18.336 spring 2009
lecture 25
Ex.: Continuity Equation
05/12/09
Traffic on road
ρ(t) + (v(x)ρ)x = 0
⇔ ρt + v(x)ρx = −v � (x)ρ
⇒ Characteristic ODE
�
ẋj = v(xj )
ρ̇j = −v � (xj )ρj
v = v(x)
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Ex.: Nonlinear conservation law
ut + ( 12 u
2 )x = 0 Burgers’ equation
�
�
ẋj = uj
⇒
if solution smooth
u̇j = 0
Problem: Characteristic curves can intersect ⇔ Particles collide
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Particle management required:
• Merge colliding particles (how?)
• Insert new particles into gaps (where/how?)
An Exactly Conservative Particle Method for 1D Scalar Conservation Laws
[Farjoun, Seibold JCP 2009]
ut + ( 12 u
2 )x = 0
Observe: If we have a piecewise linear function initially, then the
exact solution is a piecewise linear function forever
(including shocks).
1
Two choices: (A) Move shock particles using Rankine-Hugoniot condition.
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(B) Merge shock particles, then proceed in time.
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Same for insertion:
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Ex.: Shallow water equations
⎧
Dh
⎪
⎨
= −ux h
ht + (uh)x = 0
Dt
⇔
ut + uux + ghx = 0
⎪
⎩
Du
= −ghx
Dt
Df
Lagrangian derivative
= ft + u · fx
Dt
Particle Method:
⎧
⎫
⎨ ẋj = uj
⎬
ḣj = −hj (∂x u)(xj )
⎩
⎭
u̇j = −g(∂x h)(xj )
�
�
Required: Approximation to ∂x h, ∂x u at xj
Particles non-equidistant.
2
⎫
⎪
⎬
⎪
⎭
Meshfree approximation (moving least squares):
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Local fit: û(x) = ax2 + bx + c
�
Weighted LSQ-fit: mina,b,c
j:|xj −x0 |≤r
w(x) =
1
rα
or = e
−αr
|û(xj ) − u(xj )|2
w(xj − x0 )
or . . .
�
Define (∂x u)(xj ) = û (x0 )
Smoothed Particle Hydrodynamics (SPH)
Quantity f
�
f (x) =
f (x̃)δ(x − x̃)dx̃
Rd
Sequence of kernels W h
�
h
lim W (x) = δ(x), W h (x)dx = 1 ∀h.
h→0
R
Also:
W h (x) = wh (||x||),
w(d) = 0 ∀d > h ← smoothing length
Approximation I:
�
h
f (x) =
f (x̃)W h (x − x̃)dx̃
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Rd
h
⇒ lim f (x) = f (x).
h→0
Density ρ
�
Density measure µρ (A) =
ρdx
A
�
f (x̃) h
h
f (x) =
W (x − x̃) ρ(x̃)dx̃
� �� �
Rd ρ(x̃)
=dµρ (x̃)
3
Sequence of point clouds {X (n) }h∈N
X (n) = (x1 (n) , . . . , xn (n) )
Point measure
n
�
(m)
δX
=
mi (n) δx(n) −→ µρ
i=1
i
n→∞
Approximation II: f h (x) ≈
n
�
�
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f (x̃) h
W (x − x̃)dδX (n) (x̃)
ρ(x̃)
f (xi (n) ) h
W (x − xi (n) ) =: f h,n (x)
(n) )
ρ(x
i
i=1
n
�
f (xi (n) )
h,n
�W h (x − xi (n) )
⇒ �f (x) =
mi (n)
(n) ) � �� �
ρ(x
i
i=1
hard-code
n
�
m
i
fkh =
fi Wki h
ρ
i
i=1
n
�
mi
�fk h =
fi Wki h
ρ
i
i=1
=
mi (n)
Apply to Euler equations of compressible
⎧
⎫ ⎧
∂ρ
⎪
⎪ ⎪
⎪
⎪
= −� · (ρ�u) ⎪
density ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
∂t
⎪
⎪
⎪
⎨ D�u
⎬
⎨
�p
velocity
⇔
= −
⎪
⎪
⎪
ρ
Dt
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
De
p
⎪
⎪
⎪
⎪
⎪
energy ⎩
= − � · �u ⎭ ⎪
⎩
Dt
ρ
⎧
⎪
ρ˙k
⎪
⎪
⎪
⎪
⎪
⎨
u˙k
→
⎪
⎪
⎪
⎪
⎪
⎪
⎩ e˙k
⎧
⎪
ρ˙k
⎪
⎪
⎪
⎪
⎪
⎨
�u̇k
⇔
⎪
⎪
⎪
⎪
⎪
⎪
⎩ e˙k
=
=
=
=
=
=
gas dynamics:
Dρ
= �u · �ρ − � · (ρ�u)
Dt
� � � �
p
p
D�u
−
= −�
�ρ
ρ2 � �
Dt
�ρ �
De
p
p�u
= u·�
−�·
Dt
ρ
ρ
p = p(s, e)
�
�
�uk
mi �Wki −
mi u�i h �Wki
i
i
�
pk �
m i pi
−
�Wki −
mi �Wki
ρi ρi
(ρk )2 i
i
� mi pi
� mi pi�vi
�uk
�Wki −
�Wki
ρ
ρ
ρ
ρ
i
i
i
i
i
i
⎫
�
⎪
mi (�uk − �ui )�Wki
⎪
⎪
⎪
⎪
i
�
�
⎪
⎬
�
ρk
pi
−
mi
+
�W
ki
(ρk )2 (ρi )2
⎪
⎪
�i
⎪
pi
⎪
⎪
⎪
(�
u
−
�
u
)�W
mi
k
i
ki
⎭
2
(ρ
)
i
i
4
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
d
Since
dt
�
�
�
ρi
= 0 ⇒ ρk =
i
�
mi Wki
i
SPH Approximation to Euler Equations:
� 0 if adaptive)
ṁk = 0 (or =
�ẋk = �uk
�u̇k = −
�
�
mi
i
ėk =
�
ρk =
�
i
mi
pk
pi
+
2
(ρk )
(ρi )2
�
�Wki
pi
(�uk − �ui )�Wki
(ρi )2
mi Wki
i
pk = p(ρk , ek )
5
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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