Document Sample

Numerical methods IV (time stepping) by Nils Wedi (room 007; ext. 2657) In part based on previous material by Mariano Hortal and Agathe Untch Governing Equations 3 Slide 1 ECMWF What is the basis for a stable numerical implementation ? A: Removal of fast - supposedly insignificant - external and/or internal acoustic modes (relaxed or eliminated), making use of infinite sound speed (cs) and/or the hydrostatic approximation from the governing equations BEFORE numerics is introduced. B: Use of the full equations WITH a semi-implicit numerical framework, reducing the propagation speed (cs 0) of fast acoustic and buoyancy disturbances, retaining the slow convective-advective component (ideally) undistorted. C: Split-explicit integration of the full equations, since explicit NOT practical (~100 times slower) Determines the choice of the numerical scheme Governing Equations 3 Slide 2 ECMWF Choices for numerical implementation Avoiding the solution of an elliptic equation fractional step methods; Skamrock and Klemp (1992); Durran (1999) Solving an elliptic equation Projection method; Durran (1999) Semi-implicit; Durran (1999); Cullen et. al.(1994); Benard et al. (2004); Benard (2004) Preconditioned conjugate-residual solvers or multigrid methods for solving the resulting Poisson or Helmholtz equations; Skamarock et. al. (1997) Governing Equations 3 Slide 3 ECMWF Split-explicit integration Skamarock and Klemp (1992); Durran (1999); Doms and Schättler (1999); „Slow‟ part of solution „Fast‟ part of solution e.g. implemented in popular limited-area models: Deutschland Modell, WRF model Governing Equations 3 Slide 4 ECMWF Semi-implicit schemes linearised term, treated implicit non-linear term, treated explicit (i) coefficients constant in time and horizontally (hydrostatic models Robert et al. (1972), Benard et al. (2004), Benard (2004) ECMWF/Arpege/Aladin NH) (ii) coefficients constant in time Thomas (1998); Qian, (1998); see references in Bénard (2004) (iii) non-constant coefficients Skamarock et. al. (1997), (UK Met Office NH model, EULAG model) Governing Equations 3 Slide 5 ECMWF Design of semi-implicit methods Treat all terms involving the fastest propagation speeds implicitly (acoustic waves, gravity waves). Assume that the energy in those components is negligible. Consider the solvability of the resulting implicit system, which is typically an elliptic equation. Governing Equations 3 Slide 6 ECMWF Example: Shallow water equations u h u U0 g 0 t x x h U h H u 0 0 t x x Linearized: Linear analytic solution: u( x, t ) u0eit eikx U 0 gH k H denotes here a mean state depth. Phase speed: c Governing Equations 3 Slide 7 ECMWF Shallow water equations u u u u v fv 0 x y x t v v v u v fu 0 x y y t u v u v ( ) 0 x y t x y : advection : gravity-wave (or sometimes called „adjustment‟) term In the linear version: t 2 s 2 U0 V02 t s 2 t s 2 U 02 V02 advection In the atmosphere adjustment combined 2 U 0 V02 gH 300 m / s 5 in synoptic-scale models s 10 m ==> Δt≤ 236 sec ~ 4 min ECMWF Governing Equations 3 Slide 8 Explicit time-stepping • Leap-frog explicit scheme t n n 1 n 1 n n u j u j tV j u j s x j t n n 1 n 1 n n v j v j tV j v j y j s t n 1 n 1 n n j j tV j j V jn s gH x y s V j (u j , v j ) ( x , y ) x A j A j x A j x y A j A j y Aj y x j+Δy x x x j-Δx x x Stability: t s 2 x j x j-Δy x j+Δx If we treat implicitly the advection terms we do not get a Helmholtz equation Governing Equations 3 Slide 9 ECMWF Increasing the allowed timestep • Forward-backward scheme u t x 0 u 0 t x von Neumann gives t n n1 j n (u j 1 u n1 ) j j 2x u n1 u n t ( n1 n1 ) j j 1 j 1 j 2x t x sin(kx ) 2 forward backward doubles the leapfrog timestep • Pressure averaging u n1 u n1 j j 2t 1 1 1 1 {(1 2)(n1 n1 ) [(n1 n1 ) (n1 n1 )]} j j j j j 1 j 2x t x doubles the leapfrog sin(kx ) 2 timestep if ε=0 ------> leapfrog if ε=1/4 we get Governing Equations 3 Slide 10 ECMWF Split-explicit time-stepping s n 1 u u t V n u n j s j j j s n 1 n n v j v j ts V j v j s n 1 n n j j ts V j j t f n n 1 s x j u j u j s n 1 s t f n y j v j v j s t f n 1 s j j V jn s t s M t f slow ts s 2 U V 2 0 2 0 fast Stability as before but M times a simpler problem. t f s 2 Potential drawbacks: splitting errors, conservation. However recent advances for NH NWP suggested in (Klemp et. al. 2007) Note: The fast solution may be computed implicitly. Governing Equations 3 Slide 11 ECMWF Semi-implicit time-stepping t n 1 n 1 n 1 n 1 n n u j u j tV j u j 2s x ( j j ) t n 1 n 1 n n v j v j tV j v j y ( n 1 n 1 ) j j 2s t 1 n 1 n 1 n n n n 1 j j tV j j 2s (V j V j ) x y s V j (u j , v j ) ( x , y ) x Aj Aj x Aj x y Aj Aj y Aj y x j+Δy x x Solve: 2 n 1 j 4( s ) n 1 F n , n 1 j ( t ) 2 2 j-Δx x x x j x j-Δy x j+Δx x Helmholtz equation ! Stability: now only limited by the advection terms Note: if we also treat the advection terms implicitly we do not get a Helmholtz equation! Governing Equations 3 Slide 12 ECMWF Compressible Euler equations Davies et al. (2003) Governing Equations 3 Slide 13 ECMWF Compressible Euler equations Governing Equations 3 Slide 14 ECMWF A semi-Lagrangian semi-implicit solution procedure (not as implemented, Davies et al. (2005) for details) Davies et al. (1998,2005) Governing Equations 3 Slide 15 ECMWF A semi-Lagrangian semi-implicit solution procedure Governing Equations 3 Slide 16 ECMWF A semi-Lagrangian semi-implicit solution procedure Non-constantcoefficient approach! Helmholtz equation (solutions see e.g. Skamarock et al. 1997, Smolarkiewicz et al. 2000) Governing Equations 3 Slide 17 ECMWF Semi-implicit time integration in IFS Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment. In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation) and long internal gravity waves. => implicit treatment of the adjustment terms. L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at “+” that can be solved almost analytically in a spectral model. Governing Equations 3 Slide 18 ECMWF Two-time-level semi-Lagrangian semiimplicit time integration in the hydrostatic IFS DX Dt RHSx X 0.5tL X 0 0.5tL0 tRHS x1/2 tL1/2 X * For compact notation define: tt L 0.5 ( L L0 ) L1/2 “semi-implicit correction term” DX RHS 1/2 tt L => x Dt Notations: X : advected variable RHS: right-hand side of the equation L: part of RHS treated implicitly Superscripts: “0” indicates value at dep. point (t) “1/2” indicates value at mid-point (t+0.5Δt) “+” indicates value at arrival point (t+Δt) L=RHS => implicit scheme L= part of RHS => semi-implicit (β=1) L=0 => explicit (β=0) Governing Equations 3 Slide 19 ECMWF Semi-implicit time integration in IFS Dvh f k v h Rd Tv ln p Pv K v Dt Tv DT PT K T Dt 1 ( 1)q p 1 1 p (ln ps ) ( v h )d t ps 0 Dvh RHSv tt γ T Rd Tr ln ps Dt DT RHST tt D Dt (ln ps ) RHS p tt D t Governing Equations 3 Slide 20 semi-implicit corrections semi-implicit equations ECMWF Dvh RHSv tt γ T Rd Tr ln ps Dt DT RHST tt D Dt (ln ps ) RHS p tt D t Semi-implicit time integration in IFS semi-implicit equations Reference state for linearization: Tr ref. temperature Where: γX Rd X dpr d pr d 1 Rd Tr c pd pr psr ref. surf. pressure => lin. geopotential for X=T dpr X X d d 0 => lin. energy conv. term for X=D 1 dpr 1 X p X d d sr 0 Governing Equations 3 Slide 21 ECMWF Linear system to be solved D 0.5t 2 ( T Rd Tr log( ps )) D* T 0.5t D T * log( ps ) 0.5t D P* Eliminate all variables to find also a Helmholtz equation for D+ : (1 0.25 2t 2 ( RdTr )2 )D D* 0.5t2 ( T * RdTr P* ) Governing Equations 3 ~ I D D 2 γ Rd Tr Slide 22 operator acting only on the vertical I = unity operator ECMWF Semi-implicit time integration in IFS ~ I D D 2 Vertically coupled set of Helmholtz equations. Coupling through γ Rd Tr Uncouple by transforming to the eigenspace of this matrix gamma (i.e. diagonalise gamma). Unity matrix “I” stays diagonal. => ~ 1 i D D 2 One equation for each 1 i N Lev In spectral space (spherical harmonics space): n(n 1) m ~ m because 2Ynm n(n 1) Ynm 1 i Dn Dn a2 2 a Once D+ has been computed, it is easy to compute the other variables at “+”. Governing Equations 3 Slide 23 ECMWF

DOCUMENT INFO

Shared By:

Categories:

Stats:

views: | 12 |

posted: | 11/10/2009 |

language: | English |

pages: | 23 |

OTHER DOCS BY keara

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.