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

The SL success story E ECMWF forecast for today: made possible by SL methods

Outline 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 Textbooks: Falcone Ferretti: to appear Durran, Quarteroni-Valli Review papers:  Staniforth and Coté, MWR  Morton, SIAM J. Num. An  Ewing and Wang, J. Comp. Appl.Math  L.B., ETH Zurich ERCOFTAC 2004 lecture notes (

The model problem Representation formula for exact solution

The numerical method Interpolation operator Approximate foot of trajectory landing at mesh point

The real problem Large advection-diffusion-reaction systems (n>20) for atmospheric chemistry, air-water quality, biogeochemistry

Example: air pollution model 25 chemical reaction among 20 chemical species with very different reaction rates (stiff problem)

Computational 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 1Interpolation D test LinearQuadraticSpline Final/initial mass ratio D Interpolation LinearCubic, tr. I order trajectories Cubic, II order trajectories Final/initial mass ratio One dimensional test Two dimensional test

Conservative SL : remapping SL conservative methods based on remapping: Laprise 1995, Machenhauer 1997, Nair 2002, Zerroukat and Staniforth 2003, Behrens and Menstrup 2004

Flux form SL methods, I

SL flux form conservative methods: Lin Rood 1996; Fey 1998; Frolkovic 2002; B., Restelli Sacco 2006 Flux form SL methods, II

Flux 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 advection SL for d variables: method of characteristics Hyperbolic systems

Model problem Shallow water equations (2D Euler)

Hydrodynamic regimes Critical: large Froude/Mach numbers Subcritical: small Froude/Mach numbers Flow along characteristics, discontinuous solutions Flow along streamlines, continuous solutions

From SL to SISL Use advective form and apply SL + semi-implicit treatment of RHS

3D Euler equations for NWP Subcritical regime is dominant (Mach < 0.3)

Approximations for large scales Hydrostatic equilibrium Geostrophic equilibrium: zeroth order quasi-static approximation First order corrections: barotropic vorticity equation

The birth of NWP (1950)

The birth of SL (1963)

The first SL success 500 hp surface analysis (left) and Sawyer forecast (right): quantitatively correct……

The first SL success …..at much lower computational cost

The SL success story Probabilistic forecast (EPS): average and standard deviations over 50 independent runs

Environmental 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 hydraulics Correct profiles for various regimes of channel flow achieved at high Courant numbers

River hydraulics (1.5D) Flood prediction for the Adige river (De Ponti, Rosatti, B., Garegnani, IJNMF 2011)

River hydraulics Sediment transport in a curved channel (Rosatti, Chemotti,B. IJNMF 2005)

Coastal hydrodynamics (2-3D) Venice Lagoon Dx=50 m,Dt=600 s Typical velocities: m/s Complex geometry, need for appropriate trajectory computation

Coastal hydrodynamics (2-3D) High water prediction for the Venice lagoon

Nonhydrostatic 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 instabilities Warm bubble in isentropic atmosphere Dx =Dz = 20 m, Dt =1 s Final vertical velocities beyond 30 m/s

Computational gains CPU time for 1 hour CPU time for SI solver COMM time for SI solver CPU time for advection COMM time for advection SESE SI Z CPU times in seconds, comparison with split-explicit method, 3D gravity wave test case, 180*180*40 gridpoints (B., Cesari, Nuovo Cimento, 2005)

Conclusions Semi-Lagrangian methods: an accurate and efficient method for subcritical flows NWP: almost ideal application area Wide range of applications to environmental modeling