an introduction to semi-lagrangian methods iii luca bonaventura dipartimento di matematica “f....
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An introduction to semi-Lagrangian methods III
Luca Bonaventura
Dipartimento di Matematica
“F. Brioschi”, Politecnico di Milano
MOX – Modeling and Scientific Computing Laboratory
Sapienza SL workshop 2011
OutlineOutline
•From scalar advection to realistic transport models: need for mass conservative semi-Lagrangian methods
•Identify typical regimes for convenient application of SL to more complex fluid dynamics problems: NWP and the SL success story
•Some examples of other applications to environmental modeling
References on SL and alike References on SL and alike
Textbooks: Falcone Ferretti: to appearDurran, Quarteroni-Valli
Review papers: Staniforth and Coté, MWR 119 1991 Morton, SIAM J. Num. An. 1998 Ewing and Wang, J. Comp. Appl.Math. 2001 L.B., ETH Zurich ERCOFTAC 2004 lecture
notes (http://www1.mate.polimi.it/~bonavent)
The model problem The model problem
€
Dc
Dt=
∂c
∂t+ u(x, t) • ∇c = 0
c(x,0) = c0(x) x∈Rd
)),,0;((),0;(
)),0;((),( 0
tttdt
d
tctc
xXuxX
xXx
Representation formula for exact solution
The numerical methodThe numerical method
€
c(x i, tn+1) = I[c( ˆ X (x i,Δt), t
n )] = c(x*, tn )
Interpolation operator
€
I =
ˆ X (x i,Δt) = Approximate foot of trajectory landing at mesh point
The real problemThe real problem
€
∂c i∂t
+∇ • (uc i) =
∇ • D∇c i + f i(c1,K ,cn ) i =1,K ,n
•Large advection-diffusion-reaction systems (n>20) for atmospheric chemistry, air-water quality, biogeochemistry
Example: air pollution modelExample: air pollution model
25 chemical reaction among 20 chemical species with very different reaction rates (stiff problem)
Computational requirementsComputational requirements
•High order accuracy, positivity preservation
•Fully multidimensional numerical methods
•Efficiency: standard explicit time discretization often have restrictive CFL stability conditions
•Mass conservation
Mass conservation issue for SL Mass conservation issue for SL
1InterpolationD test
Linear Quadratic Spline
Final/initial mass ratio
1.00002384 1.00293481 1.00027871
2D Interpolation
Linear Cubic, tr. I order trajectories
Cubic, II order trajectories
Final/initial mass ratio
0.7668 0.8828 1.1059
One dimensional test
Two dimensional test
Conservative SL : remappingConservative SL : remapping
0Dt
Dc
dxcdxc nn 1
SL conservative methods based on remapping: Laprise 1995, Machenhauer 1997, Nair 2002, Zerroukat and Staniforth 2003, Behrens and Menstrup 2004
dtd
ddcFtA
n
nu
)()u,,(
)(n)1(
Fdcdttn
tn
u
SL flux form conservative methods:
Lin Rood 1996; Fey 1998; Frolkovic 2002; B., Restelli Sacco 2006
Flux form SL methods, IIFlux form SL methods, II
Flux form SL methods, resultsFlux form SL methods, results
Advection test with monotonic (a) SLDG scheme, (b) conventional monotonic DG, (c) exact solution (Restelli Sacco B. JCP 2006)
Beyond pure advectionBeyond pure advection
0
xt
cA
c
€
A = TLT−1
d = T−1c
0
xt
dd
• SL for d variables: method of characteristics
Hyperbolic systemsHyperbolic systems
Model problemModel problem
• Shallow water equations (2D Euler)
€
c =T
h, u[ ]
€
∂h∂t
+ u∂h
∂x+ h
∂u
∂x= 0
∂u
∂t+ u
∂u
∂x+ g
∂h
∂x= 0
€
A =u h
g u
⎡
⎣ ⎢
⎤
⎦ ⎥
Hydrodynamic regimesHydrodynamic regimes
€
u
gh≈1
€
u
gh<<1
Critical: large Froude/Mach numbers
Subcritical: small Froude/Mach numbers
Flow along characteristics,
discontinuous solutions
Flow along streamlines,
continuous solutions
€
Dh
Dt= −h
∂u
∂xDu
Dt= −g
∂h
∂x
From SL to SISLFrom SL to SISL
€
hin+1 − h*
n
Δt= −hi
n ∂un+1
∂xi
uin+1 − u*
n
Δt= −g
∂hn+1
∂xi
•Use advective form and apply SL + semi-implicit treatment of RHS
€
Dρ
Dt+ ρ∇ • u = 0
Du
Dt+ 2Ω × u +
1
ρ∇p = −gk + F
Dε
Dt= −p∇ • u −∇ • R
3D Euler equations for NWP3D Euler equations for NWP
Subcritical regime is dominant (Mach < 0.3)
Approximations for large scalesApproximations for large scales
•Hydrostatic equilibrium
€
1
ρ
∂p
∂z≈ −g
•Geostrophic equilibrium: zeroth order quasi-static approximation
p1
2 u
•First order corrections: barotropic
vorticity equation
€
Dq
Dt=Q
The first SL success The first SL success
500 hp surface analysis (left) and Sawyer forecast (right): quantitatively correct……
The SL success storyThe SL success story
Probabilistic forecast (EPS): average and standard deviations over 50 independent runs
Environmental applicationsEnvironmental applications
•Advection dominated, subcritical flows: river hydraulics, coastal modelling, high resolution atmospheric modeling
•Long time range simulations: need for maximum efficiency
•Wide range of different flow velocities: small portion of computational domain can restrict time step for standard explicit methods
River hydraulicsRiver hydraulics
Correct profiles for various regimes of channel flow achieved at high Courant numbers
River hydraulics (1.5D)River hydraulics (1.5D)
Flood prediction for the Adige river (De Ponti, Rosatti, B., Garegnani, IJNMF 2011)
River hydraulicsRiver hydraulics
Sediment transport in a curved channel (Rosatti, Chemotti,B. IJNMF 2005)
Coastal hydrodynamics (2-3D) Coastal hydrodynamics (2-3D)
•Venice Lagoon
•Dx=50 m,Dt=600 s
•Typical velocities: 0.1-1.0 m/s
• Complex geometry, need for appropriate trajectory computation
Coastal hydrodynamics (2-3D)Coastal hydrodynamics (2-3D)
High water prediction for the Venice lagoon
Nonhydrostatic flowsNonhydrostatic flows
•Idealized Foehn in a stratified atmosphere
•Dx = 2000 m, Dz = 200 m, Dt =30 s
•Semi-Lagrangian semi-implicit method (B., JCP 2000)
Thermal instabilitiesThermal instabilities
•Warm bubble in isentropic atmosphere
•Dx =Dz = 20 m, Dt =1 s
•Final vertical velocities beyond 30 m/s
Computational gainsComputational gains
CPU time for 1 hour
CPU time for SI solver
COMM time for SI solver
CPU time for advection
COMM time for advection
SE
328 91 13.4 156 40.8
SI Z
207 119 13.4 65.4 7.4
CPU times in seconds, comparison with split-explicit method, 3D gravity wave test case, 180*180*40 gridpoints (B., Cesari, Nuovo Cimento, 2005)
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