
The introduction of connected autonomous vehicles (CAVs) gives rise to mixed traffic flow on the roadway, and the coexistence of human-driven vehicles (HVs) and CAVs may last for several decades. CAVs are expected to improve the efficiency of mixed traffic flow. In this paper, the car-following behavior of HVs is modeled by the intelligent driver model (IDM) based on actual trajectory data. The cooperative adaptive cruise control (CACC) model from the PATH laboratory is adopted for the car-following model of CAVs. The string stability of mixed traffic flow is analyzed for different market penetration rates of CAVs, showing that CAVs can effectively prevent stop-and-go waves from forming and propagating. In addition, the fundamental diagram is obtained from the equilibrium state, and the flow-density chart indicates that CAVs can improve the capacity of mixed traffic flow. Furthermore, the periodic boundary condition is designed for numerical simulation according to the infinite length platoon assumption in the analytical approach. The simulation results are consistent with the analytical solutions, suggesting the validity of the string stability and fundamental diagram analysis of mixed traffic flow.
Citation: Lijing Ma, Shiru Qu, Jie Ren, Xiangzhou Zhang. Mixed traffic flow of human-driven vehicles and connected autonomous vehicles: String stability and fundamental diagram[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2280-2295. doi: 10.3934/mbe.2023107
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The introduction of connected autonomous vehicles (CAVs) gives rise to mixed traffic flow on the roadway, and the coexistence of human-driven vehicles (HVs) and CAVs may last for several decades. CAVs are expected to improve the efficiency of mixed traffic flow. In this paper, the car-following behavior of HVs is modeled by the intelligent driver model (IDM) based on actual trajectory data. The cooperative adaptive cruise control (CACC) model from the PATH laboratory is adopted for the car-following model of CAVs. The string stability of mixed traffic flow is analyzed for different market penetration rates of CAVs, showing that CAVs can effectively prevent stop-and-go waves from forming and propagating. In addition, the fundamental diagram is obtained from the equilibrium state, and the flow-density chart indicates that CAVs can improve the capacity of mixed traffic flow. Furthermore, the periodic boundary condition is designed for numerical simulation according to the infinite length platoon assumption in the analytical approach. The simulation results are consistent with the analytical solutions, suggesting the validity of the string stability and fundamental diagram analysis of mixed traffic flow.
The time-fractional partial differential equation (TFPDE) is one of the most widely studied fractional partial differential equations in mathematics and physics and has captured continued interests from researchers, due to its widespread applications [1]. For example, it is frequently used to model anomalous diffusion in porous media [2,3,4,5,6], where heterogeneities cause the diffusion rate to deviate from classical models. Additionally, the TFPDEs are employed to describe the propagation dynamics of light beams [7,8] and heat transfer in Maxwell fluids [9]. Therefore, it is very significant to study the properties of this kind of equation not only in practice but also in theory. In this paper, we consider the following TFPDEs with variable coefficients [10]:
Dαtu(x,t)=d∑i=1(ai(x,t)∂2u∂x2i+bi(x,t)∂u∂xi)+h(x,t),x∈Ω,t∈[t0,tf], | (1.1) |
subjected to the initial and boundary conditions:
u(x,t0)=f(x),x∈Ω, | (1.2) |
u(x,t)=g(t),x∈∂Ω,t0<t⩽tf, | (1.3) |
where 0<α⩽1, x is a spatial point, t is the time, t0 and tf represent the initial and final time, Ω⊂Rd, d=1,2 is a bounded domain with boundary Γ=∂Ω, ai(x,t), bi(x,t), h(x,t), f(x), and g(t) are known functions. Dαtu(x,t) is the time-fractional derivative, which has several different definitions [11,12]. Commonly used fractional derivatives include the Riemann-Liouville, Caputo, and Grunwald-Letnikov derivatives. In this study, we consider the following Caputo definition:
Dαtu(x,t)={1Γ(1−α)∫t0(t−s)−α∂u(x,s)∂sds,0<α<1,∂u(x,t)∂t,α=1. | (1.4) |
Dαtu(x,t)={1Γ(2−α)∫t0(t−s)1−α∂2u(x,s)∂s2ds,1<α<2,∂u2(x,t)∂t2,α=2. | (1.5) |
The exact analytical solutions to the above TFPDEs are possible only for a few simple cases, but are virtually impossible for most problems, which greatly limits their applications. On the contrary, this enormously promotes the development of various numerical algorithms [13,14,15,16]. It is well known that the finite element, finite difference, boundary element, and meshless/meshfree methods are commonly used approaches for solving various integer/fractional order partial differential equations [17,18,19,20]. Among them, meshless methods have become quite popular in recent years thanks to their ease of implementation and interactive applications. Representative methods in this category include the Galerkin meshless method [21,22], the meshless local Petrov-Galerkin method [23,24], corrected smoothed particle hydrodynamics [25,26], the radial basis function collocation method [27,28], the generalized finite difference method [29,30,31], the backward substitution method [32,33], and localized semi-analytical collocation methods [34,35,36,37]. In most published works, however, these methods require the discretization of the temporal derivatives by using implicit, explicit, Runge-Kutta or any other approaches [38,39], which may be computationally expensive and sometimes mathematically complicated. To address these issues, numerous scholars in recent years have proposed space-time combined meshless methods [40,41,42,43,44] for solving various partial differential equations numerically.
Inspired by previous work [41], this paper makes a first attempt to extend the recently proposed spatio-temporal meshless method (STMM) to the solutions of TFPDEs with variable coefficients. The STMM method first constructs a multiquadric radial basis function (ST-MQ-RBF) in terms of the space-time distance and then directly approximates the solution of Eqs (1.1)–(1.3) by a linear combination of the presented ST-MQ-RBFs at scattered nodes. Compared to the traditional RBF method, the main advantage of the developed approach lies in its complete avoidance of the discretization process for time-fractional-order derivatives, employing the same RBF approximation in both time and space. As a result, the method is simpler, more straightforward, easier to implement, and relatively more accurate.
The outline of the paper is as follows: In Section 2, the STMM for spatial and temporal discretization simultaneously of TFPDEs is proposed, where the numerical approximation for time-fractional derivative is described. Section 3 briefly introduces a multiple-scale technique for solving the ill-conditioned system. In Section 4, numerical examples are studied and discussed. Finally, conclusions and remarks are presented in Section 5.
The basic idea of STMM is to introduce a space-time distance into the MQ-RBF and then approximate the solutions of the partial differential equations by using the new ST-MQ-RBFs in the whole space-time domain without the differential approximation of time derivatives. For the above TFPDEs with variable coefficients, N=Ni+N0+Nb nodes (xi,ti)Ni=1 should be distributed on the space-time domain Ωst=Ω×[t0,tf] as shown in Figure 1, where Ni,N0,Nb represent the numbers of nodes in the domain, on the initial boundary and on the time boundary, respectively. For the ith node, the following MQ-RBF formulation should hold:
u(xi,ti)=N∑j=1λjϕ(xi,ti;xj,tj), | (2.1) |
where λj are the unknown coefficients, ϕ(xi,ti;xj,tj) are the MQ-RBFs given as follows:
ϕ(xi,ti;xj,tj)=√rij2+c2, | (2.2) |
in which c is the shape parameter, rij denotes the space-time distance between the points (xi,ti) and (xj,tj), which can be written as
rij=√d∑k=1(xik−xjk)2+(ti−tj)2. | (2.3) |
Like the traditional MQ-RBF method, the variables at interior points should satisfy the governing equation, and the variables at boundary points should satisfy initial and boundary conditions. With this regard, we have
N∑j=1λj(Dαtiϕ(xi,ti;xj,tj)−d∑k=1(ak(x,t)∂2u∂x2k+bk(x,t)∂u∂xk))=h(xi,ti), i=1,...,Ni, | (2.4) |
N∑j=1λjϕ(xi,t0;xj,tj)=f(xi),xi∈Ω,i=Ni+1,...,Ni+N0, | (2.5) |
N∑j=1λjϕ(xi,ti;xj,tj)=g1(ti),xi∈∂Ω,i=Ni+N0+1,...,Ni+N0+Nb. | (2.6) |
The spatial derivatives of the STRBFs in Eq (2.4) can be easily calculated. Here, we concern the calculation of the time-fractional derivative Dαtiϕ(xi,ti;xj,tj) in Eq (2.4). For α=1, it can be computed by the following expression:
Dαtiϕ(xi,ti;xj,tj)=∂ϕ(xi,ti;xj,tj)∂ti=ti−tj√rij2+c2. | (2.7) |
For 0<α<1, from the definition (1.4) of time-fractional derivative, using Gauss numerical integration, we have
Dαtiϕ(xi,ti;xj,tj)=1Γ(1−α)∫ti0(ti−s)−α∂ϕ(xi,s;xj,tj)∂sds=1Γ(1−α)∫ti0(s−tj)(ti−s)−α√d∑k=1(xik−xjk)2+(s−tj)2+c2ds |
=1Γ(1−α)∫1−1((tiξ+ti)/2−tj)(ti−(tiξ+ti)/2)−αti/2√d∑k=1(xik−xjk)2+((tiξ+ti)/2−tj)2+c2dξ=1Γ(1−α)Ng∑k=1ωkψ(ξk), | (2.8) |
where ωk and ξk are the nodes and weights of the Gaussian quadrature, Ng is the number of Gaussian nodes (in this study, Ng is set as 16), and
ψ(ξk)=((tiξk+ti)/2−tj)(ti−(tiξk+ti)/2)−αti/2√d∑k=1(xik−xjk)2+((tiξk+ti)/2−tj)2+c2. | (2.9) |
For 1<α<2,
Dαtiϕ(xi,ti;xj,tj)=1Γ(2−α)∫ti0(ti−s)1−α∂2ϕ(xi,s;xj,tj)∂s2ds=1Γ(2−α)∫ti0(d∑k=1(xik−xjk)2+2c2)(ti−s)1−α(d∑k=1(xik−xjk)2+(s−tj)2+c2)3/2ds=1Γ(2−α)∫1−1(d∑k=1(xik−xjk)2+2c2)(ti−(tiξ+ti)/2)1−αti/2(d∑k=1(xik−xjk)2+((tiξ+ti)/2−tj)2+c2)3/2dξ=1Γ(2−α)Ng∑k=1ωkψ(ξk), | (2.10) |
ψ(ξk)=(d∑k=1(xik−xjk)2+2c2)(ti−(tiξ+ti)/2)1−αti/2(d∑k=1(xik−xjk)2+((tiξ+ti)/2−tj)2+c2)3/2. | (2.11) |
By combining Eqs (2.4)–(2.6), a resultant linear system in matrix form can be generalized:
Aλ=b, | (2.12) |
where AN×N is the coefficient matrix, λ=[λ1,λ2,...,λN]T is a vector consisting of the undetermined coefficients; bN×1 is a vector consisting of the right-hand sides in Eqs (2.4)–(2.6). As long as Eq (2.12) is solved efficiently (that is to say, λ can be obtained accurately), the unknown variable at an arbitrary point (x,t) in the considered space-time box can be acquired by
u(x,t)=N∑j=1λjϕ(x,t;xj,tj). | (2.13) |
For the RBF method, in general, the algebraic equation (2.12) is ill-conditioned. A pre-conditioner is required to reduce the condition number and to yield a regular solution. The multiple scale technique [45,46,47,48,49] is an effective pre-conditioner tool and thus is applied in this paper. Here, we will give a brief review of this method.
Let A=[A1A2...AN], where {Ai}Ni=1 are column vectors of the matrix A. By the multiple scale technique, Eq (2.12) can be reformulated as follows:
⌣A⌣λ=b, | (3.1) |
where
⌣A=[A1‖ | (3.2) |
and
\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\boldsymbol{\lambda}} = {\left[ {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_1}\;\;{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_2}\;\; \cdot \cdot \cdot \;\;{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_N}} \right]^T} = {\left[ {{\lambda _1}{{\left\| {{A_1}} \right\|}_2}\;\;{\lambda _2}{{\left\| {{A_2}} \right\|}_2}\;\; \cdot \cdot \cdot \;\;{\lambda _n}{{\left\| {{A_N}} \right\|}_2}} \right]^T} . | (3.3) |
The condition number of the matrix \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\boldsymbol{A}} will be greatly reduced due to the reduction of round-off errors. After the system of Eq (3.1) is solved, the solution \boldsymbol{\lambda} = {({\lambda _1}, {\lambda _2}, ..., {\lambda _n})^T} of Eq (2.12) can be recovered from Eq (3.3) as follows:
\boldsymbol{\lambda} = {\left[ {{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_1}} \mathord{\left/ {\vphantom {{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_1}} {{{\left\| {{A_1}} \right\|}_2}}}} \right. } {{{\left\| {{A_1}} \right\|}_2}}}\;\;{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_2}} \mathord{\left/ {\vphantom {{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_2}} {{{\left\| {{A_2}} \right\|}_2}}}} \right. } {{{\left\| {{A_2}} \right\|}_2}}}\;\; \cdot \cdot \cdot \;\;{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_N}} \mathord{\left/ {\vphantom {{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_N}} {{{\left\| {{A_N}} \right\|}_2}}}} \right. } {{{\left\| {{A_N}} \right\|}_2}}}} \right]^T} . | (3.4) |
It should be pointed out that Eq (3.1) is solved using the MATLAB command 'A\B'.
In this section, two numerical examples including one- and two-dimensional TFPDEs with both regular and irregular geometries are studied by using regular and irregular nodes to illustrate the efficiency and accuracy of the proposed space-time meshless method. To evaluate the accuracy of the proposed STMM, the following L∞-error, L2-error and the root-mean-square error (RMSE) are adopted:
\begin{gathered} {L_\infty }{\text{ - error}} = \mathop {\max }\limits_{1 \leqslant j \leqslant M} \left| {{u_{num}}({x_j}, {t_j}) - {u_{exa}}({x_j}, {t_j})} \right|, \;\;\;\; \hfill \\ {L_2}{\text{ - error}} = \sqrt {\sum\limits_{j = 1}^M {{{\left( {{u_{num}}({x_j}, {t_j}) - {u_{exa}}({x_j}, {t_j})} \right)}^2}} } , \hfill \\ RMSE = \sqrt {\frac{1}{M}\sum\limits_{j = 1}^M {{{\left( {{u_{num}}({x_j}, {t_j}) - {u_{exa}}({x_j}, {t_j})} \right)}^2}} } , \hfill \\ \end{gathered} | (4.1) |
where M is the number of test points, {u_{exa}} and {u_{num}} represent the exact and numerical solutions at the jth test point, respectively.
Example 1. Consider the following two one-dimensional TFPDEs with constant and variable coefficients on the domain {\Omega ^{st}} = [0, 1] \times [0, 1]:
\mathit{Case I:}\; D_t^\alpha u(x, t) = \frac{{{\partial ^2}u(x, t)}}{{\partial {x^2}}} - \frac{{\partial u(x, t)}}{{\partial x}} + \frac{{2{t^{2 - \alpha }}}}{{\Gamma (3 - \alpha )}} + 2x - 2, | (4.2) |
\mathit{Case II:}\; D_t^\alpha u(x, t) = ({x^2} + t + 1)\frac{{{\partial ^2}u(x, t)}}{{\partial {x^2}}} - {t^2}{e^x}\frac{{\partial u(x, t)}}{{\partial x}} + \frac{{2{t^{2 - \alpha }}}}{{\Gamma (3 - \alpha )}} - 2({x^2} + t) + 2{t^2}x{e^x} - 2. | (4.3) |
The exact solution of the above two cases is u{\text{ }}(x, t) = {x^2} + {t^2} . The initial and boundary conditions of this problem are
u(x, 0) = {x^2}, \;\;x \in [0, 1] , | (4.4) |
u(0, t) = {t^2}, \;\;u(1, t) = 1 + {t^2}, \;t \in [0, 1] . | (4.5) |
First, we consider the TFPDE with constant coefficients in Case I. Taking {N_i} interior nodes as test points, the numerical accuracy of the proposed STMM for Case I is tested by using different selection strategies of shape parameter c. Table 1 lists the {L_\infty } -errors and RMSEs under different values of \alpha , and N = 1840. For details of the three strategies mentioned in Table 1, we refer to [49]. As can be seen, the three techniques of shape parameter optimization are all efficient and accurate, and thus the first approach is adopted in the following calculations. It can also be observed that the accuracy of the algorithm decreases with the increase of \alpha . Moreover, the comparison between the numerical errors obtained by the present method with those reported in [34] is shown in Table 2, where we set \delta t = 0.01, N = 5151, {N_0} = 51, {N_t} = 2 \times 100, {N_i} = 49 \times 100 and \alpha = 0.5 for the comparison. The results indicate that the proposed scheme is more accurate than that in [50]. The number of nodes was kept constant, and Table 3 compares the condition numbers of the coefficient matrix with and without the use of multiscale techniques. It can be seen that the introduction of multiscale techniques significantly reduces the condition number. It is worth noting that the computational results for both cases are essentially the same, so they are not presented here.
c | α=0.2 | α=0.6 | ||
L∞-error | RMSE | L∞-error | RMSE | |
Hardy [51] | 4.629e-04 | 2.629e-04 | 5.101e-03 | 3.685e-03 |
Fasshauer [52] | 2.177e-04 | 1.129e-04 | 5.014e-03 | 3.634e-03 |
Frank [53] | 1.809e-04 | 1.018e-04 | 5.022e-03 | 3.640e-03 |
t | Method in Ref. [50] | Present method | ||
L∞-error | L2-error | L∞-error | L2-error | |
0.1 | 6.086e-02 | 2.613e-01 | 1.709e-04 | 1.090e-04 |
0.5 | 2.958e-02 | 1.277e-01 | 2.155e-03 | 1.559e-03 |
1.0 | 2.114e-02 | 9.134e-02 | 6.688e-03 | 4.904e-03 |
Fractional orders | Condition numbers | |
With multiple scale technique | Without multiple scale technique | |
0.2 | 1.029e+09 | 1.264e+09 |
0.5 | 1.036e+09 | 1.265e+09 |
0.8 | 1.004e+09 | 1.234e+09 |
Next, we consider the TFPDE with variable coefficients in Case II. The convergence of the proposed algorithm is investigated when \alpha = 0.2 and 0.6, and the absolute error curves are illustrated in Figure 2. The results demonstrate that the STMM has high precision and good convergence. It is also noted that the smaller the \alpha , the faster the convergence. As depicted in Figure 3, the graphs of the numerical solution and absolute error with \alpha = 0.5 further confirm the validity and accuracy of the present method for solving the TFPDE with variable coefficients.
Then, Figure 4 presents the errors with respect to the nodal spacing to show the convergence in space. It can be found that as the nodal spacing decreases, the algorithm demonstrates a clear convergence trend. When the nodal spacing becomes sufficiently small, the calculation accuracy stabilizes.
Finally, we examined the impact of the number of Gaussian points on the accuracy of the algorithm. It can be seen from Table 4 that the algorithm shows a convergence trend as the number of Gauss points increases. Considering that the change in error is minimal, using 16 Gauss points is sufficient to ensure accuracy and save computation time.
{N_g} | 4 | 8 | 12 | 16 | 20 |
L∞-error | 1.192e-02 | 6.532e-03 | 4.664e-03 | 3.688e-03 | 3.097e-03 |
RMSE | 4.764e-03 | 2.548e-03 | 1.758e-03 | 1.359e-03 | 1.108e-03 |
Example 2. Consider the following two-dimensional time fractional advection-diffusion equation on \Omega = [0, \; 1] \times [0, \; 1] [54]:
D_t^\alpha u = \frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} - \frac{{\partial u}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} + 0.5\Gamma (3 + \alpha ){t^2}{e^{x + y}}, \;\;(x, y) \in \Omega , \;\;t > 0 , | (4.6) |
with the following initial and boundary conditions:
u(x, y, 0) = 0, \;\;(x, y) \in \Omega , | (4.7) |
u(x, y, t) = {t^{2 + \alpha }}{e^{x + y}}, \;\;(x, y) \in \partial \Omega , \;\;t > 0 . | (4.8) |
The exact solution of this problem can be given by
u(x, y, t) = {t^{2 + \alpha }}{e^{x + y}} . | (4.9) |
To investigate the influence of the node distribution on the numerical solution, we consider two types of node distribution (regular nodes and irregular nodes) as shown in Figure 5. When \alpha = 0.5 and a total of 4560 nodes are used, the analytical solution and absolute errors on the physical domain at t = 1.0 with regular and irregular nodal distributions are illustrated in Figure 6. These figures have proven that the newly proposed meshless approach has very good accuracy even using irregular nodal distributions.
Figure 7 presents the absolute errors at the point (0.5, 0.5, t) for the various fractional orders. Noted that the STMM has high computational accuracy, and the smaller the \alpha , the higher the accuracy, which is the same as one-dimensional problems. Furthermore, the convergence of the proposed meshless method is investigated in Table 5. As can be observed, the algorithm converges as the number of nodes increases.
N | 270 | 1540 | 4560 | 10080 | 18850 |
L∞-error | 4.908 e-02 | 3.708e-02 | 3.159e-02 | 2.758e-02 | 2.442e-02 |
RMSE | 3.703e-03 | 2.190e-03 | 1.675 e-03 | 1.422e-03 | 1.278e-03 |
Finally, the proposed methodology is tested on an irregular domain \Omega shown in Figure 8. Using the same exact solution, initial and boundary conditions, we consider the following two-dimensional time fractional PDE with variable coefficients:
D_t^\alpha u = ({x^2} + t + 1)\frac{{{\partial ^2}u}}{{\partial {x^2}}} + ({y^2} - t - 1)\frac{{{\partial ^2}u}}{{\partial {y^2}}} + {t^2}{e^x}\frac{{\partial u}}{{\partial x}} + {t^2}{e^y}\frac{{\partial u}}{{\partial y}} + h(x, y, t), \;\;(x, y) \in \Omega , \;\;t > 0 , | (4.10) |
where h(x, y, t) = 0.5\Gamma (3 + \alpha){t^2}{e^{x + y}} - ({x^2} + {y^2} + ({e^x} + {e^y}){t^2}){t^{(2 + \alpha)}}{e^{x + y}} . The boundary \partial \Omega of the computational domain is defined by the following parametric equation:
\partial \Omega = \{ (x, y)|x = \rho (\theta )\cos (\theta ), \;\;y = \rho (\theta )\sin (\theta ), \;\;0 \leqslant \theta \leqslant 2\pi \} , | (4.11) |
where \rho (\theta) = \frac{{3 + \cos (\theta - \pi /5)\sin (4\theta)}}{{5 + \sin (2\theta)}}.
Consider the regular and irregular nodal distributions on the irregular geometry shown in Figure 8. The profiles of absolute errors t = 1.0 for these two cases of nodal distribution are depicted in Figure 9, where N = 2646 and \alpha = 0.5. It can be seen that the higher accuracy is achieved for different types of node distribution, although the regular nodes are slightly better than irregular nodes.
A spatio-temporal meshless method is developed for the numerical solution of the TFPDEs with variable coefficients. The developed scheme can directly approximate the solutions of fractional partial differential equations within a space–time scale framework and is very simple mathematically, truly meshless, free of difference approximation, and computationally cost-effective. The numerical experiments further demonstrate the efficiency, precision, and convergence of the proposed STMM for the TFPDEs with variable coefficients.
The non-local nature of fractional derivatives causes the solution's behavior to be influenced by the entire temporal or spatial domain, making numerical solutions inherently challenging. The introduction of variable coefficients further increases the problem's nonlinearity, rendering traditional numerical methods cumbersome and accuracy-limited. The method presented in this paper can efficiently approximate the numerical solution for such problems in a straightforward manner without the need for specialized time discretization techniques, making it applicable to various types of variable coefficient fractional differential equations. It provides a robust tool for solving time fractional PDEs with variable coefficients, thus advancing the field and offering new possibilities for tackling complex real-world problems in engineering and science.
The present scheme can be extended to other fractional partial differential equations, such as the time-space fractional PDEs and complicated high-order and high-dimensional problems. Moreover, the study on local STMM is conducive to the solution of long-time and large-scale dynamic problems. It should be pointed out that the shape parameter in the STMM has a certain influence on the accuracy of the numerical solution, and thus the determination technique of the optimal shape parameter still needs to be discussed in depth. Otherwise, the new ST-RBFs should be studied in order to avert or greatly reduce the influence of shape parameters.
Xiangyun Qiu: Conceptualization, Methodology, Simulation analysis, Writing–original draft; Xingxing Yue: Methodology, Project administration, Supervision. Both authors have read and approved the final version of the manuscript for publication.
This work was supported by National Key R & D Program of China (Grant No. 2023YFB2503900).
The authors declare no competing financial interests.
[1] |
A. Talebpour, H. S. Mahmassani, Influence of connected and autonomous vehicles on traffic flow stability and throughput, Transp. Res. C Emerg. Tech., 71 (2016), 143–163. https://doi.org/10.1016/j.trc.2016.07.007 doi: 10.1016/j.trc.2016.07.007
![]() |
[2] |
P. Bansal, K. M. Kockelman, Forecasting americans' long-term adoption of connected and autonomous vehicle technologies, Transp. Res. A Pol., 95 (2017), 49–63. https://doi.org/10.1016/j.tra.2016.10.013 doi: 10.1016/j.tra.2016.10.013
![]() |
[3] |
G. N. Bifulco, L. Pariota, F. Simonelli, R. Di Pace, Development and testing of a fully adaptive cruise control system, Transp. Res. C Emerg. Tech., 29 (2013), 156–170. https://doi.org/10.1016/j.trc.2011.07.001 doi: 10.1016/j.trc.2011.07.001
![]() |
[4] |
J. Rios-Torres, A. A. Malikopoulos, A survey on the coordination of connected and automated vehicles at intersections and merging at highway on-ramps, IEEE Trans. Intell. Transp. Syst., 18 (2016), 1066–1077. https://doi.org/10.1109/TITS.2016.2600504 doi: 10.1109/TITS.2016.2600504
![]() |
[5] |
D. Milakis, B. Van Arem, B. Van Wee, Policy and society related implications of automated driving: A review of literature and directions for future research, J. Intell. Transp. Syst., 21 (2017), 324–348. https://doi.org/10.1080/15472450.2017.1291351 doi: 10.1080/15472450.2017.1291351
![]() |
[6] |
K. C. Dey, L. Yan, X. Wang, Y. Wang, H. Shen, M. Chowdhury, et al., A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC), IEEE Trans. Intell. Transp. Syst., 17 (2015), 491–509. https://doi.org/10.1109/TITS.2015.2483063 doi: 10.1109/TITS.2015.2483063
![]() |
[7] | Z. Wang, G. Wu, M. J. Barth, A review on cooperative adaptive cruise control (CACC) systems: Architectures, controls, and applications, in 2018 21st International Conference on Intelligent Transportation Systems (ITSC), IEEE, (2018), 2884–2891. https://doi.org/10.1109/ITSC.2018.8569947 |
[8] |
A. Kesting, M. Treiber, M. Schönhof, D. Helbing, Adaptive cruise control design for active congestion avoidance, Transp. Res. C Emerg. Tech., 16 (2008), 668–683. https://doi.org/10.1016/j.trc.2007.12.004 doi: 10.1016/j.trc.2007.12.004
![]() |
[9] |
D. Ngoduy, Analytical studies on the instabilities of heterogeneous intelligent traffic flow, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2699–2706. https://doi.org/10.1016/j.cnsns.2013.02.018 doi: 10.1016/j.cnsns.2013.02.018
![]() |
[10] | J. A. Ward, Heterogeneity, Lane-changing and Instability in Traffic: A Mathematical Approach, Ph.D thesis, University of Bristol Bristol, 2009. |
[11] |
I. A. Ntousakis, I. K. Nikolos, M. Papageorgiou, On microscopic modelling of adaptive cruise control systems, Transp. Res. Procedia, 6 (2015), 111–127. https://doi.org/10.1016/j.trpro.2015.03.010 doi: 10.1016/j.trpro.2015.03.010
![]() |
[12] |
D. Chen, S. Ahn, M. Chitturi, D. A. Noyce, Towards vehicle automation: Roadway capacity formulation for traffic mixed with regular and automated vehicles, Transp. Res. B, 100 (2017), 196–221. https://doi.org/10.1016/j.trb.2017.01.017 doi: 10.1016/j.trb.2017.01.017
![]() |
[13] |
H. Liu, X. D. Kan, S. E. Shladover, X. Y. Lu, R. E. Ferlis, Modeling impacts of cooperative adaptive cruise control on mixed traffic flow in multi-lane freeway facilities, Transp. Res. C Emerg. Tech., 95 (2018), 261–279. https://doi.org/10.1016/j.trc.2018.07.027 doi: 10.1016/j.trc.2018.07.027
![]() |
[14] |
D. F. Xie, X. M. Zhao, Z. He, Heterogeneous traffic mixing regular and connected vehicles: Modeling and stabilization, IEEE Trans. Intell. Transp. Syst., 20 (2018), 2060–2071. https://doi.org/10.1109/TITS.2018.2857465 doi: 10.1109/TITS.2018.2857465
![]() |
[15] |
H. Wang, Y. Qin, W. Wang, J. Chen, Stability of CACC-manual heterogeneous vehicular flow with partial CACC performance degrading, Transp. B Transp. Dyn., 7 (2019), 788–813. https://doi.org/10.1080/21680566.2018.1517058 doi: 10.1080/21680566.2018.1517058
![]() |
[16] |
Z. Yao, R. Hu, Y. Wang, Y. Jiang, B. Ran, Y. Chen, Stability analysis and the fundamental diagram for mixed connected automated and human-driven vehicles, Phys. A Stat. Mech. Its Appl., 533 (2019), 121931. https://doi.org/10.1016/j.physa.2019.121931 doi: 10.1016/j.physa.2019.121931
![]() |
[17] |
Z. Yao, R. Hu, Y. Jiang, T. Xu, Stability and safety evaluation of mixed traffic flow with connected automated vehicles on expressways, J. Saf. Res., 75 (2020), 262–274. https://doi.org/10.1016/j.jsr.2020.09.012 doi: 10.1016/j.jsr.2020.09.012
![]() |
[18] |
M. Shang, R. E. Stern, Impacts of commercially available adaptive cruise control vehicles on highway stability and throughput, Transp. Res. C Emerg. Tech., 122 (2021), 102897. https://doi.org/10.1016/j.trc.2020.102897 doi: 10.1016/j.trc.2020.102897
![]() |
[19] |
Z. Yao, T. Xu, Y. Jiang, R. Hu, Linear stability analysis of heterogeneous traffic flow considering degradations of connected automated vehicles and reaction time, Phys. A Stat. Mech. Its Appl., 561 (2021), 125218. https://doi.org/10.1016/j.physa.2020.125218 doi: 10.1016/j.physa.2020.125218
![]() |
[20] |
R. Luo, Q. Gu, T. Xu, H. Hao, Z. Yao, Analysis of linear internal stability for mixed traffic flow of connected and automated vehicles considering multiple influencing factors, Phys. A Stat. Mech. Its Appl., 597 (2022), 127211. https://doi.org/10.1016/j.physa.2022.127211 doi: 10.1016/j.physa.2022.127211
![]() |
[21] |
Y. C. Hung, K. Zhang, Impact of cooperative adaptive cruise control on traffic stability, Trans. Res. Rec., 2022 (2022). https://doi.org/10.1177/03611981221094822 doi: 10.1177/03611981221094822
![]() |
[22] |
D. Liu, B. Besselink, S. Baldi, W. Yu, H. L. Trentelman, Output-feedback design of longitudinal platooning with adaptive disturbance decoupling, IEEE Control Syst. Lett., 6 (2022), 3104–3109. https://doi.org/10.1109/LCSYS.2022.3181002 doi: 10.1109/LCSYS.2022.3181002
![]() |
[23] |
D. Liu, B. Besselink, S. Baldi, W. Yu, H. L. Trentelman, On structural and safety properties of head-to-tail string stability in mixed platoons, IEEE Trans. Intell. Transp. Syst., 2022 (2022), 1–13. https://doi.org/10.1109/TITS.2022.3151929 doi: 10.1109/TITS.2022.3151929
![]() |
[24] |
Z. Yao, Q. Gu, Y. Jiang, B. Ran, Fundamental diagram and stability of mixed traffic flow considering platoon size and intensity of connected automated vehicles, Phys. A Stat. Mech. Its Appl., 604 (2022), 127857. https://doi.org/10.1016/j.physa.2022.127857 doi: 10.1016/j.physa.2022.127857
![]() |
[25] | FHWA, The next generation simulation (NGSIM), 2008. Available from: http://www.ngsim.fhwa.dot.gov/ |
[26] |
V. Punzo, M. T. Borzacchiello, B. Ciuffo, On the assessment of vehicle trajectory data accuracy and application to the next generation simulation (NGSIM) program data, Transp. Res. C Emerg. Tech., 19 (2011), 1243–1262. https://doi.org/10.1016/j.trc.2010.12.007 doi: 10.1016/j.trc.2010.12.007
![]() |
[27] | M. Montanino, V. Punzo, Reconstructed NGSIM I80-1. COST ACTION TU0903 - MULTITUDE, 2013. Available from: http://www.multitude-project.eu/exchange/101.html |
[28] |
M. Montanino, V. Punzo, Trajectory data reconstruction and simulation-based validation against macroscopic traffic patterns, Transp. Res. B, 80 (2015), 82–106. https://doi.org/10.1016/j.trb.2015.06.010 doi: 10.1016/j.trb.2015.06.010
![]() |
[29] |
M. Saifuzzaman, Z. Zheng, Incorporating human-factors in car-following models: A review of recent developments and research needs, Transp. Res. C Emerg. Tech., 48 (2014), 379–403. https://doi.org/10.1016/j.trc.2014.09.008 doi: 10.1016/j.trc.2014.09.008
![]() |
[30] |
V. Milanés, S. E. Shladover, J. Spring, C. Nowakowski, H. Kawazoe, M. Nakamura, Cooperative adaptive cruise control in real traffic situations, IEEE Trans. Intell. Transp. Syst., 15 (2013), 296–305. https://doi.org/10.1109/TITS.2013.2278494 doi: 10.1109/TITS.2013.2278494
![]() |
[31] |
V. Milanés, S. E. Shladover, Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data, Transp. Res. C Emerg. Tech., 48 (2014), 285–300. https://doi.org/10.1016/j.trc.2014.09.001 doi: 10.1016/j.trc.2014.09.001
![]() |
[32] | C. Wu, A. Kreidieh, K. Parvate, E. Vinitsky, A. M. Bayen, Flow: Architecture and benchmarking for reinforcement learning in traffic control, preprint, arXiv: 1710.05465. |
[33] |
M. Zhu, X. Wang, Y. Wang, Human-like autonomous car-following model with deep reinforcement learning, Transp. Res. C Emerg. Tech., 97 (2018), 348–368. https://doi.org/10.1016/j.trc.2018.10.024 doi: 10.1016/j.trc.2018.10.024
![]() |
[34] |
M. Treiber, A. Hennecke, D. Helbing, Congested traffic states in empirical observations and microscopic simulations, Phys. Rev. E, 62 (2000), 1805–1824. https://doi.org/10.1103/PhysRevE.62.1805 doi: 10.1103/PhysRevE.62.1805
![]() |
[35] | M. Mitchell, An Introduction to Genetic Algorithms, MIT Press, 1998. |
[36] |
A. Kesting, M. Treiber, Calibrating car-following models by using trajectory data: Methodological study, Transp. Res. Rec., 2088 (2008), 148–156. https://doi.org/10.3141/2088-16 doi: 10.3141/2088-16
![]() |
[37] |
M. Treiber, A. Kesting, D. Helbing, Delays, inaccuracies and anticipation in microscopic traffic models, Phys. A Stat. Mech. Its Appl., 360 (2006), 71–88. https://doi.org/10.1016/j.physa.2005.05.001 doi: 10.1016/j.physa.2005.05.001
![]() |
[38] |
A. Kesting, M. Treiber, D. Helbing, Enhanced intelligent driver model to access the impact of driving strategies on traffic capacity, Philos. Trans. R. Soc. A, 368 (2010), 4585–4605. https://doi.org/10.1098/rsta.2010.0084 doi: 10.1098/rsta.2010.0084
![]() |
[39] |
M. Saifuzzaman, Z. Zheng, M. M. Haque, S. Washington, Revisiting the task–capability interface model for incorporating human factors into car-following models, Transp. Res. B, 82 (2015), 1–19. https://doi.org/10.1016/j.trb.2015.09.011 doi: 10.1016/j.trb.2015.09.011
![]() |
[40] |
S. E. Shladover, D. Su, X. Y. Lu, Impacts of cooperative adaptive cruise control on freeway traffic flow, Transp. Res. Rec., 2324 (2012), 63–70. https://doi.org/10.3141/2324-08 doi: 10.3141/2324-08
![]() |
[41] |
R. E. Wilson, J. A. Ward, Car-following models: Fifty years of linear stability analysis—A mathematical perspective, Transp. Plan. Technol., 34 (2011), 3–18. https://doi.org/10.1080/03081060.2011.530826 doi: 10.1080/03081060.2011.530826
![]() |
[42] | M. Treiber, A. Kesting, Traffic flow dynamics, in Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag, Berlin Heidelberg, 2013,983–1000. |
[43] |
X. Zhang, D. F. Jarrett, Stability analysis of the classical car-following model, Transp. Res. Part B Methodol., 31 (1997), 441–462. https://doi.org/10.1016/S0191-2615(97)00006-4 doi: 10.1016/S0191-2615(97)00006-4
![]() |
[44] |
G. Orosz, R. E. Wilson, G. Stépán, Traffic jams: dynamics and control, Phil. Trans. R. Soc. A, 368 (2010), 4455–4479. https://doi.org/10.1098/rsta.2010.0205 doi: 10.1098/rsta.2010.0205
![]() |
[45] |
P. Y. Li, A. Shrivastava, Traffic flow stability induced by constant time headway policy for adaptive cruise control vehicles, Transp. Res. C Emerg. Tech., 10 (2002), 275–301. https://doi.org/10.1016/S0968-090X(02)00004-9 doi: 10.1016/S0968-090X(02)00004-9
![]() |
[46] | A. R. Kreidieh, C. Wu, A. M. Bayen, Dissipating stop-and-go waves in closed and open networks via deep reinforcement learning, in 2018 21st International Conference on Intelligent Transportation Systems (ITSC), IEEE, (2018), 1475–1480. https://doi.org/10.1109/ITSC.2018.8569485 |
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t | Method in Ref. [50] | Present method | ||
L∞-error | L2-error | L∞-error | L2-error | |
0.1 | 6.086e-02 | 2.613e-01 | 1.709e-04 | 1.090e-04 |
0.5 | 2.958e-02 | 1.277e-01 | 2.155e-03 | 1.559e-03 |
1.0 | 2.114e-02 | 9.134e-02 | 6.688e-03 | 4.904e-03 |
Fractional orders | Condition numbers | |
With multiple scale technique | Without multiple scale technique | |
0.2 | 1.029e+09 | 1.264e+09 |
0.5 | 1.036e+09 | 1.265e+09 |
0.8 | 1.004e+09 | 1.234e+09 |
{N_g} | 4 | 8 | 12 | 16 | 20 |
L∞-error | 1.192e-02 | 6.532e-03 | 4.664e-03 | 3.688e-03 | 3.097e-03 |
RMSE | 4.764e-03 | 2.548e-03 | 1.758e-03 | 1.359e-03 | 1.108e-03 |
N | 270 | 1540 | 4560 | 10080 | 18850 |
L∞-error | 4.908 e-02 | 3.708e-02 | 3.159e-02 | 2.758e-02 | 2.442e-02 |
RMSE | 3.703e-03 | 2.190e-03 | 1.675 e-03 | 1.422e-03 | 1.278e-03 |
c | α=0.2 | α=0.6 | ||
L∞-error | RMSE | L∞-error | RMSE | |
Hardy [51] | 4.629e-04 | 2.629e-04 | 5.101e-03 | 3.685e-03 |
Fasshauer [52] | 2.177e-04 | 1.129e-04 | 5.014e-03 | 3.634e-03 |
Frank [53] | 1.809e-04 | 1.018e-04 | 5.022e-03 | 3.640e-03 |
t | Method in Ref. [50] | Present method | ||
L∞-error | L2-error | L∞-error | L2-error | |
0.1 | 6.086e-02 | 2.613e-01 | 1.709e-04 | 1.090e-04 |
0.5 | 2.958e-02 | 1.277e-01 | 2.155e-03 | 1.559e-03 |
1.0 | 2.114e-02 | 9.134e-02 | 6.688e-03 | 4.904e-03 |
Fractional orders | Condition numbers | |
With multiple scale technique | Without multiple scale technique | |
0.2 | 1.029e+09 | 1.264e+09 |
0.5 | 1.036e+09 | 1.265e+09 |
0.8 | 1.004e+09 | 1.234e+09 |
{N_g} | 4 | 8 | 12 | 16 | 20 |
L∞-error | 1.192e-02 | 6.532e-03 | 4.664e-03 | 3.688e-03 | 3.097e-03 |
RMSE | 4.764e-03 | 2.548e-03 | 1.758e-03 | 1.359e-03 | 1.108e-03 |
N | 270 | 1540 | 4560 | 10080 | 18850 |
L∞-error | 4.908 e-02 | 3.708e-02 | 3.159e-02 | 2.758e-02 | 2.442e-02 |
RMSE | 3.703e-03 | 2.190e-03 | 1.675 e-03 | 1.422e-03 | 1.278e-03 |