1. Finite differences
Finite differences

2. Introduction 1 2
j-1 j
j+1 N
N+1
< L >
Taylor series expansion:
Finite differences

3. Finite differences approximations=
forward approximation
Consistent if , ,…. are bounded =
backward approximation
adding both ==
centered differences
Finite differences

4. Finite differences approximations (2)
Also
===
fourth order approximation to the first derivative
===
Second order approximation to the second derivative
Finite differences

5. The linear advection equation
+ initial and boundary conditions
Analytical solution
Substituting we get
Eigenvalue problems for the operators
With periodic B.C. λ can only have
certain (imaginary) values
where k is the wavenumber
The general solution is a linear combination of several wavenumbers
Finite differences

6. The linear advection equation (2)
The solution is then:
Propagating
with speed U0
No dispersion
For a single wave of wavenumber k, the frequency is ω=kU0
Energy conservation
If periodic B.C.
Finite differences

7. Space discretization ; substituting whose solution is with c
centered second-order approximation
; substituting
whose solution is
with c U0
The phase speed c depends on k; dispersion
kΔx= π ---> λ=2Δx ==> c=0
kΔx 
Finite differences

8. Group velocity =-U0 for kΔx=π
Continuous equation
Discretized equation
=-U0 for kΔx=π
Approximating the space operator introduces dispersion
Finite differences

9. Time discretization Try Substituting we get
first-order forward approx.
Try
Substituting we get
\--v--/
α (Courant-Friedrich-Levy number)
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
Finite differences
Also another dispersion is introduced, as we have approximated the operator ∂/ ∂t

10. 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
Finite differences

11. Stability analysis Energy method define Discretized analog : En
Periodic boundary
conditions
Discretized analog : En
φn N+1≡ φn1
If En=const,
stable
Finite differences

12. 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
if En+1 > En > unstable
α > 0 ==> U0 > (upwind)
α < U0 Δt/ Δx < 1 (CFL condition)
En+1 < En
Finite differences
damped

13. Von Neumann method
Consider a single wave
if |λk| < the scheme is damping for this wavenumber 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
Finite differences

14. Matrix method Let for a two-time-level scheme
is called the amplification matrix
And call
the eigenvectors of
Expanding the initial condition
In terms of these eigenvectors
and applying n times the amplification matrix
exponential
therefore
Finite differences

15. 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:
α < downwind
α > CFL limit
α(α-1) > > unstable
Finite differences
-1/4 < α(α-1) < 0==> 0 ≤ α ≤ > stable damped scheme

16. Stability of some schemes (cont)
Leapfrog
Using von Neumann we find |α|≤1 as stability condition
Finite differences

17. Stability of some schemes (cont)
Lax Wendroff
(Taylor in t)
From
Applying Von Neumann we can find that |α| ≤ > stable
Equivalent to x
Finite differences j
j+1/2
j+1

18. Stability of some schemes (cont)
Implicit centered scheme
Krank-Nicholson
where
using von Neumann
Always neutral
Dispersion worse
than leapfrog
where
Always neutral
Dispersion
better than
implicit.
No computational
mode
Finite differences

19. “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
U0Δt
Downwind scheme
x--> point where the information
comes from (xj-U0Δt) x
j-1 j
j+1
Interval used for the
computation of ∂φ/∂x
Upwind scheme o x
x ----> α < 1
o ----> α > 1
j-1 j
j+1
CFL number ==> fraction
of Δx traveled in Δt seconds o x
Leapfrog
Implicit
j-1 j
j+1
Finite differences

20. Dispersion and group velocity
Leapfrog vg vf U0
K-N
Implicit
ωΔt
Finite differences
π/2 π

21. Effect of dispersion
Initial
Leapfrog
implicit
Finite differences

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