18.336 spring 2009
lecture 21
Systems of IVP
04/28/09
⎡
⎤
u1 (x, t)
⎢
⎥
.
Solution has multiple components: �u(x, t) = ⎣
⎦
..
m equations
um (x, t)
Uncoupled (trivial)
ut = uxx
vt = vxx
Solve independently
Triangular (easy)
(1) ut + uux = 0
(2) ρt + uρx = dρxx
Solve first (1), then (2)
velocity field
density of pollutant
Fully Coupled (hard)
�
�
ht + (uh)x = 0
shallow water equations
ut + uux + ghx = 0
� �
�
�
h
uh
⇔
+ 1 2
= 0
u t
u + gh x
2
hyperbolic conservation law
Linear Hyperbolic Systems
Linearize SW
� �
�
h
ū
+
u t
g
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¯ u:
equations around base flow h,
¯
� � �
¯
h
h
·
=0
ū
u x
Linear system:
(∗) �ut + A · �ux = 0
�
�
ū h̄
SW: A =
g ū
A ∈ Rm×m
�
¯
λ = ū ± gh
(∗) is called hyperbolic, if A is diagonalizable with real eigenvalues, and
strictly hyperbolic, if the eigenvalues are distinct.
A = R · D · R−1 ; change of coordinates: �v = R−1 · �u
⇒ �vt = D · �vx = 0 Decoupled system
(vp )t + λp (vp )x = 0 ∀p = 1, . . . , m
⇒ vp (x, t) = vp (x − λp t, 0) Simple wave
Solution is superposition of simple waves
Numerics: Implement simple waves into Godunov’s method.
1
Wave Equation
1D utt = c2 uxx
� � �
�
� �
u
0 c
u
⇔ ∂t
=
∂x
v
c 0
v
Ex.: Maxwell’s Equations
�
Et = cHx
Ht = cEx
�
Ett = (Et )t = (cHx )t = c(Ht )x = c(cEx )x = c2 Exx
⇒
Htt = · · · = c2 Hxx
Schemes based on hyperbolic systems
�
�
� �
�
�
� �
ϕ=u+v
ϕ
c 0
ϕ
ϕt − cϕx = 0
→ ∂t
·∂x
→
=
ψ =u−v
ψ
0 −c
ψ
ψt + cψx = 0
Upwind for ϕ, ψ:
⎧
ϕj n+1 − ϕj n
ϕj+1 n − ϕj n
⎪
⎪
⎨
= c
Δt
Δx
n+1
n
n
n
⎪
ψ
−
ψ
ψ
j
j − ψj−1
⎪
⎩ j
= −c
Δt
Δx
u = 12 (ϕ + ψ) and v = 12 (ϕ − ψ)
�
�
�
�
Ujn+1 − Ujn
1 ϕj n+1 − ϕj n ψj n+1 − ψj n
c ϕj+1 n − ϕj n ψj n − ψj−1 n
=
+
=
+
Δt
2
Δt
Δt
2
Δx
Δx
� n
�
n
n
n
c Uj +1 − Uj
Vj+1 n − Vj n Uj − Uj −1 Vj n − Vj−1 n
−
=
+
+
2
Δx
Δx
Δx
Δx
n
n
n
n
n
Vj+1 − Vj−1
cΔx Uj +1 − 2Uj + Uj −1
·
=c
+
2Δx
2
Δx2
Ujn+1 − Ujn−1 cΔx Vj+1 n − 2Vj n + Vj−1 n
Vj n+1 − Vj n
·
= ··· = c
+
Δt
Δx
2
Δx2
Lax-Friedrichs-like Scheme for u, v:
� �
�� �
�� �
�
� �n �
�
�
n+1 � �n
n
cΔx Uxx
U
U
0 c
U
U
1
1
·
−
=
−
+
Δt
V j
c 0 2Δx
V j+1 V j−1
Vyy
V j
2
�
��
�
artificial diffusion
Stable (check by von-Neumann stability analysis).
More accurate schemes: Use WENO and SSP-RK for ϕ, ψ.
2
Leapfrog Method
utt = c2 uxx
Ujn+1
− 2Ujn + Ujn−1
Ujn+1 − 2Ujn + Ujn−1
2
→
=
c
Δt2
Δx2
(Two-step method)
Accuracy:
utt + 12 utttt Δt2 − c2 uxx −
1 2
c uxxxx Δx2
12
= O(Δt2 ) + O(Δx
2 )
(using utt − c2 uxx = 0)
second order
Stability:
ikΔx
G2 − 2G + 1
− 2 + e−ikΔx
2 e
=
c
G
Δt2
Δx2
cΔt
r=
Δx
Courant number
⇒ G2 − 2G + 1 = 2r2 (cos(kΔx) − 1) · G
⇒ G − 2 (1 − r2 (1 − cos(kΔx))) ·G + 1 = 0
�
��
�
=a
√
⇒ G = a ± a2 − 1
If |a| > 1 ⇒ one solution
√ with |G| > 12 ⇒ unstable
If |a| ≤ 1 ⇒ G = a ± i 1 − a2 ⇒ |G| = a2 + (1 − a2 ) = 1 ⇒ stable
Have 1 − cos(kΔx) ∈ [0, 2], thus: |a| ≤ 1 ⇔ |r| ≤ 1
Leapfrog conditionally stable for |r| ≤ 1.
Staggered Grids
� � �
�
� �
u
0 c
u
∂t
=
∂x
v
c 0
v
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Collocation Grid
Staggered Grid
Central differencing
requires artificial diffusion
Central differencing
comes naturally
3
⎧ n+1
Uj − Ujn
Vj n − Vj−1 n
⎪
⎪
⎨
= c
Δt
Δx
n+1
n+1
n+1
n
⎪
U
− Vj
⎪
j+1 − Uj
⎩ Vj
= c
Δt
Δx
Both are explicit central differences.
Two step update:
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�
�
n+1
n−1
Ujn+1 − 2Ujn + Ujn+1
1 Uj − Ujn Ujn − Uj
−
=
Δt2
Δt
Δt
Δt
� n
�
� n
�
c Vj − Vj−1 n Vj n−1 − Vj−1 n−1
c
Vj − Vj n−1 Vj−1 n − Vj−1 n−1
−
−
=
=
Δt
Δx
Δx
Δx
Δt
Δt
�
�
n
n
n
n
n
n
n
Uj − Uj −1
Uj +1 − 2Uj + Uj −1
c2 Uj +1 − Uj
−
=
= c2
Δx
Δx
Δx
Δx2
Equivalent to Leapfrog.
4
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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