Further Developments of the Runge-Kutta Time Integration Scheme
44
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 1 urther Developments of the Runge-Kutta Time Integration Schem Investigation of Convergence (task 5) Gabriella Ceci, Pier Luigi Vitagliano [email protected] , [email protected]
description
Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task 5). Gabriella Ceci, Pier Luigi Vitagliano [email protected] , [email protected]. OUTLINE. OBJECTIVES AND MOTIVATIONS WORK PLAN TEST CASES DESCRIPTION - PowerPoint PPT Presentation
Transcript of Further Developments of the Runge-Kutta Time Integration Scheme
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 1
Further Developments of the Runge-Kutta Time Integration Scheme
Investigation of Convergence (task 5)
Gabriella Ceci, Pier Luigi Vitagliano
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 2
OUTLINE
• OBJECTIVES AND MOTIVATIONS
• WORK PLAN
• TEST CASES DESCRIPTION
• NEW RESULTS 2D: CONSTANT TIME STEP, NON-TVD RK3
• 3D TEST CASE: effect of different spatial scheme (3th vs 5th order)
• 3D HYDROSTATIC AND NON HYDROSTATIC MOUNTAIN FLOW
• EFFECT OF MOISTURE ON MOUNTAIN FLOW
• CONCLUSIONS
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 3
OBJECTIVES AND MOTIVATIONS
MOTIVATIONS
• ALLOWS LARGER TIME STEPS
• MORE ACCURATE
• FASTER
• CONVERGENCE PROPERTIES IN PRACTICAL APPLICATIONS UNKNOWN
OBJECTIVES
• TEST OF 3 STAGES RUNGE KUTTA TVD SCHEME WITH 5th ORDER UPWIND ADVECTION
• TEST OF NEW DYNAMICS with P' and T'
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 4
TEST CASES:
2D MOUNTAIN FLOWS WITHOUT PHYSICS
3D MOUNTAIN FLOWS WITHOUT PHYSICS
3D MOUNTAIN FLOW WITH MOISTURE
WORK PLAN
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 5
TEST CASES DESCRIPTION
• Gaussian ridge h(r)=H 2-(r/a)2
• HYDROSTATIC FLOW (aN/U) >> 1
• NON HYDROSTATIC FLOW (aN/U) ~ 1
• NON LINEAR FLOW (HN/U) ~ 1
-40 -30 -20 -10 0 10 20 30 400
1
2 Gaussian ridge - a=10 km h=500 m
• Basic flow velocity U = 10 m/s
• Brunt Väisälä frequency N = 0.01 s-1
• Rayleigh damping layer above 11 km
• Vertical resolution 100 m (195 levels)
r = (x2 + y2)½
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 6
TEST CASES DESCRIPTION
HYDROSTATIC LINEAR / NON LINEARa = 10 kmH = 10 m / 500 mTime = 60 h / 100 hdt = 2.5”Domain size 500x19.5 km2
Horizontal resolution = 4km, 2km, 1km, 500m, 250m, 125m
NON HYDROSTATICa = 500 mH = 10 mTime = 10 hdt = 2.5”Domain size 250x19.5 km2
Horizontal resolution = 1km, 500m, 250m, 125m, 62.5m
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 7
Comparison with analytical solutionlinear hydrostatic
Left: solution with a damping layer of 85 levels and nRΔt=200.Right: analytical solution following Klemp-Lilly (J.Atmos.Sc. 35, 78-107, 1978)
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 8
ISSUES WITH LATERAL BOUNDARIES
Disturbances at the side boundaries due to p’ T’ (left), removed by initialization ofreference atmosphere p0 T0 with constant Brunt-Väisälä frequency N (right)
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 9
ISSUES WITH UPPER DAMPING LAYER
Fine tuning of damping layer (both thichness and amount of damping) required to minimize wave reflection and distorsion.
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 10
NON HYDROSTATIC FLOW: w AND u
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 11
NON LINEAR HYDROSTATIC FLOW
VERY DEEP RAYLEIGH DAMPING LAYER IS REQUIRED TO OBTAIN REASONABLE SOLUTIONS
FOR HIGHER RIDGES (LEFT: 1.35 WAVE LENGTHS, RIGHT: 2 W.L.)
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 12
TIME CONVERGENCE
STEADY FLOW IS NOT OBTAINED WHEN THE RIDGE IS HIGHER THAN 500m
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 13
POST PROCESSING
• DRAG COEFFICIENT CD=∑ p'(x,0) dh/dx ∆x / PR
• MOMENTUM FLUX Mx(z)=- ρ(z) ∑ u(x,z) w(x,z) ∆x / PR
• KINETIC ENERGY= (u'(x,z)2 + w'(x,z)2)
• ABSOLUTE ERROR |-exact|
• RELATIVE ERROR |-finest mesh|
• ERROR NORM L0max |-finest mesh|
• ERROR NORM L11/N ∑ |-finest mesh|
• ERROR NORM L2[1/N ∑ (-finest mesh)2]½
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 14
OLD RESULTS: CD
DX [km]
CD
10-2 10-1 100 101 10210-3
10-2
10-1
100
|CD-CDref|2nd order
HYDROSTATIC FLOW
DX [km]
CD
10-2 10-1 100 101 10210-3
10-2
10-1
100
101
2nd order|MX-MXref||CD-CDref|
NON HYDROSTATIC FLOW
DX [km]
CD
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
2nd order|CD-CDref|
NON LINEAR HYDROSTATIC FLOW
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 15
OLD RESULTS: KINETIC ENERGY
DX
10-2 10-1 100 101 10210-5
10-4
10-3
10-2
10-1
Kinetic Energy2nd order
HYDROSTATIC FLOW
DX
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
Kinetic Energy2nd order
NON HYDROSTATIC FLOW
DX
10-2 10-1 100 101 10210-2
10-1
100
101
102
Kinetic Energy2nd order
NON LINEAR HYDROSTATIC FLOWTime=2 h
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 16
OLD RESULTS: MOMENTUM FLUX
CD
Z[m
]
0 0.1 0.2 0.3 0.4 0.5 0.60
2000
4000
6000
8000
10000
12000
DX = 4 kmDX = 2 kmDX = 1 kmDX = 0.50 kmDX = 0.25 km
NON LINEAR HYDROSTATIC FLOW
CD
Z[m
]
0.6 0.7 0.8 0.9 1 1.10
2000
4000
6000
8000
10000
12000
DX = 4 kmDX = 2 kmDX = 1 kmDX = 0.50 kmDX = 0.25 km
HYDROSTATIC MOUNTAIN CASE
CD
Alti
tud
e[m
]
-0.8 -0.6 -0.4 -0.2 00
1000
2000
3000
4000NON HYDROSTATIC FLOW
Smaller DX
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 17
NEW RESULTS
• ALL TEST CASES RUNNED AGAIN WITH CONSTANT TIME STEP = 2.5”
• TEST CASES REPEATED WITH NON-TVD 3 STAGES RUNGE KUTTA
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 18
CONVERGENCE OF VERTICAL VELOCITY w
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
L2L1L02nd order
HYDROSTATIC TESTRK3 TVD
VERTICAL VELOCITY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
L2L1L02nd order
HYDROSTATIC TESTRK3
VERTICAL VELOCITY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 19
CONVERGENCE OF VERTICAL VELOCITY w
DX [km]
Err
or
No
rm
10-3 10-2 10-1 100 10110-5
10-4
10-3
10-2
10-1
L2L1L02nd order
NON-HYDROSTATIC TESTRK3 TVD
VERTICAL VELOCITY
DX [km]
Err
or
No
rm
10-3 10-2 10-1 100 10110-5
10-4
10-3
10-2
10-1
L2L1L02nd order
NON-HYDROSTATIC TESTRK3
VERTICAL VELOCITY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 20
CONVERGENCE OF VERTICAL VELOCITY w
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
L2L1L02nd order
NON-LINEAR HYDROSTATIC TESTRK3
VERTICAL VELOCITY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
L2L1L02nd order
NON-LINEAR HYDROSTATIC TESTRK3 TVD
VERTICAL VELOCITY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 21
CONVERGENCE OF KINETIC ENERGY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
L2L1L02nd order
HYDROSTATIC TESTRK3 TVD
KINETIC ENERGY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
L2L1L02nd order
HYDROSTATIC TESTRK3
KINETIC ENERGY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 22
CONVERGENCE OF KINETIC ENERGY
DX [km]
Err
or
No
rm
10-3 10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
10-1
L2L1L02nd order
NON-HYDROSTATIC TESTRK3 TVD
KINETIC ENERGY
DX [km]
Err
or
No
rm
10-3 10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
10-1
L2L1L02nd order
NON-HYDROSTATIC TESTRK3
KINETIC ENERGY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 23
CONVERGENCE OF KINETIC ENERGY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-2
10-1
100
101
102
L2L1L02nd order
NON-LINEAR HYDROSTATIC TESTRK3 TVD
KINETIC ENERGY
DX [km]
Err
or
No
rm
10-2 10-1 100 101 10210-2
10-1
100
101
102
L2L1L02nd order
NON-LINEAR HYDROSTATIC TESTRK3
KINETIC ENERGY
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 24
CONVERGENCE OF WAVE DRAG
DX [km]
CD
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
|CD-CDref|2nd order
HYDROSTATIC FLOWRK3 TVD
DX [km]
CD
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
|CD-CDref|2nd order
HYDROSTATIC FLOWRK3
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 25
CONVERGENCE OF WAVE DRAG
DX [km]
CD
10-2 10-1 100 101 10210-3
10-2
10-1
100
101
|CD-CDref|2nd order
NON HYDROSTATIC FLOWRK3 TVD
DX [km]
CD
10-2 10-1 100 101 10210-3
10-2
10-1
100
101
|CD-CDref|2nd order
NON HYDROSTATIC FLOWRK3
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 26
CONVERGENCE OF WAVE DRAG
DX [km]
CD
10-2 10-1 100 101 10210-3
10-2
10-1
100
101
|CD-CDref|2nd order
NON LINEAR HYDROSTATIC FLOWRK3
DX [km]
CD
10-2 10-1 100 101 10210-4
10-3
10-2
10-1
100
|CD-CDref|2nd order
NON LINEAR HYDROSTATIC FLOWRK3 TVD
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 27
COMPARISON WITH ANALYTICAL SOLUTIONS
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 28
CONVERGENCE: CONCLUSIONS AFTER 2D TESTS
• 2nd ORDER SPATIAL CONVENGENCE (FAST WAVE 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 BOUNDARY CONDITION
• ISSUE IN LATERAL BOUNDARY CONDITIONS FOR p’ T’
• DIFFICOULT TO COMPARE WITH ANALYTICAL SOLUTIONS, DUE TO B.C.
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 29
3D TEST CASES
Gaussian mountain, hydrostatic flow, dry atmosphere
effect of different spatial scheme (3th vs 5th order) and grid size
Domain size: 256x128x19.5 km3
195 vertical levels
Rayleigh damping above 11 km
Basic flow velocity U = 10 m/s
Brunt Väisälä frequency N = 0.01 s-1
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 30
3D TEST CASES
0.240.200.160.120.080.040.00
-0.04-0.08-0.12-0.16-0.20-0.24
Gaussian Mountainh=300[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=1km z=100mVertical velocity w at level Z=1.5km
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 31
3D TEST CASES
11.6011.3011.0010.7010.4010.10
9.809.509.208.908.608.30
Gaussian Mountainh=300[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=1km z=100mHorizontal velocity U at level Z=1.5km
11.6011.3011.0010.7010.4010.10
9.809.509.208.908.60
Gaussian Mountainh=300[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=1km z=100mHorizontal velocity U at level Z=0.5km
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 32
3D TEST CASES
0.100.060.02
-0.02-0.06-0.10-0.14-0.18-0.22
Gaussian Mountainh=300[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=1km z=100mPressure perturbation at level Z=1.5km
0.200.00
-0.20-0.40-0.60-0.80-1.00-1.20-1.40-1.60-1.80-2.00-2.20-2.40
Gaussian Mountainh=300[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=1km z=100mTemperature perturbation at level Z=1.5km
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 33
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100mSolution at Y=0 symmetry plane - RK3 UP3
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100mSolution at Y=0 symmetry plane - RK3 UP5
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 16 km
3th order upwind 5th order upwind
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 34
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 8 km
3th order upwind 5th order upwind
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100mSolution at Y=0 symmetry plane - RK3 UP3
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100mSolution at Y=0 symmetry plane - RK3 UP5
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 35
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 4 km
3th order upwind 5th order upwind
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100mSolution at Y=0 symmetry plane - RK3 UP3
W
0.700.500.300.10
-0.10-0.30-0.50-0.70
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100mSolution at Y=0 symmetry plane - RK3 UP5
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 36
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 16 km
3th order upwind 5th order upwind
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100mSolution at ground level - RK3 UP3
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=16km z=100mSolution at ground level - RK3 UP5
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 37
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 8 km
3th order upwind 5th order upwind
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100mSolution at ground level RK3 UP3
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=8km z=100mSolution at ground level RK3 UP5
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 38
3D TEST CASES: HYDROSTATIC FLOW
Gaussian mountain height=750 m size=10 km
Horizontal resolution 4 km
3th order upwind 5th order upwind
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100mSolution at ground level - RK3 UP3
U
14.0013.0012.0011.0010.00
9.008.007.006.00
Gaussian Mountainh=750[m] a=10[km] N=0.01[1/s] U=10[m/s]
x=y=4km z=100mSolution at ground level - RK3 UP5
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 39
3D TEST CASES
SOME CONCLUSIONS
• SMALLER INFLUENCE OF DAMPING LAYER ON 3D MOUNTAIN WAVES AND DRAG
• OPTIMAL DAMPING PARAMETER t*nrdtau INCREASES TO 1000 s
• WITH POOR RESOLUTION DIFFERENT SCHEME CAN GIVE DIFFERENT SOLUTIONS
• WITH POOR RESOLUTION HIGHER ORDER UPWIND CAN IMPROVE RESULTS
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 40
3D TEST CASES: NON HYDROSTATIC FLOW
Convergence analysis
Longitude [km]-20 0 20 40
W
0.0060.0050.0040.0030.0020.0010.000
-0.001-0.002-0.003-0.004-0.005-0.006
Gaussian Mountain - Non hydrostatic flowN=0.01 a=0.5 km h=10 m U=10 m/s
x=0.125 km z=100 m t=10 sw velocity component after 4 h
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 41
3D TEST CASES: NON HYDROSTATIC FLOW
Convergence analysis
CD (momentum flux)A
ltitu
de
[km
]-1 -0.8 -0.6 -0.4 -0.2 0 0.2
0
2
4
6
8
DX= 125mDX= 250mDX= 500mDX=1000m
NON HYDROSTATIC 3D GAUSSIAN HILL
CD
Alti
tud
e[m
]
-0.8 -0.6 -0.4 -0.2 00
1000
2000
3000
4000NON HYDROSTATIC FLOW
Smaller DX
3 D 2 D
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 42
3D TEST CASES: NON HYDROSTATIC FLOW
Convergence analysis
DX
10-2 10-1 100 10110-8
10-7
10-6
10-5
10-4
NON HYDROSTATIC 3D GAUSSIAN HILL
DX [km]
Err
or
No
rm
10-3 10-2 10-1 100 101 10210-6
10-5
10-4
10-3
10-2
10-1
L2L1L02nd order
NON-HYDROSTATIC TESTRK3 TVD
KINETIC ENERGY
2 D 3 D
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 43
3D TEST CASES: EFFECT OF MOISTURE
• STEADY SOLUTION NOT ACHIEVED EVEN ON H=10m
• TEST ON H=300m RH=100% SHOWS INSTABLE LOWER LAYER
Y X
Z
QC
0.0060.0050.0040.0030.0020.0010.000
Gaussian mountain H=300 m a=10kmMesh=64x32x195 x=y=4km z=100m
t=40s Time=40hRH=100% Slice Y=0
SIMILAR TEST CASE IN 2D SHOWN BY Durran-Klemp (J.Atmos.Sc. 39, 2490-2506, 1982)
WITH 3D SIMULATION LESS INFLUENCE OF BOUNDARIES
Krakow - September, 15th 2008 COSMO WG 2 - Runge Kutta 44
3D TEST CASES: EFFECT OF MOISTURE
FURTHER WORK (?)
• MOUNTAIN HEIGHT
• TIME STEP
• SPATIAL STEP
• B.C.
Y X
Z
QV
0.0100.0090.0080.0070.0060.0050.0040.0030.0020.001
Gaussian mountain H=300 m a=10kmMesh=64x32x195 x=y=4km z=100m
t=40s Time=40hRH=100% Slice Y=0
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