出处：MIT, Numerical Methods for Partial Differential Equations
Problem Statement
Consider the traffic flow problem, described by the non-linear hyperbolic equation:
\[\dfrac{\partial \rho}{\partial t}+\dfrac{\partial\rho u}{\partial x}=0\]with $\rho=\rho(x,t)$ the density of cars (vehicles/km), and $u=u(x,t)$ the velocity. Assume that the velocity $u$ is given as a function of $\rho$:
\[u=u_{\max}\left(1-\dfrac{\rho}{\rho_{\max}}\right),\]with $u_{\max}$ the maximum speed and $0\le \rho\le\rho_{\max}$. The flux of cars is therefore given by
\[f(\rho)=\rho u_{\max}\left(1-\dfrac{\rho}{\rho_{\max}}\right).\]We will solve this problem using a first order finite volume scheme:
\[\rho_i^{n+1}=\rho_i^n-\dfrac{\Delta t}{\Delta x}\left(F_{i+\frac{1}{2}}^n-F_{i-\frac{1}{2}}^n\right).\]For the numerical flux function, we will consider two different schemes:
(a)Roe’s Scheme
The expression of the numerical flux is given by:
\[F_{i+\frac{1}{2}}^R=\dfrac{1}{2}[f(\rho_i)+f(\rho_{i+1})]-\dfrac{1}{2}|a_{i+\frac{1}{2}}|(\rho_{i+1}-\rho_i)\]with
\[a_{i+\frac{1}{2}}=u_{\max}\left(1-\dfrac{\rho_i+\rho_{i+1}}{\rho_{\max}}\right).\]Note that $a_{i+\frac{1}{2}}$ satisfies
\[f(\rho_{i+1})-f(\rho_i)=a_{i+\frac{1}{2}}(\rho_{i+1}-\rho_i).\](b)Godunov’s Scheme
In this case the numerical flux is given by:
\[F_{i+\frac{1}{2}}^G=f\left(\rho(x_{i+\frac{1}{2}}, t^{n+})\right) =\left\{\begin{aligned} &\min\limits_{\rho\in[\rho_i,\rho_{i+1}]}f(\rho), &&\rho_i < \rho_{i+1}, \\ &\max\limits_{\rho\in[\rho_i,\rho_{i+1}]}f(\rho), &&\rho_i > \rho_{i+1}. \end{aligned}\right.\]Questions
Question 1
For both Roe’s Scheme and Godunov’s Scheme, look at the problem of a traffic light turning green at time $t = 0$. We are interested in the solution at $t = 2$ using both schemes. What do you observe for each of the schemes? Explain briefly why the behavior you get arises.
Use the following problem parameters:
\[\begin{aligned} \rho_{\max}&=1.0, & \rho_L&=0.8 \\ u_{\max}&=1.0, \\ \Delta x&=\dfrac{4}{400}, & \Delta t&=\dfrac{0.8\Delta x}{u_{\max}}. \end{aligned}\]The initial condition at the instant when the traffic light turns green is
\[\rho(0)=\left\{\begin{aligned} &\rho_L, &&x < 0, \\ &0, && x\ge 0. \end{aligned}\right.\]Question 2
For the rest of this problem use only the scheme(s) which are valid models of the problem.
Simulate the effect of a traffic light at $x = -\dfrac{\Delta x}{2}$ which has a period of $T = T_1+T_2= 2$ units. Assume that the traffic light is $T_1= 1$ units on red and $T_2= 1$ units on green. Assume a sufficiently high flow density of cars (e.g. set $\rho = \dfrac{\rho_{\max}}{2}$ on the left boundary - giving a maximum flux), and determine the average flow, or capacity of cars over a time period $T$. The average flow can be approximated as
\[\dot{q}=\dfrac{1}{N_T}\sum_{n=1}^{N_T}f^n=\dfrac{1}{N^T}\sum_{n=1}^{N_T}\rho^nu^n,\]where $N_T$ is the number of time steps for each period $T$. You should run your computation until $\dot{q}$ over a time period does not change. Note that by continuity $\dot{q}$ can be evaluated over any point in the interior of the domain (in order to avoid boundary condition effects, we consider only those points on the interior domain).
Note: A red traffic light can be modeled by simply setting $F_{i+\frac{1}{2}}=0$ at the position where the traffic light is located.
Question 3
Assume now that we simulate two traffic lights, one located at $x = 0$, and the other at $x = 0.15$, both with a period $T$. Calculate the road capacity ($=$ average flow) for different delay factors. That is if the first light turns green at time $t$, then the second light will turn green at $t +\tau$. Solve for $\tau = k\dfrac{T}{10}, k = 0,\cdots,9$. Plot your results of capacity vs $\tau$ and determine the optimal delay $\tau$.
Solutions
Question 1
For both Roe’s Scheme and Godunov’s Scheme, look at the problem of a traffic light turning green at time $t = 0$. We are interested in the solution at $t = 2$ using both schemes. What do you observe for each of the schemes? Explain briefly why the behavior you get arises.
Use the following problem parameters:
\[\begin{aligned} \rho_{\max}&=1.0, & \rho_L&=0.8 \\ u_{\max}&=1.0, \\ \Delta x&=\dfrac{4}{400}, & \Delta t&=\dfrac{0.8\Delta x}{u_{\max}}. \end{aligned}\]The initial condition at the instant when the traffic light turns green is
\[\rho(0)=\left\{\begin{aligned} &\rho_L, &&x < 0, \\ &0, && x\ge 0. \end{aligned}\right.\]