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An implicit solver based on dual time stepping and finite volumes for

meteorological applications

Pier Luigi VitaglianoCIRA

COSMO WG2Conservative Dynamic Core

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OUTLINE

• Motivation• Mathematical model• Numerical schemes• Test case• Future work

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MOTIVATIONS AND GOALS

• Improve numerical efficiency• Improve conservation properties• Improve capability to deal with steeper orography

• Test a time integration scheme for meteorological applications• Test spatial schemes based on finite volumes• Issue recommendations on future implementation in COSMO

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MATHEMATICAL FORMULATION

W =

Ewvu

uHuwuv

puu

2

vHvw

pvuvv

2

wH

pwwvuww

2Fy =Fx = Fz = B=

gUggg

z

y

x

0

dVBdSnFdVWt

2

21

1 UEp pEH

wvuU ,,

EULER EQUATIONS IN CONSERVATIVE VARIABLES

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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

(W)/t + R(W) = 0 R(W) = Q – B – D

Q = fluxes D = k∆4W artificial dissipationB = source terms

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SPATIAL DISCRETISATION

Example of flux evaluation

SF mkk

mknQm =

Fmk = ½ (Fm + Fk)

Wi-1 Fi-1,i WiWi-1Wi-1Wi-1 Wi+1

• Conservation laws applied to each sub-domain (cell)• Variables stored at cell centers• Fluxes approximated at cell face centers

k m

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DUAL TIME STEPPING

Wn+1/ + ½(3Wn+1- 4Wn + Wn-1)/t + R(Wn+1) = 0add a pseudo-time derivative to the unsteady equationadvance the solution in until the residual of the unsteady equation is negligible

formulation is A-stable and damps the highest frequencyvery large physical time step t can be used

iterations in are performed by explicit Runge-Kutte schemeconvergence acceleration techniques can be adopted without loss of time accuracy:residual averaging, local time stepping, multigrid

Jameson, A., 1991: Time Dependent Calculations Using Multigrid,with Applications to Unsteady Flows Past Airfoils and Wings. AIAA Paper 91–1596

(W)/t + R(W) = 0

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DUAL TIME STEPPING

Example of time integration with DTS: a norm of the residuals of mass transport equations is monitored

Dual Time Iterations

Log

DR

RM

S

10 15 20 25 30 35 40-19

-18

-17

-16

-15

-14

-13

-12

-11

Time Step

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PRECONDITIONING

Improve convergency in dual time for low Mach number flowsCorrect ill-behaved artificial viscosity fluxes at low Mach

Difficulties rise from large ratio between acoustic wave speed and fluid speed

Premultiplying the time derivative changes the eigenvalues of the system and accelerates the convergence to steady state.

P·W/ + R(W) = 0

Turkel, E., 1999: Preconditioning techniques in computational fluid dynamics. Annu.Rev.Fluid Mech. 1999,31:385-416.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.

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PRECONDITIONING

Example of convergence to steady solution with and without Preconditioning

Iterations

log

(DR

MA

X)

0 2000 4000 6000 8000 10000-12

-10

-8

-6

-4

-2

0

2

4Mach = 0.300Mach = 0.100Mach = 0.005Mach = 0.005 prec

Elliptic Wing - Euler Solution =4 degConvergence History - Medium mesh

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Discretisation of the gravity force term

Field initialisationEffect of mesh skewnessFlux – force unbalance

Longitude [m]

Z[m

]

-50000 0 500000

5000

10000

15000

20000

W

1E-158E-166E-164E-162E-160

-2E-16-4E-16-6E-16-8E-16-1E-15

CONSOL - no inflow - vis4=0x=4km z=125m

Smkk

mk npV

g 1

XZ

-50000 0 500000

5000

10000

15000

20000

W9.0E-167.0E-165.0E-163.0E-161.0E-16

-1.0E-16-3.0E-16-5.0E-16-7.0E-16-9.0E-16-1.1E-15-1.3E-15

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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.

TEST CASE MOUNTAIN FLOW

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Mesh for test on complex orography with cold bubble

X [m]

Z[m

]

-50000 -25000 0 25000 500000

5000

10000

15000

20000

25000

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CONCLUSIONS

• COMPUTER CODE FOR TEST RUN READY

• INITIAL TESTS ON STEADY MOUNTAIN FLOW

• STUDY ON COMPLEX OROGRAPHY STARTED

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FUTURE WORK

TEST CASES:

1) Atmosphere at rest with deformed mesh (Zaengl (2004)) 2) Cold bubble (Straka et al. (1993))3) Mountain test cases:

• (Schaer et al (2002) sect. 5b) • (Bonaventura(2000))• (Klemp,Wilhelmson)

4) Linear gravity waves (Skamarock-Klemp (1994),Giraldo(2008))

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TEST CASE COLD BUBBLE

Initial Field Density contour. Step Δρ/ρSL=0.0001

Initial Field U-Velocity contour