18.336 spring 2009
lecture 13
03/19/09
Initial Value Problems (IVP)
⎧
⎨
ut = Lu
u = u0
⎩
u=g
where L
⎫
in Ω×]0, T [
← PDE
⎬
on Ω × {0}
← initial condition
⎭
on ∂Ω×]0, T [ ← boundary condition
differential operator.
⎫
Ex.: • L = �2
⎪
⎪
⎪
⎪
Poisson equation → heat equation
⎪
⎪
⎪
⎪
•Lu = b · �u
⎪
⎪
⎪
⎪
advection equation
⎬
2
2
•Lu = −� (� u)
not done yet.
⎪
⎪
biharmonic equation → beam equation �
⎪
⎪
⎪
⎪
•Lu = F |�u|
⎪
nonlinear
⎪
⎪
⎪
Eikonal equation → level set equation
⎪
⎪
⎭
• etc.
�
�
Stationary solution of IVP:
Lu = 0 in Ω
(if it exists)
u = g on ∂Ω
⎡
⎤
Later:
⎢ second order problems ⇔
systems
⎥
⎢
⎥
� � �
�
� � ⎥
⎢
⎢
⎥
∂
∂
u
0 1
u
⎢ utt = uxx
⎥
=
·
⎣
⎦
1 0
∂t v
∂x v
(wave equation)
Semi-Discretization
• In space (method of lines):
Approximate u(·, t) by �u(t)
Approximate Lu by A · �u (for linear problems) [FD, FE, spectral]
d
→ system of ODE:
�u = A · �u
dt
• In time:
Approximate time derivative by step:
d
u(x, t + Δt) − u(x, t)
u(x, t) ≈
[explicit Euler]
dt
Δt
→ Stationary problem:
unew (x) = u(x) + ΔtLu(x) = (I + ΔtL)u(x)
Need to know about ODE solvers.
1
Numerical Methods for ODE
�
ẏ(t) = f (y(t))
y(0) = ẙ
�
y(t) ∈ Rd
y n ≈ y(t), y n+1 ≈ y(t + Δt)
y n+1 − y n
Linear approximation: ẏ ≈
Δt
Explicit Euler (EE): y n+1 = y n + Δt · f (y n )
Implicit Euler (IE): y n+1 − Δt · f (y n+1 ) = y n
nonlinear system ← Newton iteration.
Local truncation error (LTE):
EE: τ n = y(t + Δt) − (y(t) + Δtf (y(t))) = 12 ÿ(t)Δt2 = O(Δt2 )
IE: τ n = O(Δt2 )
Global truncation error (GTE):
T
Over N =
time steps.
Δt
E n = y n − y(tn )
E n+1 = E n + Δt(f (y n ) − f (y(tn ))) + τ n
=⇒ |E n+1 | ≤ |E n | + ΔtL|E n | + |τ n
|
T
=⇒ |E N | ≤ eLT Δt
maxn |τ n | = O(Δt)
Time Stepping:
GTE = one order less than LTE.
Higher Order Time Stepping
• Taylor Series Methods:
Start with EE, add terms to eliminate leading order error terms.
PDE → Lax-Wendroff
• Runge-Kutta Methods:
Each step = multiple stages
�
k1
= f (y n + Δt
aij kj )
j
..
.
kr
y n+1
�
= f (y n + Δt
arj kj )
� j
= y n + Δt
bj kj
j
Butcher tableau: (cl =
�
alj )
j
2
c1
..
.
cr
a11 . . .
..
..
.
.
ar1 . . .
b1 . . .
a1r
..
c A
.
=
bT
arr
br
k1
= f (y n )
n+1
y
= y n + Δtk1
0
EE:
1
=y n+1
1 1
1
IE:
�
��
�
k1 = f (y n + Δt · k1 )
y n+1 = y n + Δt · k1
0
Explicit midpoint:
1
2
1
2
0 1
0
1 1
Heun’s:
1
2
1
2
0
1
2
1
2
RK4:
1
2
1
2
1
Implicit trapezoidal:
1
2
1
1
6
1
2
1
3
1
3
1
6
PDE → Crank-Nicolson
1
• Multistep Methods:
r
r
�
�
n+j
αj y
= Δt
βj f (y n+j )
j=0
j=0
Explicit Adams-Bashforth:
= EE
y n+1 = y n + Δtf (y n )
3
1
n+2
n+1
n+1
n
y
= y
+ Δt · [ 2 f (y ) − 2 f (y )]
..
.
Implicit Adams-Moultion:
y n+1 = y n + Δt · ( 12 f (y n ) + 12 (y n+1 ))
5
8
y n+2 = y n+1 + Δt · ( 12
f (y n+2 ) + 12
f (y n+1 ) −
..
.
O(Δt)
O(Δt2 )
= trapezoidal
1
f (y n ))
12
BDF (backward differentiation):
y n+1
= y n + Δtf (y n+1 ) = IE
3y n+2 − 4y n+1 + y n = 2Δtf (y n+2 )
..
.
3
O(Δt)
O(Δt2 )
O(Δt2 )
O(Δt3 )
Linear ODE Systems
�
ẏ = A · y
y(0) = ẙ
solution: y(t) = exp(tA) · ẙ
solution stable, if Re(λi (A)) < 0 ∀i.
=M
EE: y n+1
IE: y n+1
�
��EE �
= (I + Δt · A) ·y n
= (I − Δt
· A)−1 ·
y n
�
�
��
=MIE
Iteration y n+1 = M · y n stable, if |λi (M )| < 1 ∀i
λi eigenvalue of A
⎧
⎫
2
⎨
1 + Δt · λi eigenvalue of MEE ⎬
|1 + Δt ·
λi | < 1 if Δt <
�
�
|λi |
⇒
1
�
�
1
⎩
�
� < 1 always
eigenvalue of MIE ⎭
� 1 − Δt · λi �
1 − Δt · λi
2
EE conditionally stable: Δt <
ρ(A)
IE unconditionally stable
Message: One step implicit is more costly than one step explicit.
But: If ρ(A) large, then implicit pays!
� �� �
stiffness
Ex.: Different time scales
�
�
� �
−50 49
2
A=
, ẙ =
49 −50
0
� �
�
�
1
−1
−t
−99t
Solution: y(t) = e ·
+e
·
1
1
�
��
�
�
��
�
behavior
Ex.:
restricts Δt for EE
−2 1
. .
. .
⎢
.
.
1
⎢ 1
A=
·⎢
2
.
.. ... 1
(Δx) ⎣
1 −2
�
��
⎡
−4<λi <0
⎤
⎥
⎥
⎥
⎦
heat equation
4
⇒
ρ(A)
<
�
(Δx)2
(Δx)2
.
2
OTOH: Crank-Nicolson
�
�
�
�
Δt
Δt
n+1
I−
A ·y
= I+
A · yn
2
2
Unconditionally stable.
EE stable if Δt <
4
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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