Lecture 5
18.086
R. J. LeVeque —
AMath 585–6 Notes
Phase vs. group velocity
time = 0
•
1
Remember from physics:
group
velocity
0.5
0
phase
velocity
−0.5
−1
−3
−2
−1
0
time = 0.4
1
0.5
0
−0.5
1
2
time = 0.8
1
0.5
0
ue —
Dispersion in LW scheme
AMath 585–6 Notes
187
−0.5
−1
−3
−2
−1
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−2
−1
0
1
2
3
1
2
3
1
2
3
time = 1.2
time = 0
−3
0
1
2
−3
3
−2
−1
0
time = 1.6
time = 0.4
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−3
−2
−1
0
1
2
3
−3
−2
−1
0
time = 0.8
Figure 13.6: The oscillatory wave packet satisfies the dispersive equation u t + aux + bu
shown is a black dot, translating at the phase velocity cp (ξ0 ) and a Gaussian that is tran
group velocity cg (ξ0 ).
1
0.5
0
−0.5
−1
−3
−2
−1
0
time = 1.2
1
1
2
3
Lax equivalence theorem
•
So far we considered stability and accuracy as independent
properties, but they are linked by the
Lax equivalence theorem
For a consistent approximation of a well-posed linear problem:
stability <=> convergence
Lax equivalence thm.
•
Give an IVP
ut = Au, u(0) = u0
•
Say we have an operator S such that
•
For the analytical solution, the situation is
||R
n
t u(0)||
U (t + t) = S t U (t) = S n t U (0)
u(t + t) = R t u(t) = Rn t u(0)
 c3 ||u(0)||
•
The IVP is well-posed if
•
The discretization leading to S has order of accuracy p if
||S
tu
R
p+1
u||

c
(
t)
t
1
If p>0, the discretization is called consistent
•
The {S
t}
The {S
t}
||S
•
n
t U ||
are called stable if:
 c2 ||U ||, for all n,
are called convergent if:
t with 0  n  t
lim
t!0,n t=t
||S n t u(0)
u(t)|| = 0
Lax equivalence theorem
•
So far we considered stability and accuracy as independent
properties, but they are linked by the
Lax equivalence theorem
For a consistent approximation of a well-posed linear problem:
stability <=> convergence
Rate of convergence
•
•
•
•
We can use the previous framework to redefine the accuracy (local and global
error).
•
Nothing new… :-)
Local error:
tu
R
Global error: ||U (n t)
p+1
u||

c
(
t)
t
1
u(n t)|| = ||(S n t
Rn t )u(0)||
The global error can be estimated as (p: order of accuracy - as before!)
||(S n t
•
||S
Rn t )u(0)||  c1 c2 c3 tp ||u(0)||
Lecture
=> stability is sufficient for convergence (necessary: not shown)
2nd order PDEs (sect. 6.4):
The wave equation
2
utt = c uxx
•
Wave equation:
•
Produces waves with velocities +/- c (i.e. in both directions!)
•
General solution: u(x,t) = F1(x+ct) + F2(x-ct)
•
For given initial conditions u(x,0) and ut(x,0):
Z x+ct
1
1
u(x, t) = [u(x + ct, 0) + u(x ct, 0)] +
ut (x̃, 0)dx̃
2
2c x ct
Lecture
Numerics for the wave equation
•
@
Equivalent 1st order problem:
@t
with v1 = ut , v2 = cux
✓
v1
v2
◆
=
✓
0 c
c 0
◆
@
@x
✓
•
Can use Lax-Wendroff/Friedrichs like for 1-way wave eq!
•
But there are better suited/simpler methods
•
Again the question is: How to discretize time (2nd order!) and space
v1
v2
◆
Numerics for the wave equation
•
Consider space discretization first, i.e. transform into ODE
2
d
2 Uj+1
Uj = c
2
dt
2Uj + Uj
x2
1
= Uxx + O(Δx2)
•
Using ansatz Uj = G(t)eikj
Gdisc (t) = e±icF kt
(using method of lines)
(check this!)
x
we find
Gan (t) = e
±ickt
F = sinc(k x/2)
•
•
Discretized space already leads to dispersion (F=F(k)), i.e.
waves with different k travel at different speeds cF(k)
What happens if we also discretize time?
Lecture
Leapfrog scheme
•
Easiest numerical scheme for 2nd order problem: Leapfrog
n t)
Notation: Uj,n = U (j x,CFL
criterion for Leapfrog scheme
Uj,n+1
•
2Uj,n + Uj,n 1 !" 2!"Uj+1,n
u
=0
Equation: =+ c
2
!t
!x
t
Stability: |r| ≤ 1 (equiv. CFL condition!)
Lecture
t
n+1!
Accuracy: 2nd order
n-2!
with
physical domain
of dependence
n!
n-1!
•
2Uj,n +nU
j 1,n
+1
= !in –1 – " (!in+1 # !in–1 )
Scheme: !i
x2
numerical domain
of dependence
unstable
|uΔt/Δx|
>1
c
stable
|uΔt/Δx|
≤1
c
Lecture / see Mathematica
notebook leapfrog_stability.nb
i-3!
i-2!
i-1!
i!
i+1!
i+2!
i+3!
x