Numerical Methods for Modern Traffic
Flow Models
Alexander Kurganov
Tulane University
Mathematics Department
www.math.tulane.edu/∼kurganov
joint work with
Pierre Degond, Université Paul Sabatier, Toulouse
Anthony Polizzi, University of Michigan, Ann Arbor
1
Traffic Flow Models with Arrhenius
Look-Ahead Dynamics
[M.J. Lighthill, G.B. Whitham; 1955] proposed the simplest
deterministic continuum traffic flow model:
ut + f (u)x = 0,
f (u) = uV (u),
V (u) = Vm(1 − u)
u(x, t): a relative car density (0 ≤ u ≤ 1)
Vm ≡ const: the maximum possible speed
V (u): dimensionless velocity function that depends only on u
2
[A. Sopasakis, M.A. Katsoulakis; 2006] extended the LighthillWhitham model to a more realistic one by taking into account
interactions with other vehicles ahead (“look-ahead” rule).
The resulting scalar conservation law with a global flux is:
ut + F (u)x = 0,
F (u) = Vmu(1 − u) exp(−J ◦ u),
where the contribution of short range interactions to the flux is:
J ◦ w(x) :=
x+γ
Z
J(y − x)w(y) dy
x
γ ≡ const > 0: look-ahead distance
J: interaction potential
3
The simplest interaction potential is the constant one:
(
J(r) =
1/γ, 0 < r < γ,
0, otherwise.
Then
x+γ
Z
1
J ◦ u(x) :=
γ x
Introducing the anti-derivative, U :=
u(y) dy
Z x
−∞
u(ξ, t) dξ, we rewrite the
global flux equation as:
ut + F (u, U )x = 0


U (x + γ) − U (x) 

F (u, U ) = Vmu(1 − u)exp −
γ


4
Finite-Volume Schemes
1-D System:
1
n
ūj ≈
∆x
Z
ut + f (u)x = 0
u(x, tn) dx : cell averages over Cj := (xj− 1 , xj+ 1 )
2
Cj
2
This solution is approximated by a piecewise linear (conservative,
second-order accurate, non-oscillatory) reconstruction:
n
e n(x) = ūn
u
j + sj (x − xj )
for x ∈ Cj
5
xj−1
xj
xj+1
xj+2
For example,
sn
j = minmod θ
n
ūn
j − ūj−1
∆x
,
n
ūn
j+1 − ūj−1
2∆x
,θ
n
ūn
j+1 − ūj
∆x
!
θ ∈ [1, 2],
where the minmod function is defined as:


minj {zj },
minmod(z1, z2, ...) := maxj {zj },

0,
if zj > 0 ∀j,
if zj < 0 ∀j,
otherwise.
6
Godunov-type upwind schemes are designed by integrating
ut + f (u)x = 0
over the space-time control volumes [xj− 1 , xj+ 1 ] × [tn, tn+1]
2
2
tn+1
tn
xj−1
xj−1/2
xj
xj+1/2
xj+1
7
tn+1
tn
xj−1
1
n+1
n
ūj
= ūj −
∆x
xj−1/2
xj
xj+1/2
xj+1
tn+1
Z f (u(xj+ 1 , t)) − f (u(xj− 1 , t)) dt
tn
2
2
In order to evaluate the flux integrals on the RHS, one has to
(approximately) solve the generalized Riemann problem.
This may be hard or even impossible...
8
Nessyahu-Tadmor Scheme
The Nessyahu-Tadmor [H. Nessyahu, E. Tadmor; 1990] scheme
is a central Godunov-type scheme. It is designed by integrating
ut + f (u)x = 0
over the different set of staggered space-time control volumes
[xj , xj+1] × [tn, tn+1] containing the Riemann fans
t n+1
tn
xj
xj+1/2
xj+1
9
t n+1
tn
xj
xj+1
Z
xj+1/2
xj+1
tn+1
Z 1
n(x) dx − 1
e
ūn+1
=
u
1
j+ 2
∆x x
∆x n
t
j
f (u(xj+1, t)) − f (u(xj , t)) dt
Due to the finite speed of propagation, this can be reduced to:
ūn+1
1 =
j+ 2
n
ūn
j + ūj+1
2
1
∆t
∆x n
n+ 1
n+ 2
n
2
+
(sj − sj+1) −
f (uj+1 ) − f (uj
)
8
∆x
10
1
n+ 2
Values of u at t = t
expansion:
are approximated using the Taylor
∆t
n+ 1
e n(xj ) +
uj 2 ≈ u
ut(xj , tn)
2
n
n
e n(x) = ūn
en
• u
j + sj (x − xj ) =⇒ u (xj ) = ūj
•
ut(xj , tn) = −fx(ūn
j)
The space derivatives fx are computed using the (minmod)
limiter:
n) − f (ūn ) f (ūn ) − f (ūn )
f
(ū
j−1
j
j−1
j+1
fx(ūn
)
=
minmod
θ
,
,
j
∆x
2∆x
!
n
f (ūn
j+1 ) − f (ūj )
θ
∆x
11
Nessyahu-Tadmor Scheme for a Traffic Flow
Model with Arrhenius Look-Ahead Dynamics
1
n
ūj ≈
∆x
Z
u(x, tn) dx : cell averages over Cj := (xj− 1 , xj+ 1 )
2
Cj
2
This solution is approximated by a piecewise linear (conservative,
second-order accurate, non-oscillatory) reconstruction:
n
e n(x) = ūn
u
j + sj (x − xj )
for x ∈ Cj
For example,
n
n
n
n
n − ūn
ū
−
ū
ū
−
ū
ū
j−1
j
j
j−1
j+1
j+1
sn
=
minmod
θ
,
,
θ
j
∆x
2∆x
∆x
!
θ ∈ [1, 2]
12
The Nessyahu-Tadmor (NT) scheme is designed by integrating
ut + F (u, U )x = 0
over the space-time control volumes [xj , xj+1] × [tn, tn+1]:
1
n+1
ū 1 =
j+ 2
∆x
xj+1
Z
xj
e n(x) dx −
u
1
∆x
tn+1
Z F (u(xj+1, t), U (xj+1, t))
tn
−F (u(xj , t), U (xj , t)) dt
Due to the finite speed of propagation, this can be reduced to:
1
1
1
1 ū
+
ū
∆x
∆t
n+
n+
n+
n+
j
j+1
n − sn ) −
ūn+1
=
+
(s
F (uj+12 , Uj+12 ) − F (uj 2 , Uj 2 )
1
j
j+1
j+ 2
2
8
∆x
13
n+ 1
2
u and U at t = t
are approximated using the Taylor expansion:
∆t
n+ 1
e n(xj ) +
uj 2 ≈ u
ut(xj , tn)
2
1
n+ 2
e n(x ) + ∆t U (x , tn)
≈U
Uj
t j
j
2
n(x ) = ūn
n(x − x ) =⇒ un := u
e
e n(x) = ūn
• u
+
s
j
j
j
j
j
j
e n from xmin to x we obtain the piecewise
• By integrating u
quadratic approximation of the antiderivative U :
e n(x) =
U
Zx
e n(ξ) dξ = ∆x
u
j−1
X
i=0
Zx
ūn
i +
i=0
xmin
= ∆x
j−1
X
x
j− 1
2
n
ūn
i + ūj (x − xj− 1 ) +
2
e n(ξ) dξ
u
sn
j
2
(x − xj− 1 )(x − xj+ 1 ), x ∈ Cj
2
2
In particular,
e n(x ) = ∆x
Ujn := U
j
j−1
X
i=0
∆x n (∆x)2 n
n
ūi +
ūj −
sj
2
8
14
∆t
n+ 1
n
2
e
uj
≈ u (xj ) +
ut(xj , tn)
2
1
∆t
n+ 2
n
e
Uj
≈ U (xj ) +
Ut(xj , tn)
2
n)
,
U
ut(xj , tn) = −Fx(un
j
j
(1)
The space derivative Fx is computed using the (minmod) limiter:
n
Fx(un
j , Uj ) = minmod θ
n
n
n
F (ūn
j , Uj ) − F (ūj−1 , Uj−1 )
,
∆x
n , U n ) − F (ūn, U n) !
n ) − F (ūn , U n )
F
(ū
F (ūn
,
U
j−1 j−1
j
j
j+1 j+1
j+1 j+1
,θ
2∆x
∆x
Finally, we integrate equation (1) with respect to x to obtain:
n
Ut(xj , tn) = −F (un
j , Uj )
15
Numerical Examples
Example 1 — Red Light Traffic
(
u(x, 0) =
1,
0,
4<x<6
otherwise
This corresponds to a situation in which the traffic light is located
at x = 6 and it is turned from red to green at t = 0.
First, fix γ = 1
16
∆ x=1/5
∆ x=1/10
1−ORDER
2−ORDER
REFERENCE
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
4
6
8
10
4
∆ x=1/20
∆ x=1/40
∆ x=1/400
0.8
0.6
0.4
0.4
0.2
0.2
0
6
8
10
6
8
10
0.8
0.6
4
1−ORDER
2−ORDER
REFERENCE
0
3.9
∆ x=1/20
∆ x=1/40
∆ x=1/400
4
4.1
4.2
17
Impact of the look-ahead dynamics: global vs. non-global fluxes
GLOBAL FLUX
NON−GLOBAL FLUX
0.8
0.6
0.4
0.2
0
4
6
8
10
18
We finally demonstrate the dependence of the computed solution
on the look-ahead distance γ:
• as γ decreases, the drivers can see less and the original traffic
jam dissipates more slowly
• when γ is large, the effect of the global flux is small
1
γ=0.1
γ=0.2
γ=0.5
γ=1
0.8
0.6
1
0.8
0.6
0.4
0.4
0.2
0.2
0
0
4
6
8
10
γ=10
γ=5
γ=2
γ=1
4
6
8
10
19
Formal limits of the global flux as γ → ∞ and γ → 0
(x)
The global factor exp − U (x+γ)−U
γ
→ 1 as γ → ∞ assuming
the total number of cars on the freeway is bounded. Therefore,
F (u, U ) −→ f (u) = Vmu(1 − u) as γ → ∞
that is, the global model reduces to the original, non-global one.
U (x + γ) − U (x)
γ
!
→ Ux(x) = u(x) as γ → 0. Therefore,
F (u, U ) −→ f (u)e−u = Vmu(1 − u)e−u as γ → 0
e−u: “slow down” factor in the limiting low visibility case.
20
Example 2 — Traffic Jam on a Busy Freeway
(
u(x, 0) =
1,
0.75,
16 < x < 18
otherwise
Dispersive effect is much more prominent here since the waves
are moving in the direction opposite to the vehicles’ velocity.
21
Even in the case of a large γ = 10, the global and non-global
flux solutions are significantly different:
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
0
10
−10
t=10
0
10
−10
t=6
0
10
−10
t=3
0
10
t=1
1
0.9
0.8
−10
0
10
−10
0
10
−10
0
10
−10
0
22
10
For a smaller γ = 5, the dispersive waves are getting together
by time t = 10:
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
0
10
−10
0
10
−10
0
10
−10
0
10
For γ = 3, the larger waves catch the smaller ones much faster
now and a dispersive “wave package” is formed at about t = 3:
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
0
10
−10
0
10
−10
0
10
−10
0
10
23
When γ = 1, a dispersive “wave package” develops right away
and is later transformed into one wave:
t=10
t=6
t=3
t=1
1
0.9
0.8
−5
0
5
10
15
−5
0
5
10
15
−5
0
5
10
15
−5
0
5
10
15
When γ = 0.5, the dispersive effect can be observed at small
times only:
t=10
t=6
t=3
t=1
1
0.9
0.8
0
5
10
15
0
5
10
15
0
5
10
15
0
5
10
24
15
For even smaller γ = 0.1, no dispersive effect can be observed:
t=10
t=6
t=3
t=1
1
0.9
0.8
0
5
10
15
0
5
10
15
0
5
10
15
0
5
10
15
A certain smoothing effect can be observed though the solution
seems to contain a sub-shock at the left edge of the wave:
∆ x=1/200
∆ x=1/400
1
1
0.95
0.95
0.9
0.9
0.85
0.85
0.8
0.8
0.75
0
0.75
0
2
4
6
2
4
6
25
• Large and intermediate values of γ =⇒ dispersive effect
• Small values of γ =⇒ smoothing effect starts dominating
The nature of these phenomena can be heuristically explained
by expanding U (x + γ) in
ut + F (u, U )x = 0, F (u, U ) = Vmu(1 − u)exp −
U (x + γ) − U (x)
γ
!
into the Taylor series about x:
γ2
!
γ
= 0, TS := − u + ux +
ut + Vm
uxx + . . .
x
2
6
m
!
2
γ
γ
2
TS
2
TS
ut + Vm(1 − 3u + u )e
ux = Vm(u − u )e
uxx +
uxxx + . . .
2
6
h
(u − u2)eTS
i
Our conjecture: for small γ initially smooth solutions may remain
smooth for all t. However, the viscosity coefficient in this case
is small so that smooth solutions may develop sharp gradients.
26
Example 3 — Numerical Breakdown Study
To check our conjecture, we consider the smooth initial data:
2
−(x−17)
u(x, 0) = 0.75 + 0.25e
∆ x=1/800
γ=1
1
γ=0.1
γ=1
0.95
0.9
∆ x=1/1600
∆ x=1/3200
0.86
0.84
0.85
0.82
0.8
0.8
0.75
−2
0
2
4
6
8
1
2
3
4
5
27
For small γ = 0.1, the solution does not seem to break down at
any stage of its evolution:
t=20
t=14
t=8
t=2
1
0.9
0.8
−10
0
10
−10
0
10
−10
0
10
−10
0
10
For large γ = 10, the solution of the global model clearly breaks
down in finite time like the solution of the non-global model:
0.88
0.86
γ=10
NON−GLOBAL FLUX
0.84
0.82
0.8
0.78
0.76
−15
−10
−5
0
5
10
28
Linear Interaction Potential
A more realistic interaction potential is a linear one:
(
!
r
1
J(r) = ϕ
,
γ
γ
ϕ(r) =
2 − 2r, 0 < r < 1
0,
otherwise
that is,
J(r) =


2


γ
!
1−


 0,
r
, 0<r<γ
γ
otherwise
In this case,
J ◦ u(x, t) =
Z∞
J(y − x)u(y, t) dy =
x
x+γ
Z
2
γ x
!
1−
y−x
u(y, t) dy.
γ
or after the integration by parts,

J ◦ u(x, t) =
x+γ
Z
2 1

γ γ x

U (y, t) dy − U (x, t)

29
It is easy to show that
J ◦ u(x, t) −→ u(x, t) as γ → 0
This suggests that for small γ, the modified model is not
expected to be much different from the original one.
• Indeed, when the visibility is very bad, all vehicles located within
the interval (x, x + γ] are expected to have almost the same
influence on the driver of the car located at x.
• However, when γ is not too small, the use of the linear
interaction potential is motivated by the fact that the vehicles
located further away from x (but still visible to the driver there)
typically have smaller influence on the driver’s behavior then the
vehicles that are nearby.
30
Formula

x+γ
Z
2 1
J ◦ u(x, t) = 
γ γ x

U (y, t) dy − U (x, t)

can be rewritten as
"
J ◦ u(x, t) =
2 U(x + γ, t) − U(x, t)
− U (x, t)
γ
γ
#
where U denotes the antiderivative of U :
U(x, t) :=
Zx
U (ξ, t) dξ
−∞
31
The modified model is:
ut + F (u, U, U)x = 0
"
2 U(x + γ, t) − U(x, t)
− U (x, t)
F (u, U, U) = Vmu(1 − u) exp −
γ
γ
Compare with
ut + F (u, U )x = 0
F (u, U ) = Vmu(1 − u) exp −
U (x + γ) − U (x)
γ
!
32
#!
Nessyahu-Tadmor Scheme for a Traffic Flow
Model with Linear Interaction Potential
ut + F (u, U, U)x = 0
ūn+1
1 =
j+ 2
ūj + ūj+1
∆x n
+
(sj − sn
j+1 )
8
 2

−λ 
F
(u,
U,
U)

n+ 1
2
(xj+1, t
)
− F (u, U, U)

n+ 1
2
(xj , t
)



Therefore, to complete the description of the NT scheme we
need to provide a recipe for the calculation of the intermediate
1
point values {U (xj , tn+ 2 )}.
33
From the Taylor expansion
1
n+ 2
U(xj , t
∆t
n
) ≈ U(xj , t ) +
Ut(xj , tn)
2
The approximate values of {U (xj , tn)} can be obtained by
integrating the piecewise quadratic function
e n = ∆x
U
j−1
X
i=0
n(x − x
ūn
+
ū
)+
i
j
j− 1
2
sn
j
2
(x − xj− 1 )(x − xj+ 1 ), x ∈ Cj
2
2
which results in a piecewise cubic approximation of U(x, tn):
Ue n(x) =
+
ūn
j
2
Zx
e n(ξ) dξ = ∆x
U
xmin
(x − xj− 1 )(x − xj+ 1 ) +
2
2
j−1
X
i=0
sn
j
6
win + wjn(x − xj− 1 )
2
(x − xj− 1 )(x − xj )(x − xj+ 1 ), x ∈ Cj
2
2
34
Finally, integrating equation
n
Ut(xj , tn) = −F (un
j , Uj )
we obtain
Ut(xj , tn) = −
Zx
F (u(ξ, tn), U (ξ, tn), U(ξ, tn)) dξ
xmin
which can be calculated by exact integration of the piecewise
linear approximation of F .
35
Back to Example 2 — Traffic Jam on a Busy Freeway
(
u(x, 0) =
1,
0.75,
16 < x < 18
otherwise
36
γ = 10
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
1
0
10
−10
1
t=10
0
10
−10
1
t=6
0
10
−10
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
-20 -15 -10 -5
0
5
10
15
0.75
20
-20 -15 -10 -5
0
5
10
15
0.75
20
-20 -15 -10 -5
0
5
10
15
0
10
t=1
0.75
20
-20 -15 -10 -5
0
Linear interaction potential
37
5
10
15
20
γ=5
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
1
0
10
−10
1
t=10
0
10
−10
1
t=6
0
10
−10
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
-20 -15 -10 -5
0
5
10
15
0.75
20
-20 -15 -10 -5
0
5
Linear interaction potential
10
15
0.75
20
-20 -15 -10 -5
0
5
10
15
0
10
t=1
0.75
20
-20 -15 -10 -5
0
38
5
10
15
20
γ=3
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
−10
1
0
10
−10
1
t=10
0
10
−10
1
t=6
0
10
−10
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
20
-10
0.75
20
-10
0.75
20
-10
0.75
-10
-5
0
5
10
15
-5
0
5
10
Linear interaction potential
15
-5
0
5
10
15
0
10
t=1
-5
0
5
39
10
15
20
γ=1
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
−5
1
0
5
10
15
−5
1
t=10
0
5
10
15
−5
1
t=6
0
5
10
15
−5
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
0.75
20
-5
0.75
20
-5
0.75
-5
0
5
10
15
0
5
10
Linear interaction potential
15
0
5
10
15
20
0
5
10
15
0
5
10
15
t=1
-5
40
20
γ = 0.5
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
0
5
10
1
15
0
5
10
1
t=10
15
0
1
t=6
5
10
15
0
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
0.75
20
0.75
20
0.75
0
5
10
15
0
5
10
Linear interaction potential
15
0
5
10
15
20
5
10
15
5
10
15
t=1
0
41
20
γ = 0.1
Constant interaction potential
t=10
t=6
t=3
t=1
1
0.9
0.8
0
5
10
1
15
0
5
10
1
t=10
15
0
1
t=6
5
10
15
0
1
t=3
0.95
0.95
0.95
0.95
0.9
0.9
0.9
0.9
0.85
0.85
0.85
0.85
0.8
0.8
0.8
0.8
0.75
0.75
20
0.75
20
0.75
0
5
10
15
0
5
10
Linear interaction potential
15
0
5
10
15
20
5
10
15
5
10
15
t=1
0
42
20
Models of Formation and Evolution
of Traffic Jams
[H.J. Payne; 1971,1979] → first “second-order” fluid dynamics
models of traffic flow were proposed
[C. Daganzo; 1995] → These models admit absurd results:
• negative car density
• backward car movement
since cars in traffic have properties usual fluids do not have
[A. Aw, M. Rascle; 2000, 2002] → A new “second-order” model
was derived from the microscopic Follow-the-Leader model
43
Aw-Rascle Model
(
ρt + (ρu)x = 0
(ρw)t + (ρwu)x = 0,
w = u + p(ρ)
ρ(x, t): a relative car density (0 ≤ ρ ≤ 1)
u > 0: dimensionless velocity
w: drivers “preferred velocity”
p(ρ): velocity offset (0 ≤ p(ρ) ≤ ∞)
Example ([A. Aw, M. Rascle; 2000, 2002])
p(ρ) = ργ
Not the best choice since the region [0, ρ∗]×[0, u∗] is not invariant
44
(
ρt + (ρu)x = 0
(ρw)t + (ρwu)x = 0,
w = u + p(ρ)
m
(
∂tρ + ∂x(ρu) = 0
(∂t + u∂x)(u + p(ρ)) = 0
m

 ρt + (ρu)x = 0
0 (ρ)u
 u + u2
=
ρp
x
t
2 x
Compare with the pressureless gas dynamics (PGD) system:


ρ
+
(ρu)
=
0
 t
 ρt + (ρu)x = 0
x
⇐⇒
 u + u2
 (ρu) + (ρu2)x = 0
=
0
t
t
2 x
45
Example [F. Berthelin, P. Degond, M. Delitala, M. Rascle; 2008]
p(ρ) =
1
1
− ∗
ρ ρ
!−γ
for ρ ≤ ρ∗,
γ>0
ρ∗: maximal density
Notice that
p(ρ) → ∞ as ρ → ρ∗
For the modified Aw-Rascle model the region [0, ρ∗] × [0, u∗] is
invariant
46
Modeling Traffic Jams
[F. Berthelin, P. Degond, M. Delitala, M. Rascle; 2008]


 ∂tρε + ∂x(ρεuε) = 0

 (∂t + uε∂x)(uε + εp(ρε)) = 0,
p(ρε) =
1 − 1 −γ
ρε
ρ∗
p
0
ρ∗
ρ
! No velocity offset unless ρ ∼ ρ∗
! This is a very stiff problem for small ε
47
ε→0
For a nonsingular p(ρ) (e.g., p(ρ) = ργ ), εp(ρε) → 0 and

 ρt + (ρu)x = 0
∂tρε + ∂x(ρεuε) = 0
2
−→
ε
ε
ε

(∂t + u ∂x)(u + εp(ρ )) = 0
=0
ut + u2
x
!−γ
1
1
b
For p(ρ) =
− ∗
, εp(ρε) → p(x,
t) ≥ 0
ρ ρ
(
(P GD)
b
! p(x,
t) may become positive and finite
The formal limit of the Rescaled Modified Aw-Rascle Model is
the Constrained PGD system:


∂tρ + ∂x(ρu) = 0

(

ε
ε
ε

∂tρ + ∂x(ρ u ) = 0
b =0
−→ (∂t + u∂x)(u + p)
ε
ε
ε


(∂t + u ∂x)(u + εp(ρ )) = 0

 0 ≤ ρ ≤ ρ∗, pb ≥ 0, (ρ∗ − ρ)pb = 0
48
Constrained PGD System


∂ ρ + ∂x(ρu) = 0


 t
b =0
(∂t + u∂x)(u + p)



 0 ≤ ρ ≤ ρ∗, pb ≥ 0, (ρ∗ − ρ)pb = 0
!
This system is still ill posed since the velocity offset pb is
undefined inside the clusters, that is, in the regions ρ = ρ∗
! Rigorous passage to the ε → 0 limit shows that
• ux = 0 inside the clusters
• When two cluster collide, they form a new cluster moving
with the velocity of the cluster on the right.
49
Numerical Methods for the
Constrained PGD System
Method 1: Finite-Volume Upwind Scheme
The idea: at each time step solve the PGD system and correct
the velocities at the edges of clusters
Assume that at some time level t the computed solution
{ρj (t), uj (t)} is available. Then it is evolved to t + ∆t:
(
ρj (t)
)
→
uj (t)
d
dt
ρj (t)
uj (t)
(
!
=−
ρen(·, t)
e n(·, t)
u
)

ρ

 H 1 (t)
j+ 2
→
u

 Hj+ 1 (t)
2
Hj+ 1 (t) − Hj− 1 (t)
2
2
∆x





(
→
ρj (t + ∆t)
)
uj (t + ∆t)

ρ
H
(t)
 j+ 12

Hj+ 1 (t) :=  u

H
(t)
2
j+ 1
2

,
50
To incorporate the constraint, a special treatment of clusters is
proposed.
First, clusters are marked using Ij
ρ
ρ∗
x
Ij
0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 0
51
Piecewise Linear Reconstruction
e
ρ(t)
= ρj (t) + (ρx)j (x − xj )
e
u(t)
= uj (t) + (ux)j (x − xj )
for x ∈ Cj
Since the eigenvalues of the PGD are equal to u, they are
nonnegative and thus we only need to compute the left-sided
point values at each cell interface:
ρ−
:= ρj (t) + ∆x
2 (ρx )j
u−
:= uj (t) + ∆x
2 (ux )j
1 (t)
j+ 2
1 (t)
j+ 2
Velocity Correction behind the Cluster:
If Ij (t) = 0 and Ij+1(t) = 1 and
ρ−
−
(t)u
1
j+ 2
−
(t)
>
ρ
1
j+ 2
−
(t)u
3
j+ 2
3
j+ 2
(t) then correct the velocity
ρ−
u−
1
j+ 2
(t) :=
−
3 (t)uj+ 3 (t)
j+ 2
2
ρ−
1
j+ 2
(t)
52
CFL Condition
d
ρj (t) = −
dt
ρ
ρ
(t) − H 1 (t)
1
j+ 2
j− 2
H
∆x
,
H
ρ
−
−
(t)
=
ρ
(t)u
1
1 (t)
j+ 1
j+
j+
2
2
2
For the Forward Euler method,
−
−
n − λ ρ− u −
ρn+1
=
ρ
−
ρ
u
1 j+ 1
1 j− 1
j
j
j+ 2
j− 2
2
2
!
≤ ρ∗
provided either
ρ−
−
1 uj+ 1
j+ 2
2
− ρ− 1 u −
1
j− 2 j− 2
≥0
or
λ≤ −
ρ 1 u−
j− 2
ρ∗ − ρn
j
1
j− 2
− ρ−
−
1 uj+ 1
j+ 2
2
Question: Will the time steps be too small?
53
Cluster Velocity Correction
After the evolution step, the new solution {ρj (t+∆t), uj (t+∆t)}
and cluster markers {Ij (t + ∆t)} are available
Assume that Ij−q = 0, Ij−q+1 = . . . = Ij = 1, Ij+1 = 0. Then
If uj (t + ∆t) > uj+1(t + ∆t) and ρj+1(t + ∆t) > 0 set
uj (t + ∆t) := uj+1(t + ∆t)
If uj (t + ∆t) > uj+1(t + ∆t) and ρj+1(t + ∆t) = 0 set
uj (t + ∆t) := w
w: preferred velocity
54
Example 1 — Riemann Initial Data
1
t=0
1
t=0.2
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Density
0.6
t=0
0.6
t=0.2
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Velocity
55
Example 2 — Random Initial Velocity
0.9
1
0.85
0.8
0.8
0.75
0.6
0.7
0.4
0.65
0.6
0.2
0.55
0
0.5
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
Initial density (left) and velocity (right)
56
8
9
10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
4
5
6
7
8
9
10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Density at times t = 25, 50, 75, and 100
57
Method 2:
Finite-Volume Cluster Propagation
The approximate solution at t = tn is represented as a piecewise
constant function, interpreted as a set of clusters of density ρn
j
n
n
moving with the velocities uj (CFL condition: λ max uj ≤ 1)
ρ
j
ρ∗
x
ρ
ρ∗
x
58
The evolved clusters are then projected onto the original grid
ρ
ρ∗
x
ρ
ρ∗
x
59
We have obtained the scheme for the PGD.
Example — PGD, Riemann Initial Data (same as in Example 1)
12
12
t=0
12
t=0.2
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
t=0.4
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Density
0.5
t=0
0.5
t=0.2
0.5
0.45
0.45
0.45
0.4
0.4
0.4
0.35
0.35
0.35
0.3
0.3
0.3
0.25
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Velocity
t=0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
60
Backward Mass Transfer
ρ
ρ∗
x
ρ
ρ∗
x
61
Example 1 — Riemann Initial Data
1
t=0
1
t=0.2
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Density
0.6
t=0
0.5
0.4
0.3
0.2
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
t=0.2
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Velocity
62
Example 2 — Random Initial Velocity
0.9
1
0.85
0.8
0.8
0.75
0.6
0.7
0.4
0.65
0.6
0.2
0.55
0
0.5
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
Initial density (left) and velocity (right)
63
8
9
10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
4
5
6
7
8
9
10
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Density at times t = 70, 140, 210, and 280
64
Future Plans
• Second-order extension
Propagation method
of
the
Finite-Volume
Cluster
• Extension to the case ρ∗ = ρ∗(u)
• 2-D extension to the models of crowd dynamics
65