 symmetry plane - RK3 UP5
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
3th order upwind
5th order upwind
3D TEST CASES: HYDROSTATIC FLOW
Krakow - September, 15th
2008
19
Deutscher Wetterdienst
Gaussian mountain height=750 m size=10 km
Horizontal resolution 16 km
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100m
Solution at Y=0 symmetry plane - RK3 UP3
Gaussian Mountain
h=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100m
Solution at Y=0 symmetry plane - RK3 UP5
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
W
0.70
0.50
0.30
0.10
-0.10
-0.30
-0.50
-0.70
3th order upwind
5th order upwind
3D TEST CASES: HYDROSTATIC FLOW
20
Deutscher Wetterdienst
P. L. Vitagliano, G. Ceci
CONVERGENCE OF KINETIC ENERGY
10
NON-HYDROSTATIC TEST
RK3 TVD
KINETIC ENERGY
-1
L2
L1
L0
2nd order
10 -2
-3
10
10 -4
10
-5
10
-6
10
L2
L1
L0
2nd order
10 -2
Error Norm
Error Norm
10
10
NON-HYDROSTATIC TEST
RK3
KINETIC ENERGY
-1
-3
10 -4
-3
10
-2
10
-1
10
0
10
1
10
2
DX [km]
FE 13 – 22.03.2016
10
-5
10
-6
10
-3
10
-2
10
-1
10
DX [km]
21
0
10
1
10
2
Deutscher Wetterdienst
CONVERGENCE OF KINETIC ENERGY
10
NON-LINEAR HYDROSTATIC TEST
RK3 TVD
KINETIC ENERGY
2
L2
L1
L0
2nd order
10 0
10
-1
10
-2
10
-2
FE 13 – 22.03.2016
10
-1
L2
L1
L0
2nd order
10 1
Error Norm
10 1
Error Norm
10
NON-LINEAR HYDROSTATIC TEST
RK3
KINETIC ENERGY
2
0
10
DX [km]
10
1
10
2
22
10 0
10
-1
10
-2
10
-2
10
-1
0
10
DX [km]
10
1
10
2
Deutscher Wetterdienst
Conclusions from Convergence Tests:
2D mountain flow
• slightly less than 2nd order spatial convergence (fast waves scheme dominates)
• TVD and non-TVD 3 stages Runge Kutta show similar behaviour
• time step has minor effect (if any) on spatial convergence
• important issues with upper and lateral boundary condition
• difficult to compare with analytical solutions, due to b.c.
3D mountain flow
•smaller influence of damping layer on 3d mountain waves and drag
• optimal damping parameter dt*nrdtau increases to 1000 s
• with poor resolution different scheme can give different solutions
• with poor resolution higher order upwind can improve results
FE 13 – 22.03.2016
23
Deutscher Wetterdienst
Task 6: deep valleys
Goal:
detection of the reason for the unrealistic ‚cold pools‘ in Alpine valleys
+ Task 7: Different filter options for orography
(Förstner, Torrisi, Reinhardt, deMorsier)
Status:
The reason for the cold pools was identified: metric terms of the pressure gradient
Dynamical Bottom boundary condition (DBBC) (A. Gassmann (2004), COSMONewsl.)
and a slope-dependent orography-filtering cures the problem to a certain extent.
example: For the COSMO-DE first the orography is filtered globally to remove
scales approximately smaller than 4-. In a second step a stronger filter (5-) is
used for all points with a step of the orography still bigger than 625 m.
FE 13 – 22.03.2016
24
Deutscher Wetterdienst
(G. Zängl)
Spurious noise over mountains in a resting atmosphere




Tests reveal a 2Δz structure in the horizontal and vertical wind
field
Depending on the difference between base state and actual
temperature profile, it can take more than 12 h until the noise
reaches a significant amplitude
Afterwards, it rapidly grows within a time scale of a few hours
until some sort of saturation is reached
Tests indicate that a modified discretization of the dw/dz term in
the pressure tendency equation may damp the noise
Setup of test experiments: mountain with h = 1500 m, a = 5 km; Δx = 1
km, no ambient winds; results are shown for t = 24 h
FE 13 – 22.03.2016
25
Deutscher Wetterdienst
Results with implicit 2nd-order vertical advection
θ (contour interval 1 K), u (colours)
standard discretization
FE 13 – 22.03.2016
with damping discretization
26
Deutscher Wetterdienst
Results with implicit 2nd-order vertical advection
θ (contour interval 1 K), w (colours)
standard discretization
FE 13 – 22.03.2016
with damping discretization
27
Deutscher Wetterdienst
Results for quasi-linear flow over a mountain, h = 300 m, u = 10 m/s
θ (contour interval 1 K), u (colours)
standard discretization
FE 13 – 22.03.2016
with damping discretization
28
Deutscher Wetterdienst
Spurious noise over mountains in a resting atmosphere



In the modified version, the term is not only evaluated between
half-levels but also between full-levels (which damps 2Δz
waves), followed by a weighting of both terms
A weight of 0.05 of the damping discretization turned out to
suffice for eliminating the noise
Normally very small impact on flow dynamics, but stability
problems over steep topography in the presence of strong winds
FE 13 – 22.03.2016
29
Deutscher Wetterdienst
Task 8: Higher order discretization in the vertical for RK-scheme
(Baldauf)
Improved vertical advection for the dynamic var. u, v, w, T (or T‘), p‘
motivation: resolved convection
 vertical advection has increased importance => use scheme of higher order
(compare: horizontal adv. from 2. order to 5. order)
 => bigger w (~20 m/s) => Courant-crit. is violated =>
implicit scheme or CNI-explicit scheme
up to now: implicit (Crank-Nicholson) advection 2. order (centered differences)
new:
implicit (Crank-N.) advektion 3. order  LES with 5-banddiagonal-matrix
but: implicit adv. 3. order in every RK-substep; needs ~ 30% of total computational time!
 planned: use outside of RK-scheme (splitting-error?, stability with fast waves?)
Work to do: best combination with time integration scheme?
FE 13 – 22.03.2016
30
Deutscher Wetterdienst
Comparison of the two implicit vertical advection schemes
Test with constant vertical velocity; initial cone distribution
implicit cent. diff. 2nd order
FE 13 – 22.03.2016
implicit cent. diff. 3rd order
31
Deutscher Wetterdienst
Task 9: Physics coupling scheme
(deMorsier, Förstner)
original idea: problems with reduced precipitation could be due to a
nonadequate coupling between physics scheme and dynamics
Problems in new physics-dynamics coupling (NPDC) (=WRF-like coupling):
 Negative feedback between NPDC and operational moist turbulence
parameterization (not present in dry turbulence parameterization)
 2-z - structures in the specific cloud water field (qc)
 2-z - structures in the TKE field, unrealistic high values, where qc > 0
Work to do:
Is the problem cured now also in the moist turbulence case with the
improvement of the TKE-Diffusion (solution of O. Fuhrer)?
FE 13 – 22.03.2016
32
Deutscher Wetterdienst
Task 10: Testing of alternative fast wave scheme
(Torrisi, Gassmann)
Goals:
• p‘T‘-RK-scheme
• ‚shortened-RK2‘-scheme (Gassmann)
• this allows the use of the ‚radiative upper boundary condition‘ (RUBC)
Properties of A. Gassmann dyn. core:
• Splitting up of vertical advection of p*/T into fast/slow mode
equations and consistent boundary conditions
• Vertical average to half levels: mass weighted mean (in RK
simple mean) and base-state consistent formulation of the
discrete w-equation
• Different horizontal pressure gradient discretization
• Divergence in conservative flux-form
• Slightly different buoyancy term
• No artificial divergence damping
FE 13 – 22.03.2016
33
Deutscher Wetterdienst
Status:
• The fast waves part (Gassmann) is combined with the Leapfrog scheme in LM 3.21
• Original Gassmann dynamical core poses stability problems in several cases!
• Gassmann fast waves part in RK3 worked in only 1 case
• ‚shortened RK2‘-scheme (Gassmann (2002), Gassmann and Herzog (2007)) is
implemented into LM 3.21 using the fast waves solver of RK3 and the RK3
advection/physics subroutines
• Preliminary investigation of this dynamical core (L. Torrisi)
tested in real cases for a five days period: similar results to the RK3 splitting method
• Separate inspection of divergence in conservation form and vertical staggering
• Implementation questions pointed out:
•Splitting of contravariant vertical velocity
poses problems in formulation of lower boundary conditions
Overall assessment: needs too much work to bring to operational use
FE 13 – 22.03.2016
34
Deutscher Wetterdienst
Task 13: Divergence damping in a true 3D-version
(Baldauf)
Description:
Cases occured, where the up to now used 'quasi-2D' divergence filtering lead to
unstable results. But a complete abandoning of the divergence filtering (as
proposed by A. Gassmann for her dynamical core) also leads to several
instabilities. This was also shown by stability analyses of the RK-core by M.
Baldauf. P. Prohl (DWD) could demonstrate, that the Bryan-Fritsch- test case of a
rising warm bubble is unstable with 'quasi-2D' divergence damping but becomes
stable only with a full 3D (=isotropic) version (realised with a preliminary explicit
formulation). For operational use an implicit version of 3D divergence damping is
necessary.
FE 13 – 22.03.2016
35
Deutscher Wetterdienst
Task 14: DFI for RK
(L. Torrisi)
An initialization scheme
Twice Digital Filter Initialization
Adiabatic backward integration
Diabatic
forward integration
*
FE 13 – 22.03.2016
36
Deutscher Wetterdienst
Digital Filter Initialization in RK core
(L. Torrisi)
• Some modifications (mostly in the adv functions ) are needed to run DFI with RK
core.
They are in:
- dfi_initialization.f90: add initialization of rho_snow
- src_runge_kutta.f90: correction in wind Rayleigh damping
- src_advection_rk.f90: changes in cfl control and changes in adv function interfaces
- fast_waves_rk.f90: changes in adv function interfaces
- numeric_utilities_rk.f90: changes in adv function interfaces and correction to run with DFI
• All the odd order advection operators are changed to run in the backward
integration of the DFI. The odd order advection operators implicitly contain a
dissipative term that needs a special treatment in the backward integration of the
DFI.
The dissipative terms are treated as the horizontal diffusion operator in the backward
integration of DFI (when dt<0 , -1 is multiplied to the dissipative term).
For example:
5th order velo*ds/dx operator = 6th order velo*ds/dx operator + dissipative term * SIGN(1.,dt)
* Sign(1.,dt)
FE 13 – 22.03.2016
37
Deutscher Wetterdienst
Digital Filter Initialization in RK core

DFI seems to work well using a 7 km grid spacing
2h
FE 13 – 22.03.2016
2h
38
Deutscher Wetterdienst
COSMO-IT (2.8km)
Digital Filter Initialization in RK core
 Using a 2.8km grid spacing
DFI works only with explicit
vertical advection
FE 13 – 22.03.2016
1h 39
1h
Deutscher Wetterdienst
Optimization of horizontal advection:
(M. Baldauf)
up to COSMO 4.3:
'advection operators' = a subroutine acting on every single grid point
 compiler has problems to optimize loops
since COSMO 4.4:
advection routines using 'field operations'
(and additionally the DFI modifications of Lucio Torrisi)
Efficiency gain for routine COSMO-DE at DWD (IBM):
• speedup of the horizontal advection alone: ~ 3 times faster
• overall reduction of model run time: ~ 1 Min. / 20 Min. ~ 5%
Furthermore, some inconsistencies using metrical factors could be repaired:
acrlat(j,1)  acrlat(j,2) lent to an error of ~ -0.05% in the term v dw/dy
FE 13 – 22.03.2016
40
Deutscher Wetterdienst
Introduction of RK-scheme into operational models
DWD:
COSMO-DE (2.8km): since 16.04.2007
COSMO-EU (7km): planned for ~Q4/2008 (if pressure bias problems removed)
(weak artificial horizontal diffusion, SL-scheme, new aver. reference pressure)
MeteoCH:
COSMO-S2: operational since April 2008
COSMO-S7:
CNMCA:
COSMO-IT (2.8km): since Oct. 2007
COSMO-ME (7km): in next future
FE 13 – 22.03.2016
41
Deutscher Wetterdienst
Publications of the PP 'Runge-Kutta'
Reviewed articles
• M. Baldauf (2008): Stability analysis for linear discretisations of the advection equation with RungeKutta time integration, J. Comput. Phys. 227, 6638-6659
Other articles
• M. Baldauf (2008): A Tool for Testing Conservation Properties in the COSMO-Model (LM), COSMONewsletter 7, 7-17
• J. Förstner, M. Baldauf, A. Seifert (2006), Courant Number Independent Advection of the
Moisture Quantities for the LMK, COSMO-Newsletter 6, 51-64
• L. Torrisi (2006): Sensitivity experiments with the Runge-Kutta time integration scheme, COSMONewsletter No. 6
Final report: Draft version (12.09.2008) available
FE 13 – 22.03.2016
42
Deutscher Wetterdienst
Thanks to all contributing scientists (in alphabetical order):
Michael Baldauf1,
Gabriella Ceci2,
Jochen Förstner1,
Oliver Fuhrer4,
Almut Gassmann5,
Hans-Joachim Herzog1,
Guy deMorsier4,
Thorsten Reinhardt 7,
Gdaly Rivin6,
Lucio Torrisi3,
Pier Luigi Vitagliano2,
Günther Zängl1
1
Deutscher Wetterdienst (DWD), Germany
Centro Italiano Ricerche Aerospaziali (CIRA), Italy
3 Centro Nazionale di Meteorologia e Climatologia Aeronautica (CNMCA), Italy
4 MeteoSchweiz, Switzerland
5 Max-Planck-Insitut, Hamburg, Germany
6 Federal Service for Hydrometeorology and Environmental Monitoring, Russia
7 Universität Köln, Germany
2
FE 13 – 22.03.2016
43
Deutscher Wetterdienst
FE 13 – 22.03.2016
44
Deutscher Wetterdienst
List of people contributing to the project during Oct. 2007 - Sept. 2008:
(alphabetical order)
•
•
•
•
•
•
•
Michael Baldauf
Gabriella Ceci
Oliver Fuhrer
Lucio Torrisi
Pier Luigi Vitagliano
Gdaly Rivin
Günther Zängl
(DWD, D)
(CIRA, I)
(MeteoCH, CH)
(CNMCA, I)
(CIRA, I)
(Roshydromet, RU)
(DWD,D)
Additional meeting of PP-RK-Group during the LM-User-Workshop, Langen,
05.03.2008
FE 13 – 22.03.2016
45
Task 3:
Deutscher Wetterdienst
Weisman-Klemp (1982)-test case
without physical parameterisation
(only advection &
Condensation/Evap.)
total moisture mass
M =  x dV
Semi-Lagrange-Adv. for qx
with multiplicative filling
x :=  (qv + qc )
(Mn-Mn-1) / t
Res.
violation in moisture conservation (?)
total surface flux
timestep
04.09.2007
46
Deutscher Wetterdienst
Convergence tests
G. Ceci, P. L. Vitagliano
HYDROSTATIC LINEAR / NON LINEAR
a = 10 km
H = 10 m / 500 m
Time = 60 h / 100 h
dt = 2.5”
Domain size 500x19.5 km2
Horizontal resolution = 4km, 2km, 1km, 500m, 250m, 125m
NON HYDROSTATIC
a = 500 m
H = 10 m
Time = 100 h
dt = 2.5”
Domain size 250x19.5 km2
Horizontal resolution = 1km, 500m, 250m, 125m, 62.5m
• All test cases runned again with constant time step = 2.5”
• Test cases repeated with non-TVD 3-stage RK
FE 13 – 22.03.2016
48
Deutscher Wetterdienst
P. L. Vitagliano, G. Ceci
CONVERGENCE OF VERTICAL VELOCITY w
HYDROSTATIC TEST
RK3 TVD
VERTICAL VELOCITY
10 -2
10 -2
L2
L1
L0
2nd order
10 -3
Error Norm
10 -4
10 -4
10
-5
10
-6
10
L2
L1
L0
2nd order
Error Norm
10 -3
HYDROSTATIC TEST
RK3
VERTICAL VELOCITY
-2
FE 13 – 22.03.2016
10
-1
0
10
DX [km]
10
1
10
2
49
10
-5
10
-6
10
-2
10
-1
0
10
DX [km]
10
1
10
2
Deutscher Wetterdienst
CONVERGENCE OF VERTICAL VELOCITY w
NON-HYDROSTATIC TEST
RK3 TVD
VERTICAL VELOCITY
10 -1
10 -1
L2
L1
L0
2nd order
Error Norm
10 -3
10
-4
10
-5
10
L2
L1
L0
2nd order
10 -2
Error Norm
10 -2
NON-HYDROSTATIC TEST
RK3
VERTICAL VELOCITY
10 -3
-3
FE 13 – 22.03.2016
10
-2
-1
10
DX [km]
10
0
10
1
50
10
-4
10
-5
10
-3
10
-2
-1
10
DX [km]
10
0
10
1