1. Introduction

The dynamics of traffic flow have been extensively studied in the literature. The conservation equations of the model (dynamics) come from a nonlinear partial differential equation. The solution develops under the initial conditions of the vehicle input density [1]. These initial conditions allow decoupling each of the equations for each input and solving independently. The function that determines the initial condition for the one-dimensional problem is known and defined on the solution interval, giving the solution to the Cauchy problem [2], [3]. Velocity is a function that depends on density. This scalar conservation law must be complemented with adequate initial conditions and with boundary conditions as developed in this document.

Traffic flow models and simulation tools are often used for traffic state estimation and prediction [4]. Traffic flow models such as the kinematic-wave model, "higher order" models, and car tracking models are more or less accurate representations of reality. In a simulation tool, the model equations are solved using numerical methods, again with near precision. On the other hand, the development of the finite element method in a spatial dimension allows calculating, from the point of numerical analysis, a system of linear equations that allows finding the value or prediction of flow in each of the points in space. The system of linear equations has the property of forming a tridiagonal positive semidefinite matrix whose inverse does not entail a higher computational cost. The formulation of the numerical solution to the traffic problem is obtained by comparing it to the fluid mechanics problem, reaching highly accurate results [5], [6]. By the similarity between fluid dynamics, traffic flow dynamics, and Newton's second law, the one-dimensional finite element method for velocity estimation is proposed as a numerical solution as an application of motion fluids mechanics [7]. Due to the nonlinearity in the temporal variable, the behavior of the solution from specific points of the spatial variable is evaluated in several temporal lines.

The theorems that allow us to guarantee the existence and uniqueness of the solution given by the Dirichlet [8], [9], [10], Robin [11] and Neumann boundary conditions will be presented in section 2 of this document.

2. Content

2.1. Problem model

In the domain Ω=(𝑥₀,𝑥_𝑛), find 𝑢 with 𝑢(𝑥₀)=𝑢(𝑥_𝑛)=𝑢_𝑐 for a given function 𝑓, such they satisfy the Dirichlet initial conditions.

Dirichlet minimization statement [5].

To find 𝜙 ∈𝑋 such that:

Noticing,

Should be minimized,

Without losing generality, for any 𝑙(𝑣) ∈ 𝐻⁻¹(Ω), then it is required to find 𝑢 ∈𝐻₀¹(Ω) [6] such that:

For example, 𝑙(𝑣)=⟨𝛿_𝑥0,𝑣⟩=𝑣(𝑥₀) is admissible.

With the regularity theorems for boundary problems, the model takes the following form [7]:

Theorem 1. Let Ω∈ ℝ^𝑛 be an open of class 𝐶² with bounded frontier, 𝑢 ∈𝐿²(Ω) and 𝑓 ∈𝐻₀¹(Ω) be the solution of the Dirichlet problem. So, 𝑓 ∈𝐻₀²(Ω) given the fact that:

With a constant 𝐶>0 that only depends on Ω.

Furthermore, if Ω is of class 𝐶^𝑚+2 and 𝑢 ∈𝐻^𝑚(Ω), with 𝑚≥1 integer, then 𝑓 ∈𝐻^𝑚+2(Ω) and there is a constant 𝐶_𝑚>0 that only depends on 𝑚 and 𝛺, in such a way that:

If we have that 𝑚 > 𝑛/2 then 𝑓 ∈ 𝐶²($\stackrel{-}{\mathrm{\Omega }}$) giving way to the following two theorems given the initial conditions of the problem [11].

Theorem 2. Let Ω ∈ ℝ^𝑛 be an open of class 𝐶² with bounded frontier, 𝑢 ∈ 𝐿²(Ω) 𝜎 ∈ 𝐶²(𝛺̅) and 𝑓 ∈ 𝐻₀¹(𝛺) be the solution to the Robin problem. So, 𝑓 ∈ 𝐻²₀(𝛺) given the fact that:

With a constant 𝐶 > 0 that only depends on Ω.

On the other hand, if Ω is of class 𝐶^𝑚+2 and 𝑢 ∈ 𝐻^𝑚(Ω) and 𝜎(𝑥) ∈ 𝐶^𝑚+1(𝛺̅), with 𝑚 ≥ 1 integer, then 𝑓 ∈ 𝐻^𝑚+2(Ω) and there is a constant 𝐶𝑚 > 0 that only depends on 𝑚 𝑦 𝛺, such that:

If 𝑚 > 𝑛/2 then 𝑓 ∈ 𝐶²($\stackrel{-}{\mathrm{\Omega }}$), giving way to the following two theorems given the initial conditions of the problem [12].

If we have 𝑚 > $\frac{n}{2}$ then 𝑢 ∈ 𝐶² ($\stackrel{-}{\mathrm{\Omega }}$).

Theorem 3. Let Ω ∈ ℝ^𝑛 an open of class 𝐶² with bounded, 𝑢 ∈𝐿²(Ω) and 𝑓 ∈𝐻₀¹(𝛺) the solution of the Neumann problem. So, 𝑓 ∈𝐻₀²(𝛺) given the fact that:

With a constant 𝐶>0 that only depends on Ω.

Furthermore, if Ω is of class 𝐶^𝑚+2 and 𝑢 ∈𝐻^𝑚(Ω), with 𝑚≥1 integer, then 𝑓 ∈𝐻^𝑚+2(Ω) and there is a constant 𝐶_𝑚>0 that only depends on 𝑚 and 𝛺, such that:

If we have that m>n/2 then f 6 C²( $\stackrel{-}{\mathrm{\Omega }}$ ), giving way to the following two theorems given the initial conditions of the problem [13].

2.2. Solution using finite elements

The domain is discretized through the triangulation 𝑇_ℎ, in the space of 𝑋_ℎ where it is generated by the basis 𝑋_ℎ=𝑠𝑝𝑎𝑛{𝜑₁,…,𝜑_𝑛} [14], [15], [16].

Be

Set u_hj such that:

Since any v ∈ X_h can be written as a linear combination of the form:

Knowing that: , then we have:

Written in matrix form, we get:

Taking the following expansion, the above equations (24) can be developed as follows:

We have:

Then taking:

We have:

And so on 𝑣^𝑡𝐴ℎ𝑢ℎ = 𝑣^𝑡𝐹ℎ, ∀𝑣 ∈ ℝ^𝑛 , with 𝐴ℎ𝑢ℎ = F_h

The elements of the Matrix 𝐴ℎ [17] are constructed with 𝜑𝑖 as linear functions (Figure 1) and the derivatives of $\mathrm{\phi }\mathrm{i}\frac{d\phi i}{dx}$ for i=1 without losing generality (Figure 2).

Figure 1
Functions in a linear way [17].

Figure 2
Derivative of the shape functions on the intervals [17].

The A_hij elements are given by:

The nonzero elements are j = i - + 1

The border lines

The formation of the elements of the A_h matrix characterizes this matrix as being positive definite [18] (A_h> 0), diagonally dominant, sparse, and tridiagonal as shown below:

The "charge" vectors are constructed in the general case as:

Figure 3 describes the integration of 𝑓 on 𝑖 𝑒 𝑖 + 1. On the other hand, we have that 𝑢ℎ ∈ ℝ^𝑛 satisfies:

Figure 3
Integration of 𝑓 in the elements 𝑖, 𝑖 + 1.

To form the orthogonal basis of linear functions, we have the “mass” matrix, which has the property 𝑀ℎ ∈ ℝ^𝑛𝑥𝑛 ≻ 0 to be positive semidefinite [19], the terms of this matrix are obtained by integrating the functions in a way, like this:

If 𝑣 ≠ 0, then 𝜑𝑖 are the bases in linear form. Therefore, for linear elements the nodal linear basis is given by the “mass” matrix [20]:

In the temporary variable it is discretized with the recurrence equation:

In the case of linear basis elements with equal time division, the finite element approximation provides a coupled linear system of equations for each of the 𝑓𝑖 with 𝑖 = 2, … , 𝑛. In this way we arrive at the numerical solution of the finite element expansion, as shown below:

3. Analysis and results

Figure 4 shows the dynamics of the density. In the case study of this document, this function is taken as the initial condition for the Cauchy problem, posed through equations (9) and (24).

Figure 4
Initial density conditions.

v is the known velocity (initial conditions), and u is the density (vehicles/m).

The behavior of the velocity at different points in the spatial domain is shown in Figure 5, i.e., (t,x),(t,0) blue curve, (t, 1) black curve, (t, 5) green curve, and (t, 10) red curve.

Figure 5
Velocity v(t).

Using Figure 6, the numerical solution of the density of vehicles per meter can be verified. This solution is subject to the initial conditions established by the function proposed in Figure 4.

Figure 6
Input density 𝑓(𝑥).

4. Conclusions

Due to the velocity limit that must be met for the theorems outlined in this document, the solution is bounded between 0 and 1 vehicles/meter as shown in Figures 4 to 6. On the other hand, the non-linear solution given by the temporal variable is estimated by means of finite elements for 5 specific points in the space of the velocity dynamics.

The theorems described above are satisfied if Ω = ℝ^𝑛 and the solution obtained by any numerical method converges to the initial conditions and meets the initial conditions determined according to the problem posed (Dirichlet, Robin, or Neumann).

Due to the velocity of propagation, to solve the Cauchy problem for traffic flow in a whole network, it is proposed to build a local solution in a neighborhood. The solution found meets the conditions of the problem posed with equations (9) and (24) and the numerical solution given by equation (39).

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Notes

Author notes

^aEmails: femesa@utp.edu.co; ^bdmdevian@utp.edu.co

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