lighthill-whitham-richards model · lwr model relating speed, volume, and density the slope of the...
TRANSCRIPT
![Page 1: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/1.jpg)
Lighthill-Whitham-Richards Model
CE 391F
January 24, 2013
LWR Model
![Page 2: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/2.jpg)
OUTLINE
![Page 3: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/3.jpg)
1 Review
2 Fundamental relationship and fundamental diagram
3 PDE formulation of LWR model
LWR Model Outline
![Page 4: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/4.jpg)
REVIEW
![Page 5: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/5.jpg)
Fundamental quantities...
q
k
u
u
Headways
N
LWR Model Review
![Page 6: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/6.jpg)
These quantities can be visualized on a trajectory diagram.
LWR Model Review
![Page 7: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/7.jpg)
The flow q is the rate trajectories cross a horizontal line.
LWR Model Review
![Page 8: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/8.jpg)
The density k is the rate trajectories cross a vertical line.
How is speed represented in a trajectory diagram?
LWR Model Review
![Page 9: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/9.jpg)
RELATING SPEED,VOLUME, AND DENSITY
![Page 10: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/10.jpg)
How are these three quantities related to each other?
I sit at the side of the road for one hour, while cars drive by at 70 mi/hr.If the density is 10 veh/mi, how many vehicles pass by?
In one hour, every car within 70 mi will drive by. Since the density is 10veh/mi, I will see 700 vehicles.
LWR Model Relating speed, volume, and density
![Page 11: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/11.jpg)
This is a demonstration of the fundamental relationship between speed,flow, and density:
q = uk
Think units: [veh/hr] = [mi/hr][veh/mi]
LWR Model Relating speed, volume, and density
![Page 12: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/12.jpg)
How are flow and density related?
If q = 0, either k = 0 or u = 0.
If u = 0, then the density is at its maximum value kj , the jam density.
So, q = 0 if k = 0 or k = kj .
The major assumption in the LWR model is that q is a function of k alone.(k determines u, so q(k) = u(k) · k)
LWR Model Relating speed, volume, and density
![Page 13: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/13.jpg)
Therefore, Q(k) must be a concave function with zeros atk = 0 and k = kj
This is the fundamental diagram, denoted Q(k). The fundamental diagramcan be calibrated to data, resulting in different traffic flow models.
LWR Model Relating speed, volume, and density
![Page 14: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/14.jpg)
Therefore, Q(k) must be a concave function with zeros atk = 0 and k = kj
Also notice that there is some point at which flow is maximal. This maxi-mum flow is the capacity qmax , which occurs at the critical density kc .
LWR Model Relating speed, volume, and density
![Page 15: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/15.jpg)
We can get speed information from the fundamentaldiagram: q = uk , so u = q/k .
LWR Model Relating speed, volume, and density
![Page 16: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/16.jpg)
This is the same slope as on a trajectory diagram.
LWR Model Relating speed, volume, and density
![Page 17: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/17.jpg)
What happens if something interrupts this flow?
LWR Model Relating speed, volume, and density
![Page 18: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/18.jpg)
Shockwaves
LWR Model Relating speed, volume, and density
![Page 19: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/19.jpg)
Let’s say one car stops for a while. What happens to thenext vehicles?
LWR Model Relating speed, volume, and density
![Page 20: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/20.jpg)
LWR Model Relating speed, volume, and density
![Page 21: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/21.jpg)
LWR Model Relating speed, volume, and density
![Page 22: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/22.jpg)
LWR Model Relating speed, volume, and density
![Page 23: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/23.jpg)
We can identify regions of constant density. Theboundaries between these regions are shockwaves.
LWR Model Relating speed, volume, and density
![Page 24: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/24.jpg)
Let’s label these regions I, II, III, and IV.
LWR Model Relating speed, volume, and density
![Page 25: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/25.jpg)
Let’s look at a vertical slice of the trajectory diagram.
LWR Model Relating speed, volume, and density
![Page 26: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/26.jpg)
How quickly are these shockwaves moving? (In other words, when will thelink fill up?)
LWR Model Relating speed, volume, and density
![Page 27: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/27.jpg)
Consider an arbitrary shockwave.
LWR Model Relating speed, volume, and density
![Page 28: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/28.jpg)
What rate are vehicles entering the shockwave?
LWR Model Relating speed, volume, and density
![Page 29: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/29.jpg)
What rate are vehicles leaving the shockwave?
LWR Model Relating speed, volume, and density
![Page 30: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/30.jpg)
These flow rates should be identical, since vehicles are not appearing ordisappearing at the shockwave. Thus
(uA − uAB)kA = (uB − uAB)kB
or
uAB =qA − qBkA − kB
LWR Model Relating speed, volume, and density
![Page 31: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/31.jpg)
The slope of the shockwaves on a trajectory diagram is theslope of the line connecting the points on the fundamentaldiagram.
LWR Model Relating speed, volume, and density
![Page 32: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/32.jpg)
LWR Model Relating speed, volume, and density
![Page 33: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/33.jpg)
PDE FORMULATION
![Page 34: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/34.jpg)
In general, the values of q, k , and u can vary with x and t, subject to thefundamental relationship and fundamental diagram. Denote these q(x , t),k(x , t), and u(x , t).
We can also define cumulative counts N as a function of x and t. Whileactual vehicle trajectories are discrete, we can “smooth” them so N(x , t)is continuous.
LWR Model PDE Formulation
![Page 35: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/35.jpg)
Flow and density are related to the cumulative counts:
q(x , t) =∂N(x , t)
∂t
k(x , t) = −∂N(x , t)
∂x
LWR Model PDE Formulation
![Page 36: Lighthill-Whitham-Richards Model · LWR Model Relating speed, volume, and density The slope of the shockwaves on a trajectory diagram is the slope of the line connecting the points](https://reader035.vdocuments.site/reader035/viewer/2022071417/61143e9732a37d646c17a96c/html5/thumbnails/36.jpg)
The “solution” to the LWR problem is to find the values N(x , t), q(x , t),and k(x , t) such that
q(x , t) =∂N(x , t)
∂t
k(x , t) = −∂N(x , t)
∂x
q(x , t) = Q(k(x , t))
given boundary conditions (typically values of N for certain x and t)
Over the next few weeks, we will see several ways to solve this formulation.
LWR Model PDE Formulation