1. An Introduction to Numerical Methods
For Weather Prediction by
Joerg Urban
office 012
Based on lectures given by Mariano Hortal
Numerical methods

2. Shallow water equations in 1 dimension
Non linear equations h
advection
adjustement
diffusion
u … velocity along x direction
h … absolute height
g … acceleration due to gravity
K …diffusion coefficient
Velocity is equal
in layers vertical
direction (shallow)
Numerical methods

3. Linearization u=U0+u’ h=H +h’
Const. + perturb. in the x-comp. of velocity
Const. + perturb. in the height of the free surface
Small perturbations
Substitute and drop products of perturbations
Numerical methods

4. Classification of PDE’s
Boundary value problems D
open domain ΓD
f, g: known function; L, B: differential operator
φ: unknown function of x
boundary
Initial value problems (most important to us)
φ: unknown function of t t
t=0
Numerical methods

5. Classification of PDE’s (II)
Initial and boundary value problems
Eigenvalue problems
φ: unknown eigenfunction
λ: eigenvalue of operator L
Numerical methods

6. Existence and uniqueness of solutions
; y(t0)=y (initial value problem)
Does it have a solution?
Does it have only one solution?
Do we care?
If it has one and only one solution it is called a well posed problem
Numerical methods

7. Picard’s Theorem Let and be continuous in the rectangle
then, the initial-value problem
Has a unique solution y(t)
on the interval
Finding the solution (not analytical)
Numerical methods (finite dimensions)
Numerical methods

8. Discretization Finite differences Spectral Finite elements
Transform the continuous differential equation
into a system of ordinary algebraic equations
where the unknowns are the numbers fj
Numerical methods

9. The Lax-Richtmeyer theorem
Convergence
Consistency
Stability
Lax-Richtmeyer theorem
Discretized solution > continuous solution
discretization finer and finer
Discretized equation > continuous equation
Discretized solution bounded
If a discretization scheme is consistent and stable
then it is convergent, and vice versa
Numerical methods

10. Finite Differences - Introduction1 2
j-1 j
j+1 N
N+1
< L >
Taylor series expansion:
Numerical methods

11. Finite differences approximations
forward approximation
Consistent if … are bounded
backward approximation
adding both
centered differences
Consistent if … are bounded
Numerical methods

12. Finite differences approximations (2)
Also
fourth order approximation to the first derivative
Using the Taylor expansion again we can easily get the
second derivative:
second order approximation of the second derivative
Numerical methods

13. The linear advection equation
+ initial and boundary conditions
We start with a guess:
Substituting we get:
Eigenvalue problems for
With periodic B.C. λ can only have certain (imaginary) values where k is the wave number
The general solution is a linear combination of several wave numbers
Numerical methods

14. The linear advection equation (2)
The analytic solution is then:
Propagating with speed U0
No dispersion
For a single wave of wave number k the frequency is ω=kU0
Energy:
For periodic B.C.:
Numerical methods

15. Space discretization centered second-order approximation Try:
results in
whose solution is
with c
The phase speed c depends on k dispersion U0
kΔx= π ---> λ=2Δx ==> c=0
kΔx π
Numerical methods

16. Group velocity =-U0 for kΔx=π
Continuous equation
Discretized equation
=-U0 for kΔx=π
Approximating the space operator introduces dispersion
Numerical methods

17. Time discretization Try Substituting we get
In addition to our 2nd order centered approx. for the space derivative we use a
1st order forward approx. for the time derivative:
Try
Courant-Friedrich-
Levy number
Substituting we get
If b>0, φjn increases exponentially with time (unstable)
If b<0, φjn decreases exponentially with time (damped)
If b=0, φjn maintains its amplitude with time (neutral)
ω=a+ib
Numerical methods
Also another dispersion is introduced, as we have approximated the operator ∂/ ∂t

18. Three time level scheme (leapfrog)
This scheme is centered (second order accurate) in both space and time
exponential
Try a solution of the form
If |λk| > 1 solution unstable
if |λk| = 1 solution neutral
if |λk| < 1 solution damped
Substituting
physical mode
Δx--->0
Δt --->0
computational mode
Numerical methods

19. Stability analysis Energy method We have defined
We have discretized t and
hence the discretized analog of E(t) is En
For periodic
boundary conditions
φn N+1≡ φn1
If En=const,
than the scheme is stable
Numerical methods

20. Example of the energy method
upwind if U0>0
downwind if U0<0 x
j-1 j
α=0 => U0= no motion
α= Δt= Δx/U0
En+1=En if
En+1 > En unstable
α > 0 => U0 > upwind
α < U0 Δt/ Δx < 1 CFL cond. damped
En+1 < En if
Numerical methods

21. Von Neumann method
Consider a single wave
if |λk| < the scheme is damping for this wave number k
if |λk| = k the scheme is neutral
if |λk| > for some value of k, the scheme is unstable
alternatively
if Im(ω) > scheme unstable
if Im(ω) = scheme neutral
if Im(ω) < scheme damping
Vf= ω/k vg=∂ω/∂k
Numerical methods

22. Stability of some schemes
Forward in time, centered in space (FTCS) scheme
Upwind or downwind
using Von Neumann, we find
scheme unstable
upwind if U0 > 0
downwind if U0 < 0
Using Von Neumann, we find
α < downwind
α > CFL limit
α(α-1) > unstable
-1/4 < α(α-1) < 0 => 0 ≤ α ≤ stable damped scheme
Numerical methods

23. Stability of some schemes (cont)
Leapfrog
Using von Neumann we find |α|≤1 as stability condition
As a reminder α is the Courant-Friedrich-Levy number:
Numerical methods

24. Stability of some schemes (cont)
Lax Wendroff scheme
From a Taylor expansion in t we get:
Substitution the advection equation we:
Discretization:
Applying Von Neumann we can find that |α| ≤ > stable
Numerical methods

25. Stability of some schemes (cont)
Implicit centered scheme
We replace the space derivation by the average value of the centred
space derivation at time level n-1 and n+1
using von Neumann
Always neutral, however an
Expensive implicit equation
need to solved
Dispersion worse than leapfrog
Numerical methods

26. “Intuitive” look at stability
If the information for the future time step “comes from” inside the interval used for the computation of the space derivative, the scheme is stable.
Otherwise it is unstable
Downwind scheme (unstable)
x, x’ … point where the information
comes from (xj-U0Δt)
Interval used for the
computation of ∂φ/∂x
U0Δt
j-1 j
j+1 x
Upwind scheme (conditionally stable)
x if α < 1
x’ if α > 1
x’ x
j-1 j
j+1
Leapfrog (conditionally stable)
CFL number ==> fraction
of Δx traveled in Δt seconds
x’ x
Implicit (unconditionally stable)
j-1 j
j+1
Numerical methods

27. Dispersion and group velocity
Vf= ω/k vg=∂ω/∂k
Leapfrog vg vf U0
K-N
ωΔt
Implicit
Numerical methods
π/2 π

28. Effect of dispersion
Initial
Leapfrog
implicit
Numerical methods

29. Two-dimensional advection equation
Using von Neumann, assuming a solution of the form
we obtain
using
we obtain, for |λ| ≤ 1 the condition
where Δs= Δx= Δy
This is more restrictive than in one dimension by a factor
Numerical methods

30. Non linear advection equation Continuous form
u(x,t)
u(x,0) x x
Change in shape even
for the continuous form
One Fourier component ukeikx no longer moving with constant
speed but interacting with other components
Fourier decomposition valid at each individual time
but it changes amplitude with time
No analytical solution!
Numerical methods

31. Energy conservation Define again: == Discretization in space
periodic B.C.
First attempt:
Second attempt:
terms joined by arrows
cancel from
consecutive j’s
Numerical methods

32. Aliassing Consider the product Minimum wavelength
Aliasing occurs when the non-linear interactions in the advection term
produce a wave which is too short to be represented on the grid.
Consider the product
in the interval 0≤x ≤2π
Minimum wavelength 1 2 n
N+1
Maximum wave number representable with the discretized grid
< L >
Numerical methods

33. Aliassing (cont.) wave number k  wave number 2kM-k
Trigonometrical manipulations lead to: sin(kxj)=-sin[(2kM-k)xj]
Therefore, it is not possible to distinguish wave numbers k and (2kM-k)
on the grid.
wave number k  wave number 2kM-k x x x
Numerical methods

34. Non-linear instability
If k1+k2 is misrepresented as k1 there is positive feedback, which causes instability
k1 = 2kM - (k1+k2) > 2k1=2kM-k2
2kM 2k1  kM
2Δx ≤ λ1 ≤ 4Δx
These wavelengths keep storing energy and total energy is not conserved
We can remove energy from the smallest wavelengths by
- Fourier filtering
- Smoothing
- Diffusion
- Use some other discretization (e.g. semi-Lagrangian)
Numerical methods