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ECMWF Numerical methods 3 Slide 1 Numerical methods III (Advection: the semi-Lagrangian technique) Based on previous material by Mariano Hortal and Agathe Untch by Nils Wedi (room 007; ext. 2657)

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ECMWF Numerical methods 3 Slide 2 Advection: The semi-Lagrangian technique material time derivative or time evolution along a trajectory thus avoiding quadratic terms; x x x x x x x x x x x x x x x x x x From a regular array of points we end up after Δt with a non-regular distribution Semi-Lagrangian: (usually) tracking back Solution of the one-dimensional advection equation : origin point interpolation computing the origin point via trajectory calculation disadvantage: not flux-form!

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Stability in one dimension Linear advection equation without r.h.s. j x p α Origin of parcel at j: X * =X j -U 0 Δt multiply-upstream p: integer Linear interpolation α is not the CFL number except when p=0, then=> upwind Von Neumann : |λ|1 if 0 α 1 (interpolation from two nearest points) Damping! e.g. McDonald (1987)

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ECMWF Numerical methods 3 Slide 4 Cubic spline interpolation S(x) is a cubic polynomial - S(x j )=φ j at the neighbouring grid points - S(x)/ x is continuous - d 2 S/dx 2 dx is minimal Then: S(x)=D j-1 (x j -x) 2 (x-x j-1 )/(Δx) 2 -D j (x-x j-1 ) 2 (x j -x) /(Δx) 2 + +φ j-1 (x j -x) 2 [2(x-x j-1 )+ Δx] /(Δx) 3 + φ j (x-x j-1 ) 2 [2(x j -x)+ Δx] /(Δx) 3 where (D j-1 +4D j +D j+1 )/6=(φ j+1 - φ j-1 )/2 Δx

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ECMWF Numerical methods 3 Slide 5 Shape-preserving interpolation Creation of artificial maxima /minima x x x x x x: grid points x: interpolation point Shape-preserving and quasi-monotone interpolation - Spline or Hermite interpolation x x derivatives interpolation x x modified derivatives x x - Quasi-monotone interpolation φ max φ min x x

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ECMWF Numerical methods 3 Slide 6 Interpolation in the IFS semi-Lagrangian scheme with the weights ECMWF model uses quasi-monotone quasi-cubic Lagrange interpolation xxxx x xxxx xxxx xxxx x x x x y x Number of 1D cubic interpolations in two dimensions is 5, in three dimensions 21! To save on computations: cubic interpolation only for nearest neighbour rows, linear interpolation for rest => quasi-cubic interpolation => 7 cubic + 10 linear in 3 dimensions. Cubic Lagrange interpolation:

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ECMWF Numerical methods 3 Slide 7 3-t-l Semi-Lagrangian schemes in 2-D L: linear operator N: non-linear function Interpolating x x G o x x I Two interpolations needed Ritchie scheme U=U*+UV=V*+V G o x 2VΔt 2V*Δt o I Non-interpolating Average the non-linear terms between points G and o The three of them are second-order accurate in space-time

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ECMWF Numerical methods 3 Slide 8 Stability of 2-D schemes In the linear advection equation the interpolating scheme is stable provided the interpolations use the nearest points In the linear shallow-water equations (with rotation), treating the linear terms implicitly, the stability limit is In the two non-interpolating schemes the stability is given by Δt f 1 Coriolis term (kU+lV) Δt 1 Which is always true due to the definition of U and V

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ECMWF Numerical methods 3 Slide 9 Treatment of the Coriolis term Implicit treatment : Advective treatment: In three-time-level semi-Lagrangian: In two-time-level semi-Lagrangian: treated explicitly with the rest of the rhs Extrapolation in time to the middle of the trajectory leads to instability (Temperton (1997)) Two stable options: Helmholtz eqs partially coupled for individual spectral components => tri-diagonal system to be solved. (Vector R here is the position vector.)

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ECMWF Numerical methods 3 Slide 10 Trajectory calculation M i j A i j D i j Tangent plane projection Semi-Lagrangian advection on the sphere X Y Z A V x D Momentum eq. is discretized in vector form (because a vector is continuous across the poles, components are not!) Trajectories are arcs of great circles if constant (angular) velocity is assumed for the duration of a time step. Interpolations at departure point are done for components u & v of the velocity vec- tor relative to the system of reference local at D. Interpolated values are to be used at A, so the change of reference system from D to A needs to be taken into account.

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ECMWF Numerical methods 3 Slide 11 Iterative trajectory calculation x x x x x assumed constant during 2Δt trajectory straight line or great circle can be taken as V0ΔtV0Δt r0r0 V1ΔtV1Δt r1r1 or implicit

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ECMWF Numerical methods 3 Slide 12 Iterative trajectory computation (1 dimension) r (n+1) =g-V 0(n) Δt Where, for simplicity, we have taken a 2-time-level scheme and taken the velocity at the departure point of the trajectory Assume that V varies linearly between grid-points V=a+b.r b = r (n+1) = g - aΔt - Δt b r (n) For this procedure to converge, it must have a solution of the form r = λ n + K; (| λ| < 1) Substituting, we get K=(g - a Δt)/(1 + b Δt) and λ = -b Δt more generally for three dimensions this translates to the determinant of a matrix: Physical meaning: To prevent trajectory intersections !!! It is in general less restrictive than the CFL condition and it does not depend on the mesh size. Lipschitz condition (Smolarkiewicz and Pudykiewicz, 1992) |b|Δt < 1

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ECMWF Numerical methods 3 Slide 13 Mass conservation Semi-Lagrangian formulations are based on the advective form of the equations but can be made mass conserving (e.g. Zerroukat 2007; Kaas 2008) 2 fundamental issues: The iterative trajectory calculation should (but normally does not) involve the continuity equation. The conservation properties of the interpolation operator are important.

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ECMWF Numerical methods 3 Slide 14 the r.h.s. R can be evaluated by interpolation to the middle of the trajectory or averaged along the trajectory: R M (t)={R D (t)+R A (t)}/2 Three-time-level schemes - centered (second-order accurate) scheme - split in time (first-order accurate) - R at the departure point (first-order accurate) Semi-Lagrangian advection with rhs

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ECMWF Numerical methods 3 Slide 15 Example Let us apply each of the above schemes to the equation whose analytical solution (with appropriate initial and boundary conditions) is: Z = Re( Ae -ikx e ωt ) with ω=ikU 0 -k 2 K WARNING: the three-time-level scheme applied to the diffusion eq. has an absolutely unstable numerical solution! With the values A=1, k=2π/100, K=10 -2, the r.m.s. error with respect to the analytical solution (before the unstable numerical solution grows too much) grows linearly with time. After 200 sec of integration, the error is: 5×10 -4 Δt for the split treatment 5×10 -4 Δt for r.h.s. at departure point 5×10 -8 (Δt) 2 for the centered scheme

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ECMWF Numerical methods 3 Slide 16 Semi-Lagrangian advection with rhs Unstable! => noisy forecasts Two-time-level (second-order accurate) schemes : Forecast of temperature at 200 hPa (from 1997/01/04) with Extrapolation in time to middle of time interval

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ECMWF Numerical methods 3 Slide 17 Stable extrapolating two-time-level semi-Lagrangian (SETTLS): Forecast 200 hPa T from 1997/01/04 using SETTLS With and Taylor expansion to second order

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ECMWF Numerical methods 3 Slide 18 Trajectory computation with SETTLS Mean velocity during Δt The Taylor expansion from which we started is: which represents a uniformly accelerated movement Note: The middle of the trajectory is not the average between the departure and the arrival points.

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