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Folie 1

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Priority Project CDC Task 2: The compressible approach COSMO-GM, 06.-10.09.2010, Moscow Pier Luigi Vitagliano (CIRA), Michael Baldauf (DWD) – PowerPoint PPT presentation

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Title: Folie 1


1
Priority Project CDC Task 2 The compressible
approach COSMO-GM, 06.-10.09.2010,
Moscow Pier Luigi Vitagliano (CIRA), Michael
Baldauf (DWD)
2
Task 2.3 Fully 3D, i.e. non-direction splitted,
conservative advection scheme At MeteoCH, a
diploma thesis (M. Müllner) has been started,
toimplement MPDATA into COSMO. No serious
problems expected.
3
Task 2.2 Complete Finite-Volume solver for the
EULER equations
  • MOTIVATIONS AND GOALS
  • Improve numerical efficiency
  • Improve conservation properties
  • Capability to deal with steep orography
  • Test a new time integration scheme
  • Test spatial schemes based on finite volumes

4
  • Dry Euler equations (without Coriolis force) in
    conservative form

W
Fy
Fx
Fz
B
E Ekin Eint ½ U2 cp T
5
  • SPATIAL DISCRETISATION
  • Finite Volumes approach
  • Integral form allows discontinuities in the flow
    field
  • Conservation laws applied to each sub-domain
    (cell)
  • Variables stored at cell centers
  • Fluxes approximated at cell face centers

?(VW)/?t R(W) 0
R(W) Q B D
Q fluxes
D k?4 artificial dissipation
B source terms
6
  • DUAL TIME STEPPING

?Wn1/?t ½(3Wn1- 4Wn Wn-1)/Dt R(Wn1) 0
formulation is A-stable and damps the highest
frequency ? very large physical time step Dt can
be used
  • Solution of the implicit equation system
  • add a pseudo-time t derivative to the unsteady
    equation
  • integration in ? is performed by an explicit
    Runge-Kutte scheme
  • advance the solution in t until the residual of
    the unsteady equation is negligible
  • convergence acceleration techniques can be
    adopted without loss of time accuracyresidual
    averaging, local time stepping, multigrid

Jameson, A., 1991 Time Dependent Calculations
Using Multigrid,with Applications to Unsteady
Flows Past Airfoils and Wings. AIAA Paper 911596
7
  • DUAL TIME STEPPING

Example of time integration with DTS a norm of
the residuals of mass transport equations is
monitored
8
  • PRECONDITIONING

Improve convergency in dual time for low Mach
number flows Correct ill-behaved artificial
viscosity fluxes at low Mach
Difficulties rise from large ratio between
acoustic wave speed and fluid speed Premultiplyin
g the time derivative changes the eigenvalues of
the system and accelerates the convergence to
steady state.
Turkel, E., 1999 Preconditioning techniques in
computational fluid dynamics. Annu.Rev.Fluid
Mech. 1999,31385-416. Venkateswaran, S., P. E.
O. Buelow, C. L. Merkle, 1997 Development of
linearized preconditioning methods for enhancing
robustness and efficiency of Euler and
Navier-Stokes Computations, AIAA Paper 97-2030.
9
  • PRECONDITIONING

Example of convergence to steady solution with
and without Preconditioning
10
  • For task 2.2 the following idealised dry test
    cases where defined (only a reduced set of the
    test cases from task 3.1)
  • Atmosphere at rest (G. Zaengl (2004) MetZ)
  • test balance of pressure gradient forces, metric
    correction terms and buoyancy
  • Cold bubble (Straka et al. (1993)) strong
    nonlinear, unstationary test, well established
    reference solution
  • Mountain flow tests stationary test, well known
    (partly analytic) solutions
  • Schaer et al (2002) sect. 5b
  • Bonaventura (2000) JCP
  • Linear Gravity waves (Skamarock, Klemp (1994),
    Giraldo (2008)) unstationary test, wave
    expansion, analytic solution available

11
ATMOSPHERE AT REST
Pressure gradient discretisation
Field initialisation Effect of mesh skewness Flux
force unbalance
standard formulation pmk ½ (pm pk)
pressure gradient correction pmk ½ pm
(?p/?z)m?zm pk (?p/?z)k ?zk
(?p/?z)m ?mg
12
ATMOSPHERE AT REST
INITIAL FIELD W component
13
ATMOSPHERE AT REST
pressure gradient correction
standard pressure gradient
W component after 90000 seconds ?t3 sec
14
ATMOSPHERE AT REST
Solution with pressure gradient correction after
90000 seconds ?t300 sec
15
ATMOSPHERE AT REST
  • CONCLUSIONS
  • Solution is not affected by physical time step
    nor by CFL
  • Pressure gradient correction improves
    initialisation of the fields, but has only a
    smaller positive influence on accuracy after 1
    day simulation (but helps convergence).The
    induced vertical velocities are in the same order
    of magnitude than in COSMO.
  • Some issues with boundary conditions

16
  • TEST CASE
  • MOUNTAIN FLOW

Linear, hydrostatic case
Flow over a gaussian mountain simulated with a
test code based on finite volumes conservative
schemes. Vertical velocity component. The dashed
line shows the lower boundary of the Rayleigh
damping layer, which prevents the wave reflection.
17
STRAKA TEST
  • Implemented viscous fluxes with constant ?
  • Implemented reference atmosphere with constant
    ?T/?z

INITIAL FIELD
18
STRAKA TEST
?t10 sec
SOLUTION AFTER 600 sec ?x50 m
reference solution by Straka et al (1993)
4.8 km
19.2 km
19
STRAKA TEST
  • good agreement with reference solution by Straka
    et al. (1993)
  • time step from 0.25 to 100 sec possible
  • mesh size from 25 m to 200 m
  • solution diffused with larger time steps

20
Gravity wave test (Skamarock, Klemp (1994) MWR)
21
  • Conclusions
  • most of the test cases were carried out
    successfully
  • atm. at rest
  • cold bubble
  • linear, hydrostatic mountain flow
  • linear gravity wave test
  • some idealised tests are still missing
  • not all mountain flow tests availablemost
    probably due to initialisation/setup problems of
    the test cases
  • a lot of work had to be done with the
    implementation of buoyancy termsinto the model
    this looks promising
  • dual time stepping is a promising time
    integration approachstatements about efficiency
    compared to current COSMO are not so easy until
    now but scalability is probably not an obstacle

22
  • Scalability on future supercomputing platforms
  • no tests made yet (toy model)
  • a 3D implicit solver is used, but in an
    'explicit' manner due to the dual time stepping
    ? should not pose serious problems
  • preconditioning only local operations ? linear
    speedup expected(Choi, Merkle (1993) JCP)

23
Task 2.2.2 It has to be clarified how the moist
equations should be formulated. An adequate test
case should be performed Weisman, Klemp (1982)
(warm , moist bubble test) Task 2.2.3 The
properties of the A-grid formulation concerning
wave propagation should be investigated.
Therefore a wave analysis on the A-grid will be
performed.
24
Task 2.5 Testing the dual time stepping in
COSMO The dual time stepping (DTS) scheme is
generally able to integrate implicit equation
systems. Therefore it can be used to integrate
the COSMO equations by abandoning the
time-splitting procedure. Fast processes have to
be formulated implicitely, but with the same
spatial discretizations as they are used now for
the Runge-Kutta scheme. Such an implementation is
not expected to be more efficient, but possibly
could solve problems connected with steep
terrain. This task therefore serves as an
intermediate step towards task 2.6 This
preliminary DTS implementation can be at first
tested with the implemented idealised test cases
(see task 3.1). This testing can be performed by
a press of a button in COSMO. Deliverables
COSMO model using DTS scheme.
25
Task 2.6 Implement the Finite Volume solver into
COSMO Finally the scheme developed in task 2.2
will be implemented into COSMO. Again with the
implemented idealised test cases (see task 3.1) a
testing to find elementary bugs can be performed
by a press of a button in COSMO. Deliverables
COSMO model using the compressible, implicit
Finite Volume dyn. core Task 2.7 Perform
realistic test cases After finishing task 2.6
real case simulations with full physics
parameterisations with COSMO are possible.
Stand-alone runs for several weather regimes can
be performed for both dynamical cores (FV, RK) at
different resolutions. One has to obey that
physical parameterizations have to be adapted to
the new dynamical core. This probably requires
support from the physical parameterization
working group. Deliverables report about the
behaviour of real case test simulations with
COSMO
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