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

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

3. Stability in one dimension
e.g. McDonald (1987)
Linear advection equation without r.h.s. p
Origin of parcel at j: X* =Xj-U0Δt x α j
“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!

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. 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 x x
interpolation
φmax
φmin x x
- Quasi-monotone interpolation

6. Interpolation in the IFS semi-Lagrangian scheme
ECMWF model uses
quasi-monotone quasi-cubic Lagrange interpolation
Cubic Lagrange interpolation:
with the weights
Number of 1D cubic interpolations
in two dimensions is 5,
in three dimensions 21! x y
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.

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

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’

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

10. Semi-Lagrangian advection on the sphere
Momentum eq. is discretized in vector form (because a vector is continuous across
the poles, components are not!)
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.
Trajectories are arcs of great circles if
constant (angular) velocity is assumed
for the duration of a time step. X Y Z A V x D M i j A D
Tangent plane projection
Trajectory calculation

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

12. Iterative trajectory computation (1 dimension)
r(n+1)=g-V0(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 = λn + K; (| λ| < 1)
Substituting, we get K=(g - a Δt)/(1 + b Δt) and λ = -b Δt
 |b|Δt < 1
more generally for three dimensions this translates to
the determinant of a matrix:
Lipschitz condition
(Smolarkiewicz and Pudykiewicz, 1992)
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.

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.

14. Semi-Lagrangian advection with rhs
• Three-time-level schemes
- centered (second-order accurate) scheme
- split in time (first-order accurate)
- R at the departure point (first-order accurate)
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

15. Example Z = Re( Ae-ikx eωt)
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:
5×10-4 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

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

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

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.