In order to control traffic congestion
,
many mathematical models have been
used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation
,
which provides
a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density
At present, we cannot think of our life even in a single day without vehicles like buses, trucks, cars, auto-rickshaws, CNG, etc. to go from one place to another place. With the increasing population, the demand for vehicles is also increasing day by day. This increase in vehicles creates a traffic congestion problem. Traffic congestion is one of the greatest problems in Bangladesh like some other countries of the world. In all the major cities of our country, especially Dhaka, traffic congestion is a must. Due to the unavailability of adequate space for the extension of the transport system, we face the traffic congestion problem. Besides this, a lot of reasons are responsible for traffic congestion in Bangladesh such as plying useless and unsuitable transports, inexperienced drivers, bad and limited roads, improper traffic system and lack of proper training on traffic system, lack of awareness, etc. Traffic congestion wastes our valuable time, creates time delays, decreases fuel economy and causes the risk of a vehicle collision. Around 5 million working hours are eaten up due to the traffic congestion in Dhaka city every day, costing the country USD 11.4 billion annually. Heavy traffic congestion can cause serious physical and mental problems, including stress and aggression resulting in road rage. Traffic management needs to be developed at the core of traffic congestion. Therefore, to eliminate such huge traffic congestion, efficient traffic control systems and management are essential.
For proper management of highway traffic, an efficient modeling approach and planning under statistical observations of highway traffic are highly required. To an observer, the traffic stream is viewed as a property of the spacing of the vehicles such that the closer they are together, the slower they move. Mathematically, it indicates an inverse relationship between speed and density. To solve the traffic congestion problem mathematicians and create a traffic flow model on the assumption that there are some relations between traffic density and velocity. Greenshields introduced the relation between traffic density and velocity (1934) [
The mathematical modeling of traffic flow often rests on a fluid flow analogy, treating the traffic stream as a one-dimensional movement of fluid, considered as a continuum model of traffic flow [
To solve traffic congestion mathematically, Lighthill and Whitham (1955) and Richards (1956) [
We discretize the diffusion-type traffic flow model equation by finite difference method to get different numerical schemes [
A well-known first-order LWR traffic flow model is used to study traffic flow characteristics such as traffic velocity v ( t , x ) , flux (traffic flow rate) q ( t , x ) and traffic density u ( t , x ) . All these parameters are functions of space, x ∈ ℝ and time, t ∈ ℝ + . This model is based on the conservation of mass and reads as,
∂ u ∂ t + ∂ q ∂ x = 0 ; t > 0 , − ∞ < x < ∞ (2.1)
This first order model does not explain the traffic diffusivity. For this reason, Khune added a diffusion term in 1984 and initiated using the methods of nonlinear dynamics for analyzing the equation [
∂ u ∂ t + ∂ ∂ x ( q − D ∇ u ) = 0 (2.2)
The term − D ∇ u known as the Fickian term is used to account for external noise, i.e. the term represents road conditions, engine power, braking variability, changes in wind, drivers’ response to the stimuli [
∂ u ∂ x + ∂ q ∂ x = D ∂ 2 u ∂ x 2 ; t > 0 , − ∞ < x < ∞ (2.3)
This is known as second-order parabolic diffusion type traffic flow model.
We consider the diffusion-type traffic flow model as a Cauchy problem,
∂ u ∂ t + ∂ q ∂ x = D ∂ 2 u ∂ x 2 ; t > 0 , − ∞ < x < ∞ (3)
Initial condition,
u ( x , 0 ) = u 0 ( x ) ; a ≤ x ≤ b
By Cole-Hopf transformation, the analytical solution of diffusion type traffic flow model [
u ( t , x ) = D u max v max ∂ ∂ ( ln ⌊ 1 4 π ∫ − ∞ ∞ exp ( − ( v max t − x − z ) 2 4 D − v m D m ∫ 0 z u 0 ( x ′ ) d x ′ ) d ⌋ )
where u 0 ( x ) = u ( 0 , x ) is the initial condition.
This analytical solution may cause huge errors because it depends on the initial value function over an integrodifferential equation and it is very difficult to evaluate the analytical solution because of the above integrodifferential equation.
To avoid this complexity and estimate the relative error of each scheme, we use the diffusion rate D = 0 . In this case the analytical solution of diffusion type traffic flow model with linear velocity density relation is,
u ( t , x ) = u 0 ( x ( t ) − v max ( 1 − 2 u u max ) t )
And with exponential velocity density relation the analytical solution in implicit form is,
u ( t , x ) = u 0 ( x ( t ) − v max exp ( − u u max ) ( 1 − u u max ) t )
where u 0 ( x ) = u ( 0 , x ) is the initial condition.
For the complexity of analytical solution, we need to solve the diffusion type traffic flow model by another method. Here, we use finite difference method to solve the PDE (2.3) as a numerical solution [
∂ u ∂ t + ∂ q ∂ x = D ∂ 2 u ∂ x 2 (4.1)
With initial condition u ( x , 0 ) = u 0 ( x ) ; a ≤ x ≤ b .
Boundary condition, u ( t , a ) = u a ( t ) ; t 0 ≤ t ≤ T u ( t , b ) = u b ( t ) ; t 0 ≤ t ≤ T .
Where, q ( u ) = u v max exp ( − u u max ) .
To solve the partial differential equation of the model numerically, we divide the domain in space using a mesh x 0 , x 1 , x 2 , ⋯ , x M and in time using a mesh t 0 , t 1 , t 2 , ⋯ , t N . Here, we consider a uniform partition both in space and time, so that the difference between two consecutive spatial (mesh) points will be ∆x and between two consecutive temporal (mesh) points will be ∆t, i.e.
x j = x 0 + j Δ x , j = 0 , 1 , 2 , ⋯ , M
And t n = t 0 + n Δ t , n = 0 , 1 , 2 , ⋯ , N .
For spatial non-uniform mesh:
Let, Δ x j = x j + 1 − x j , j = 0 , 1 , 2 , ⋯ , M are the spatial step size.
For spatial non-uniform mesh:
Let, Δ t n = t n + 1 − t n , n = 0 , 1 , 2 , ⋯ , N are the temporal step size.
Therefore,
x j = x 0 + j Δ x , j = 0 , 1 , 2 , ⋯ , M
t n = t 0 + n Δ t , n = 0 , 1 , 2 , ⋯ , N
To obtain this scheme, we discretize the time derivative ∂ u ∂ t by forward difference formula, first order space derivative ∂ q ∂ x by backward difference formula and ∂ 2 u ∂ x 2 by the second-order central difference formula at any point ( t n , x j ) , j = 0 , 1 , 2 , ⋯ , M − 1 ; n = 0 , 1 , 2 , ⋯ , N − 1 .
And considering u ( t n , x j ) , we have,
∂ u ( t n , x j ) ∂ t ≈ u j n + 1 − u j n Δ t (4.2)
∂ q ( t n , x j ) Δ x ≈ q j n − q j − 1 n Δ x (4.3)
∂ 2 u ( t n , x j ) ∂ x 2 ≈ u j + 1 n − 2 u j n + u j − 1 n ( Δ x ) 2 (4.4)
Now, using (4.2), (4.3) and (4.4) in Equation (4.1), we obtain,
u j n + 1 = u j n − Δ t Δ x ( q j n − q j − 1 n ) + D Δ t ( Δ x ) 2 ( u j + 1 n − 2 u j n + u j − 1 n ) (4.5)
where, q j n = u j n ∗ v max exp ( − u j n u max ) .
And q j − 1 n = u j − 1 n ∗ v max exp ( − u j − 1 n u max ) .
Equation (4.5) is known as the explicit upwind difference scheme or FTBSCS scheme of diffusion type traffic flow model.
To obtain this scheme, we discretize the time derivative ∂ u ∂ t by forward difference formula, first order space derivative ∂ q ∂ x by central difference formula and ∂ 2 u ∂ x 2 by the second-order central difference formula at any point ( t n , x j ) , j = 0 , 1 , 2 , ⋯ , M − 1 ; n = 0 , 1 , 2 , ⋯ , N − 1 .
And considering u ( t n , x j ) , we have,
∂ u ( t n , x j ) ∂ t ≈ u j n + 1 − u j n Δ t (4.6)
∂ q ( t n , x j ) ∂ x ≈ q j + 1 n − q j − 1 n 2 Δ x (4.7)
∂ 2 u ( t n , x j ) ∂ x 2 ≈ u j + 1 n − 2 u j n + u j − 1 n ( Δ x ) 2 (4.8)
Using (4.6), (4.7) and (4.8) in (4.1), we obtain,
u j n + 1 = u j n − Δ t 2 Δ x ( q j + 1 n − q j − 1 n ) + D Δ t ( Δ x ) 2 ( u j + 1 n − 2 u j n + u j − 1 n ) (4.9)
where, q j − 1 n = u j − 1 n ∗ v max exp ( − u j − 1 n u max ) .
And q j + 1 n = u j + 1 n ∗ v max exp ( − u j + 1 n u max ) .
Equation (4.9) is known as the explicit centered difference scheme or FTCSCS scheme of diffusion type traffic flow model.
For explicit Lax-Wendroff scheme of diffusion type traffic flow model, we discretize the part ∂ u ∂ t + ∂ q ∂ x = 0 in half time step Lax-Friedrich scheme [
To get Lax-Friedrich scheme, we discretize the time derivative ∂ u ∂ t by forward difference formula, first order space derivative ∂ q ∂ x by central difference formula and ∂ 2 u ∂ x 2 by the second-order central difference formula at any point ( t n , x j ) , j = 0 , 1 , 2 , ⋯ , M − 1 ; n = 0 , 1 , 2 , ⋯ , N − 1 and considering u ( t n , x j ) = u j n , we have,
u j n + 1 = 1 2 ( u j + 1 ` n + u j − 1 n ) − Δ t 2 Δ x ( q j + 1 n − q j − 1 n ) (4.10)
Equation (4.10) is known as Lax-Friedrich scheme.
Now, take half time step in Lax-Friedrich scheme, we have,
u j + 1 2 n + 1 2 = 1 2 ( u j + 1 n + u j n ) − Δ t 2 Δ x ( q j + 1 n − q j n ) (4.11)
u j − 1 2 n + 1 2 = 1 2 ( u j n + u j − 1 n ) − Δ t 2 Δ x ( q j n − q j − 1 n ) (4.12)
To get leapfrog scheme, we discretize both time derivative ∂ u ∂ t and space derivative ∂ q ∂ x by central difference formula,
∴ u j n + 1 = u j n − 1 − Δ t Δ x ( q j + 1 n − q j − 1 n ) (4.13)
Now, we take half time step Leapfrog scheme in (4.13), we get,
u j n + 1 = u j n − Δ t Δ x ( q j + 1 2 n + 1 2 − q j − 1 2 n + 1 2 ) . (4.14)
Using exponential velocity density relationship q = q ( u ) = u ∗ v exp ( u u max ) in (4.14), we have,
u j n + 1 = u j n − v max Δ t Δ x ( u j + 1 2 n + 1 2 exp ( − u j + 1 2 n + 1 2 u max ) − u j − 1 2 n + 1 2 exp ( − u j − 1 2 n + 1 2 u max ) ) (4.15)
This is known as the Leapfrogs scheme for half time step.
We use central difference formula for second-order space derivative ∂ 2 u ∂ x 2 , we have,
∂ 2 u ( t n , x j ) ∂ x 2 ≈ u j + 1 n − 2 u j n + u j − 1 n ( Δ x ) 2 (4.16)
Now, combining (4.15) and (4.16) and using (4.11) and (4.12), we will get the required Lax-Wendroff scheme for diffusion type traffic flow model,
u j n + 1 = u j n − v max Δ t Δ x ( u j + 1 2 n + 1 2 exp ( − u j + 1 2 n + 1 2 u max ) − u j − 1 2 n + 1 2 exp ( − u j − 1 2 n + 1 2 u max ) ) + u j + 1 n − 2 u j n + u j − 1 n ( Δ x ) 2 (4.17)
where,
u j + 1 2 n + 1 2 = 1 2 ( u j + 1 n + u j n ) − Δ t 2 Δ x ( q j + 1 n − q j n )
u j − 1 2 n + 1 2 = 1 2 ( u j n + u j − 1 n ) − Δ t 2 Δ x ( q j n − q j − 1 n )
q j n = u j n ∗ v max exp ( − u j n u max )
q j + 1 n = u j + 1 n ∗ v max exp ( − u j + 1 n u max )
q j − 1 n = u j − 1 n ∗ v max exp ( − u j − 1 n u max )
Scheme (4.17) is known as the Lax-Wendroff scheme diffusion-type traffic flow model with exponential velocity density relation.
By Von Neumann Stability analysis, we obtain the stability condition of explicit upwind, explicit centered and explicit second-order Lax-Wendroff scheme [
The Stability Condition of Explicit Upwind Difference Scheme is,
0 ≤ γ ≤ 1 and − γ ≤ α ≤ 1 − 2 γ
The Stability condition of explicit centered difference scheme,
0 ≤ α ≤ 1 and 0 ≤ γ ≤ 1 2
And the stability condition of Lax-Wendroff scheme is,
0 ≤ γ ≤ 1 and 0 ≤ α ≤ 1
where, α = max ( max ( q ′ ( u j n ) ) ) Δ t Δ x and γ = D Δ t ( Δ x ) 2 .
For the simulation of a diffusion-type traffic flow model using different schemes, we use the periodic initial and left boundary conditions [
In Figures 1-6, we present numerical experiments of density, velocity and flux profile for 3 minutes or 180 seconds of diffusion type traffic flow model to visualize the behavior of density, velocity and flux with respect to space and time using explicit upwind, explicit centered and Lax-Wendroff schemes. We also computed the relative error of diffusion type traffic flow model by using each of the explicit upwind, explicit centered and Lax-Wendroff schemes. Here, we observe that each of the three schemes gives better flow with the progress of time. The relative error of diffusion type traffic flow model using these three schemes is pretty good.
Under the assumption mentioned in 6, we present a comparison density profile using explicit upwind, explicit centered, and Lax-Wendroff schemes. We also present a comparison of relative error using these three schemes.
In
Lax-Wendroff scheme gives better solution than explicit upwind and explicit centered difference schemes.
In
The numerical solution using explicit upwind, explicit centered, and Lax-Wendroff schemes is presented in our paper. From the comparison of density profiles using explicit upwind, explicit centered, and Lax-Wendroff schemes, it is seen that Lax-Wendroff gives better flow than explicit upwind and explicit centered difference schemes. From the comparison of relative error, we have seen that the Lax-Wendorff scheme gives less error than the explicit upwind and explicit centered difference schemes. So, from the numerical as well as analytical analysis and comparative result discussion, it can be concluded that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.
The authors declare no conflicts of interest regarding the publication of this paper.
Haque, M., Sultana, M. and Andallah, L.S. (2023) Numerical Simulation of Diffusion Type Traffic Flow Model Using Second-Order Lax-Wendroff Scheme Based on Exponential Velocity Density Function. American Journal of Computational Mathematics, 13, 398-411. https://doi.org/10.4236/ajcm.2023.133023