The semi-Lagrangian technique

1. The semi-Lagrangiantechnique

2. x x x x x x x x x The semi-Lagrangian technique material time derivative or time evolution along a trajectory no quadratic terms x From a regular array of points we end up after Δt with a non-regular distribution x x x x x x x x Semi-Lagrangian: tracking back Solution of the one-dimensional advection equation: origin point computing the origin point

3. Linear advection equation without r.h.s. Stability in one dimension p Origin of parcel at j: X*=Xj-U0Δt x j multiply upstream α 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

4. Cubic spline interpolation S(x) is a cubic polynomial - S(xj)=φj at the neighbouring grid points - ∂S(x)/ ∂x is continuous - ∫d2S/dx2 dx is minimal Then: S(x)=Dj-1(xj-x)2(x-xj-1)/(Δx)2-Dj(x-xj-1)2(xj-x) /(Δx)2 + +φj-1(xj-x)2[2(x-xj-1)+ Δx] /(Δx)3+ φj(x-xj-1)2[2(xj-x)+ Δx] /(Δx)3 where (Dj-1+4Dj+Dj+1)/6=(φj+1- φj-1)/2 Δx

5. Cubic Lagrange interpolation Q(x) is a cubic polynomial - Q(xj)= φj at 4 nearest grid-points

6. x x x x x x Shape-preserving interpolation • Creation of artificial maxima /minima x: grid points x x x x: interpolation point x x • Shape-preserving and quasi-monotone interpolation - Spline or Hermite interpolation derivatives modified derivatives interpolation - Quasi-monotone interpolation x φmax x φmin

7. o o x x 3-t-l Semi-Lagrangian schemes in 2-D L: linear operatorN: non-linear function • Interpolating G x x x x x x x x x x x x I x Two interpolations needed • Ritchie scheme U=U*+U’ V=V*+V’ G x x x x x x x x x x x x I’ 2V*Δt o’ 2V’Δt • Non-interpolating Average the non-linear terms between points G and o’ The three of them are second-order accurate in space-time

8. Bicubic: underlined points Details about interpolation 12-point interpolation in 2-D; 32-point interpolation in 3-D: red points Bilinear: shadowed points x x x x x xxxx x xxxx x xxxx x xxxx x G x o

9. 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, treating the linear terms implicitly, the stability limit is • In the two non-interpolating schemes the stability is given by • Advective treatment of Coriolis term Δt f ≤1 Coriolis term (kU’+lV’) Δt ≤1 Which is always true due to the definition of U’ and V’

10. Spherical geometry in semi-Lagrangian advection Z V x G j j j Y i G X O I i i Trajectory calculation Tangent plane projection

11. V1Δt Iterative trajectory calculation V0Δt x x x x x x x x x x x x x x x x x x x x x x x x x r0 r1 can be taken as trajectory straight line or great circle or as implicit assumed constant during 2Δt

12. Iterative trajectory computation (1 dimension) rn+1=g-VnΔt 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=dV/dr (divergence) r n+1 = g - aΔt - Δt b rn For this procedure to converge, it must have a solution of the form r = λn + K; (| λ| < 1) Substituting, we get K=(g - a Δt)/(1 + b Δt) and λ = -b Δt therefore we must have The condition means thet the parcels do not overtake eachother is less restrictive than the CFL condition, in general doesn’t depend on the mesh size

13. Semi-Lagrangian equation with right-hand-side • Three-time-level schemes - centered (second-order accurate) scheme the r.h.s. R can be evaluated by interpolation to the middle of the trajectory or averaged along the trajectory: RM(t)={RD(t)+RA(t)}/2 - split in time (first-order accurate) - R at the departure point (first-order accurate)

14. 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 eωt) with ω=ikU0-k2K 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: 5×10-4Δt for the split treatment 5×10-4Δt for r.h.s. at departure point 5×10-8 (Δt)2 for the centered scheme

15. Semi-Lagrangian equation with right-hand-side (cont) • Two-time-level schemes - centered scheme with - split or R at the departure point similar to the 3-time-level case - SETTLS scheme Taylor expansion around the departure point and

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