Solving the Payne-Whitham traffic flow model as a hyperbolic system of conservation laws with relaxation

Solving the Payne-Whitham traffic flow model as a hyperbolic system of conservation laws with relaxation W.L. Jin and H.M. Zhang August 3 Abstract: In this paper we study the Payne-Whitham (PW) model as a hyperbolic system of conservation laws with relaxation. After studying the Riemann problem for the homogeneous version of the PW model we introduce three first-order numerical solution methods for solving the system. In these methods the homogeneous part of the PW model is approximated by Godunovtype difference equations and different treatments of the source term are used. Numerical results show that solutions of the PW model with these methods are close to those of the LWR model when the PW model is stable and that the PW model can simulate cluster effect in traffic when it is unstable. The PW model is also studied for roadways with inhomogeneities. Keywords: The Payne-Whitham (PW) model hyperbolic system of conservation laws with relaxation Riemann problem Godunov s method. Introduction Continuum models of vehicular traffic on long crowded roads have been studied and applied in simulation for decades. In these models traffic conditions at a location x and time t are measured by density ρ travel speed v and flow-rate q for which a basic relationship is that q = ρv at any location and any time. Traffic conservation a principle governing the evolution of traffic conditions can be described by a partial differential equation ρ t + q x =. () In the first continuum model by Lighthill and Whitham (955) and Richards (956) the LWR model traffic is restricted to equilibrium states. That is in the LWR model q observes the equilibrium relation q = f (ρ) which is also known as the fundamental diagram of traffic flow. Correspondingly we have the speed-density relationship v = v (ρ) f (ρ)/ρ. Hence the LWR model can be written as ρ t + (ρv (ρ)) x =. () Generally the equilibrium velocity is assumed to be decreasing with respect to density; i.e. v (ρ) < ; the fundamental diagram is concave; i.e. f (ρ) <. University of California Davis 9566

In reality however traffic is generally observed in nonequilibrium. Nonequilibrium traffic flow models are therefore suggested. Among them the Payne-Whitham (PW) model (Payne 97; Whitham 974) is perhaps the best known. In the PW model besides equation () a second equation which defines traffic acceleration is introduced v t + vv x + c ρ ρ x = v (ρ) v (3) τ where the constant c > is the traffic sound speed and τ the relaxation time. With q = ρv the PW model can be written as ( ρ q which is in the conservative form ) t + ( ) q q ρ + c ρ x = ( f (ρ) q τ ) (4) u t + f(u) x = s(u) (5) where u = (ρ q) is the state variable. Thus the PW model is a hyperbolic system of conservation laws with relaxation in the sense of Whitham (959; 974) and Liu (987). Schochet (988) has shown that as τ the system (4) admits the limit ρ t + (ρv ) x = νρ xx which is the viscous version of the LWR model. Therefore the LWR model is called the degenerate system or subsystem of the PW model. The system (4) has two wave speeds called frozen characteristic speeds by Pember (993) λ (ρ v) = v c. Since v and c > λ < λ and λ >. Besides the PW model has a sub-characteristic λ (ρ) = v (ρ) + ρv (ρ) which is the wave speed of the LWR model can be positive or negative and smaller than λ. Whitham (974) showed that the stability condition for the linearized system with a relaxation term is λ < λ < λ. (6) Liu (987) showed that if the condition (6) is always satisfied then the corresponding PW model is stable under small perturbations and the time-asymptotic solutions of system (4) are completely determined by the equilibrium LWR model. Chen Levermore and Liu (994) showed that if condition (6) is satisfied then solutions of the system tend to solutions of the equilibrium equations as the relaxation time tends to zero. Note that since λ < λ for the PW model instability can only occur when λ < λ is violated (Zhang 999). For solving the PW model numerically Leo and Pretty (99) applied Roe s flux difference splitting algorithm in which an approximate Riemann solver is used. In this paper we will study the PW model as a hyperbolic system of conservation laws with relaxation. In our study the Riemann problem for the homogeneous version of the PW model is solved exactly. Based on this we apply the Godunov methods which are well-developed in the area of computational fluid dynamics for finding the numerical solutions of the PW model. These methods can also be directly applied for the PW model of traffic on roadways with inhomogeneities such as lane-drops.

The remaining part of this paper is organized as follows. In Section we analytically solve the homogeneous version of the PW model. In Section 3 we study several different Godunov methods for the PW model. In Section 4 we present numerical results for the PW model with stable or unstable initial conditions and for a roadway with lane-drops. In Section 5 we conclude this paper with our findings of the numerical methods and the properties of the PW model. Solving the homogeneous version of the PW model The homogeneous version of the PW model can be written as ( ) ( ) ρ q + q q ρ + c ρ = (7) or t x u t + f(u) x = (8) which is ( a hyperbolic system ) of conservation laws. For (8) the Jacobian matrix can be written as f = q + c ρ q for which the two real distinct eigenvalues are λ = v c and the ρ corresponding right eigenvectors are r = ( λ ) r = ( λ ). Furthermore the system (8) is genuinely nonlinear since λ i r i = c ρ. Discussions of general hyperbolic systems of conservation laws can be found in (Smoller 983). Here we follow discussions by Zhang (998 999 ) which are carried out for a model similar to the PW model.. The Riemann problem The Riemann problem for (8) around x = has the following jump initial conditions: u(x t = ) = { ul = ( q l ) if x < u r = ( q r ) if x >. (9) This problem is solved by shock waves and/or rarefaction waves. We begin by studying shock waves. As we know there are two distinct types of shock waves for (8); namely -shocks and -shocks. Lax s (97) entropy condition states that -shocks satisfy and -shocks satisfy λ (u r ) < s < λ (u l ) s < λ (u r ) () λ (u r ) < s < λ (u l ) s > λ (u l ). () 3

In both () and () the shock wave speed s is determined by the Rankine-Hugoniot jump condition These two equations yield s[ρ] = [q] [ ] q s[q] = ρ + c ρ. For -shocks the first inequality of () yields q r = ±c ρr. () or q r c < s = q r < q l c q r q r c < < q l q r c hence q l q / r c < < q l q / r c. Since c > we obtain c > q l q r > c >. Hence < q l > q r. (3) Moreover s λ (u r ) = q r q r c = q r q l ( ) c < c < ; i.e. if u l and u r satisfy (3) then they also satisfy the second part of the entropy condition (). Therefore the states which can be connected to u l by a -shock on the right must lie on the curve (denoted by S ) S : q ρ = c ρ ρρl s (ρ; u l ) ρ > q ρ < q l. (4) 4

Finally by direct calculations we find ds dρ = c (ρ + ) ρ ρ < d s dρ = c (ρ + 3 ) 4ρ ρ > ; i.e. S is decreasing and convex. With similar analyses we obtain the curve (denoted by S ) consisting of those states which can be connected to u l by a -shock on the right S : q ρ = c ρ ρρl s (ρ; u l ) ρ < q ρ < q l (5) and ds dρ = c (ρ + ) ρ ρ > d s dρ = c (ρ + 3 ) 4ρ ρ <. Besides shock wave solutions the Riemann problem for (8) admits two families of rarefaction waves which are continuous in the form u(x t) = u(x/t) and satisfy that λ i (u(x/t)) increases as x/t increases. Let ξ = x/t u(x t) = u(ξ) satisfies the following ordinary differential equation or ξu ξ + fu ξ = ( f ξi)u ξ =. If u ξ then ξ is the eigenvalue for f and u ξ = (ρ ξ q ξ ) is the corresponding eigenvector. Related to the two real and distinct eigenvalues λ of f there exist two families of rarefaction waves: -rarefaction waves and -rarefaction waves. For -rarefaction waves the eigenvector u ξ satisfies ( λ q + c ρ q ρ λ )( ρξ q ξ ) = ( which yields λ ρ ξ + q ξ =. Since ρ ξ we obtain q ξ /ρ ξ = λ. Hence dq dρ = λ = q ρ c. Solving this equation we obtain the -rarefaction wave curve R : q ρ = c ln(ρ/ ) r (ρ; u l ). (6) Here R is the curve consisting of states which can be connected to u l by a -rarefaction wave on the right. Since λ (u) > λ (u l ) the states on this curve must satisfy q/ρ > q l / and from (6) we have ρ <. We can also find dr dρ = c ρ < and d r dρ = c ρ >. 5 )

and Similarly we obtain the -rarefaction wave curve R : q ρ = c ln(ρ/ ) r (ρ; u l ) ρ > q ρ > q l (7) dr dρ = c ρ > and d r dρ = c ρ <. The four basic types of wave solutions are depicted in Figure. From the figure we can see that the state space is partitioned into four regions and when u r locates in each of these regions the Riemann problem is solved by a wave combining two basic waves. Totally therefore there are eight types of wave solutions to the Riemann problem including -shock waves (S ) -shock waves (S ) -rarefaction waves (R ) -rarefaction waves (R ) S -S (left S and right S ) waves R -R (left R and right R ) waves R -S (left R and right S ) waves and S -R (left S and right R ) waves. These wave solutions are self-similar in the sense that u(x/t) is constant when x/t=const. Note that the curves S and R have a second-order contact at u l ; i.e. their first two derivatives are equal at this point and the same is true for the two corresponding curves in the second characteristic family.. The time average of u at x = In this subsection we compute the averages of ρ and q at the boundary x = over a time interval ( t) from the Riemann solutions for (8) with initial data (u l u r ). These averages denoted by ρ and q are defined by ρ = t t ρ(x = t)dt and q = t t q(x = t)dt. As we know the Riemann problem has eight types of self-similar wave solutions. Therefore the time average (ρ q ) is equal to the boundary state u(x = t > ) which is a constant. For the eight types of wave solutions conditions of their existence and the corresponding boundary states (ρ q ) are discussed one by one as follows.. S If (u l u r ) satisfy S : q r = c ρr > q r < q l the Riemann problem for (8) with initial data (u l u r ) is solved by a -shock and the shock speed is s = q r. For S -type solutions (ρ q ) is summarized in Table. 6

s = qr q l ρ q S s > q l s q r Table : The boundary state for S -type solutions s = q r q l ρ q S s > q l Table : The boundary state for S -type solutions. S If (u l u r ) satisfy S : q r = c ρr < q r < q l the Riemann problem for (8) with initial data (u l u r ) is solved by a -shock. Since the wave speed s = q r > (ρ q ) for this type of solutions is the same as ( q l ) and shown in Table. 3. R If (u l u r ) satisfy R : q r = c ln( / ) < q r > q l the Riemann problem for (8) with initial data (u l u r ) is solved by a -rarefaction and the characteristic velocity is determined by the first eigenvalue of the system: λ (ρ q) = q ρ c. If λ (u l ) the boundary state (ρ q ) is the same as the left state i.e. ( q l ); similarly if λ (u r ) it is the right state; otherwise (ρ q ) solves the following equations: Solving them we obtain λ (ρ q ) = q q ρ ρ c = = c ln(ρ / ). ρ = e +q l/(c ) (8) q = c ρ. (9) For this type of solutions (ρ q ) is summarized in Table 3. 7

λ λ (u l ) ρ q q l R λ (u r ) q r o.w. given in (8) and (9) Table 3: The boundary state for R -type solutions λ ρ q R λ > q l Table 4: The boundary state for R -type solutions 4. R If (u l u r ) satisfy R : q r = c ln( / ) > q r > q l the Riemann problem for (8) with initial data (u l u r ) is solved by a -rarefaction. Since the characteristic velocity i.e. the second eigenvalue of the system is λ (ρ q) = q/ρ + c > (ρ q ) for this type of solutions is the same as the left state and shown in Table 4. 5. R -R If (u l u r ) satisfy and R : R : q m = c ln( / ) < q m > q l () q r q m = c ln( / ) > q r > q m () where u m = ( q m ) is an intermediate state the Riemann problem for (8) with initial data (u l u r ) is solved by a -rarefaction + -rarefaction wave. Adding () to () we find hence c ln ρ m = q r where < and < and = e q r/ρr q l / c q m = q l / c ln( / ). Note that (ρ q ) is only related to the -rarefaction wave connecting u l and u m. Therefore the boundary state is summarized in Table 5. 8

λ λ (u l ) ρ q q l R -R λ (u m ) q m o.w. given in (8) and (9) Table 5: The boundary state for R -R -type solutions λ λ (u l ) ρ q q l R -S λ (u m ) q m o.w. given in (8) and (9) Table 6: The boundary state for R -S -type solutions 6. R -S If (u l u r ) satisfy and R : q m = c ln( / ) < q m > q l S : q r q m = c ρr < q r < q m where u m = ( q m ) is an intermediate state the Riemann problem for (8) with initial data (u l u r ) is solved by a -rarefaction+-shock wave. These two equations yield c ln( / ) c ρr + q r = () where < <. From () we can obtain and then compute q m as follows q m = q l / c ln( / ). Hence (ρ q ) can be determined from the -rarefaction wave connecting u l and u m and is summarized in Table 6. 7. S -S If (u l u r ) satisfy S : q m = c ρm > q m < q l and S : q r q m = c ρr < q r < q m 9

s = qm q l ρ q S -S s > q l s q m Table 7: The boundary state for S -S -type solutions where u m = ( q m ) is an intermediate state the Riemann problem for (8) with initial data (u l u r ) is solved by a -shock+-shock wave. These two equations yield c ρm c ρr + q r = (3) where > and >. From (3) we can obtain and then compute q m as follows q m = q l c ρm. Hence (ρ q ) can be determined from the -shock connecting u l and u m and is summarized in Table 7. 8. S -R If (u l u r ) satisfy S : q m = c ρm > q m < q l and R : q r q m = c ln( / ) > q r > q m where u m = ( q m ) is an intermediate state the Riemann problem for (8) with initial data (u l u r ) is solved by a -shock+-rarefaction wave. These two equations yield c ρm c ln( / ) + q r = (4) where > >. From (4) we can obtain and then compute q m as follows q m = q l c ρl. Hence (ρ q ) can be determined from the -shock connecting u l and u m and is summarized in Table 8. Remarks: For R -S -type solutions () cannot be analytically solved to obtain. Note that is the solution of f 6 (ρ) = where (letting β = ( q r )/c ) f 6 (ρ) = ln(ρ/ ) ρ ρr ρ + β > ρ >

s = qm q l ρ q S -R s > q l s q m Table 8: The boundary state for S -R -type solutions whose derivative is f 6(ρ) = ρ + ρ + ρ ρ >. Similarly for S -S -type solutions is the solution of f 7 (ρ) = where f 7 (ρ) = ρ ρρl ρ ρr ρ + β ρ > and f 7(ρ) = ρ + ρ ρ + ρ + ρ ρ > ; and for S -R -type solutions is the solution of f 8 (ρ) = where f 8 (ρ) = ρ ρρl ln( /ρ) + β < ρ < and f 8(ρ) = ρ + ρ ρ + ρ >. To solve f i ( ) = (i = 6 7 8) therefore we can apply Newton s method; i.e. given kth candidate ρ k for (k + )th candidate can be computed as ρ k+ = ρ k f i(ρ k ) f i (ρ i = 6 7 or 8. k) Also for each type of solutions ρ k+ should be inside the given domain of. 3 Numerical solution methods for the PW model The Godunov (959) method is efficient for solving hyperbolic systems of conservation laws. For the homogeneous version of the PW model therefore we will use the Godunov-type difference equations to approximate it. The PW model however has one relaxation term. Thus different treatments of the effects of the source term result in different methods.

A general system (5) can be approximated by the following Godunov-type finite difference equation U j+ i = U j i t j j (f(u x i+/ ) f(ui / )) + t s(u) (5) in which U j i = (ρ j i qj i ) is the average of u(x t) over ith cell at time j t s(u) is the average of the source term over ((i /) x (i + /) x) (j t (j + ) t) and the boundary state U j i+/ = (ρ j i+/ q j i+/ ) is computed as shown in the preceding section from the Riemann problem for (8) with initial condition { ul if x x u i+/ (x t j ) = i+/ < (6) u r if x x i+/ where u l = U j i and u r = U j+ i. When the source term is approximated implicitly we call the numerical solution method as the implicit method. In this method the PW model can be approximated by ρ j+ i ρ j i k + q j i+/ q j i / h = (7) q j+ i q j i + k q j i+/ + c ρ j ρ j i+/ i+/ q j h i / ρ j i / c ρ j i / = f (ρ j+ i ) q j+ i (8) τ from which we can write the evolution equations for the PW model as ρ j+ i q j+ i = = ρ j i k h (q j i+/ q j i / ( + k τ ) qj i k q j i+/ h ρ j i+/ + k } τ f (ρ j+ i ). ) (9) + c ρ j i+/ q j i / ρ j i / c ρ j i / (3) Besides the implicit method an explicit treatment was used for the source term in (Pember 993) where u is approximated by the averages of boundary states and s(u) = j (s(ui / ) + s(u j i+/ )). Here the corresponding method is called the explicit method. A third treatment of the source term is used in the fractional step splitting method in which each time step t is split into three steps. In the first and third fractional steps a first-order implicit method is used to integrate ( ) ( ) ρ = q t f (ρ) q τ for time steps of t/. In the second step we solve the corresponding homogeneous system of (4) for a time step of t.

4 Numerical simulations of the PW model In this section we perform numerical simulations of the PW model for traffic on a ring road. The model parameters are chosen as follows (Kerner and Konhäuser 994; Herrman and Kerner 998): the unit length l =.8 km the relaxation time τ = 5 s the free flow speed v f = 5l/τ the jam density ρ j = 8 veh/km and the sound speed c =.48445l/τ; the equilibrium speed-density relationship v (ρ) = 5.46[( + exp{[ρ/ρ j.5]/.6}) 3.7 6 ]l/τ. The equilibrium functions v (ρ) and f (ρ) are shown in Figure in which the two critical densities ρ c =.73ρ j =3 veh/km and ρ c =.396ρ j =7 veh/km determined by λ = λ i.e. ρv (ρ)+c =. The length of the ring road is L = 8l and the time interval [ T ] where T = 5τ. In the numerical studies we partition the roadway into N cells and the time interval into K steps with N/K = /5; e.g. if N = and K = 5 the cell length is x = 8l km and the length of each time step t = τ s. Since λ v f + c < 7.5l/τ we find that the CFL (Courant Friedrichs and Lewy 98) condition number λ t/ x.9375 <. 4. Convergence rates for different methods when the PW model is stable In this subsection convergence rates for the three methods are computed by comparing solutions of the PW model when it is stable at time T = 5τ for two different number of cells into which the roadway is partitioned e.g. N and N. Denote solutions as (Ui N ) N i= for N cells and (U i N ) N i= for N cells respectively and define a difference vector (e N N ) N i= between these two solutions as e N N i = N (Ui + Ui N ) Ui N i = N (3) the relative error with respect to L - L - or L -norm can be computed as ɛ N N = e N N. (3) Finally the convergence rate r is obtained by comparing two relative errors: r = log ( ɛn N ). (33) ɛ4n N Here U can be ρ or v. Here we use the following global perturbation as the initial traffic conditions: ρ(x ) = ρ h + ρ sin(πx/l) x [ L] v(x ) = v (ρ h ) + v sin(πx/l) x [ L]. (34) Setting ρ h = veh/km ρ = 3 veh/km and v =. km/s the initial conditions are inside the stable region of the PW model. Setting N =64 8 56 5 and 4 the relative errors and convergence rates for the implicit method the explicit method and the fractional step splitting method are respectively given in Tables 9 and. From these tables we can see that the convergence rates in the L norm are close to one but lower in the L and L norms all rates increase with respect to N and the fractional step splitting method has the highest convergence rates. 3

ρ 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L.95e-.79.e-.88 6.e-.93 3.e- L.57e-.64.65e-.76 9.78e-.85 5.4e- L 5.48e-.37 4.4e-.56.88e-.73.74e- v 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L 4.e-5.78.45e-5.87.34e-5.93 7.4e-6 L 5.6e-5.6 3.65e-5.74.9e-5.84.e-5 L.3e-4.35.e-4.55 6.98e-5.7 4.5e-5 Table 9: Convergence rates for the implicit method ρ 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L.6e-.77.e-.87 6.64e-.93 3.49e- L.67e-.6.74e-.74.4e-.84 5.83e- L 5.59e-.34 4.4e-.53 3.5e-.7.88e- v 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L 4.46e-5.76.63e-5.86.45e-5.9 7.7e-6 L 5.8e-5.6 3.83e-5.7.3e-5.8.3e-5 L.3e-4.3.4e-4.5 7.6e-5.69 4.5e-5 Table : Convergence rates for the explicit method ρ 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L.8e-.85.e-.9 5.3e-.96.73e- L.43e-.7.5e-.8 8.58e-.89 4.64e- L 5.3e-.4 3.96e-.6.57e-.77.5e- v 8-64 Rate 56-8 Rate 5-56 Rate 4-5 L 3.9e-5.85.7e-5.9.5e-5.96 5.93e-6 L 5.3e-5.69 3.9e-5.8.89e-5.88.3e-5 L.4e-4.43 9.3e-5.6 6.4e-5.77 3.55e-5 Table : Convergence rates for the fractional step splitting method 4

4. Comparison of solutions by different methods Still using (34) as initial conditions and setting N = and K = 5 we obtain solutions of the PW model with the implicit method shown in Figure 3 from which we find that traffic approaches equilibrium after a sufficiently long time. Comparison of solutions at different times with different methods are shown in Figure 4 from which we can see that solutions by these methods are close to each other. For the same fundamental diagram initial ρ N and K we solve the LWR model and obtain similar solutions as those of the PW model as shown in Figure 5. The figure shows that when the PW model is stable it behaves similarly as the corresponding LWR model in large time. 4.3 Simulation of unstable traffic Unstable phenomena such as breakdowns and vehicle clusters can be often observed in traffic flow. The PW model is capable of simulating unstable traffic under unstable initial traffic conditions which can be seen from the following example. For initial conditions (34) when ρ h = 33 veh/km ρ = 3 veh/km and v =.l/τ λ < λ is violated. Therefore traffic is unstable. We solve the PW model under these initial conditions by using the implicit method with N = and K = 6. The solutions are shown in Figure 6. We can see from the figure that for unstable initial conditions a cluster traveling in a constant speed forms on the ring road. As we know clusters cannot be observed from solutions of the corresponding LWR model. Instability that results in cluster solutions therefore is an important property of the PW model that distinguishes it from the LWR model. Detailed discussions of the structure and properties of cluster solutions can be found in (Jin and Zhang ). 4.4 Simulation of traffic with lane-drops In this subsection we use the PW model to simulate traffic on a ring road with a bottleneck at [3l 4l]: a(x) = { x [3l 4l) [ 3l) or (4l 8l] where a(x) is the number of lanes. For the inhomogeneous roadway the PW model can be modified as ( ρ q ) t + ( ) q q ρ + c ρ x = ( ρv (ρ/a(x)) q τ ). (35) Since the only difference of this model from (4) is the source term the Godunov methods can be directly applied. For the following initial traffic conditions ρ(x ) = a(x)(ρ h + ρ sin πx L ) x [ L] v(x ) = v (ρ h ) + v sin πx L x [ L] (36) 5

in which ρ h = veh/km/lane ρ = 3 veh/km/lane and v =. km/s solutions of the PW model as well as those of the corresponding LWR model are shown in Figure 7. (For detailed discussions for solving the LWR model for inhomogeneous roadways please refer to (Jin and Zhang ).) From the figure we can see that for both models a congested region forms upstream to the bottleneck and traffic flow-rate approaches constant after a sufficient long time. However the constant flow-rate for the LWR model which is capacity flow of one lane is higher than that for the PW model. 5 Conclusions In this paper we studied first the Riemann problem of the homogeneous version of the PW model and then different numerical methods for solving the PW model. These methods combine a firstorder Godunov-type approximation of the homogeneous version of the system and different treatments of the source term. They can replace many ad-hoc numerical methods for the PW model that do not fully make use of the property that the PW model is a hyperbolic system of conservation laws with relaxation. From the numerical results we can see that when the PW model is stable its solutions are close to those of the LWR model and different numerical methods including the implicit method the explicit method and the fractional step splitting method produce similar results. We also show when the PW model is unstable it is capable of simulating cluster effects in traffic. The PW model can also be applied without major modifications for inhomogeneous roadways. References [] G.Q. Chen C.D. Levermore and T.P. Liu Hyperbolic conservation laws with stiff relaxation terms and entropy Communications on Pure and Applied Mathematics Vol. XLVII pp. 787-83 994. [] R. Courant K. Friedrichs and H. Lewy ber die partiellen differenzengleichungen der mathematischen physik Math. Ann. pp. 3-74 98. [3] S.K. Godunov A difference method for numerical calculations of discontinuous solutions of the equations of hydrodynamics Mat. Sb. 47 pp. 7-36 959. (In Russian.) [4] M. Herrmann and B.S. Kerner Local cluster effect in different traffic flow models Physica A 55 pp. 63-98 998. [5] W.L. Jin and H.M. Zhang The inhomogeneous kinematic wave traffic flow model as a resonant nonlinear system University of California at Davis submitted to Transportation Science. [6] W.L. Jin and H.M. Zhang The formation and structure of vehicle clusters in the Payne- Whitham traffic flow model submitted to Transportation Research B. [7] B.S. Kerner and P. Konhäuser Structure and parameters of clusters in traffic flow Physical Review E Vol. 5 No. pp. 54-83 994. 6

[8] P.D. Lax Hyperbolic systems of conservation laws and the mathematical theory of shock waves SIAM Philadelphia Pennsylvania 97. [9] C.J. Leo and R.L. Pretty Numerical simulation of macroscopic continuum traffic models Transportation Research B Vol. 6 No. 3 pp. 7-99. [] M.J. Lighthill and G.B. Whitham On kinematic waves: II. A theory of traffic flow on long crowded roads Proc. Royal Society Vol. 9 (78) of A pp. 37-345 955. [] T.P. Liu Quasi-linear hyperbolic systems Commun. Math. Phys. 68 pp. 4-7 979. [] T.P. Liu Hyperbolic conservation laws with relaxation Comm. Math. Phys. 8 pp. 53-75 987. [3] H.J. Payne Models of freeway traffic and control in Mathematical Models of Public Systems Vol. of Simulation Councils Proc. Ser. pp. 5-6 97. [4] R.B. Pember Numerical methods for hyperbolic conservation laws with stiff relaxation I. Spurious solutions Siam J. Applied Mathematics Vol. 53 No. 5 pp. 93-33 993. [5] P.I. Richards Shock waves on the highway Operations Research Vol. 4 pp. 4-5 956. [6] S. Schochet The instant response limit in Whitham s nonlinear traffic model: uniform wellposedness and global existence Asymptotic Analysis pp. 63-8 988. [7] J. Smoller Shock waves and reaction-diffusion equations New York Springer-Verlag 983. [8] G.B. Whitham Some comments on wave propagation and shock wave structure with application to magnetohydronamics Comm. Pure Appl. Math. Vol. XII pp. 3-58 959. [9] G.B. Whitham Linear and nonlinear waves John Wiley and Sons New York 974. [] H.M. Zhang A theory of nonequilibrium traffic flow Transportation Research B Vol. 3 No. 7 pp. 485-498 998. [] H.M. Zhang An analysis of the stability and wave properties of a new continuum theory Transportation Research B Vol. 33 No. 6 pp. 387-398 999. [] H.M. Zhang Structural properties of solutions arising from a non-equilibrium traffic flow theory Transportation Research B Vol. 34 pp. 583-63. [3] H.M. Zhang A finite difference model of nonequilibrium traffic flow Transportation Research B Vol. 35 No. 5 pp. 337-365. 7

q/ρ R R U l S S ρ Figure : Basic curves in the ρ q/ρ plane 8

5 4 v * (ρ) / (l / τ) 3 ρ c ρ c...3.4.5.6.7.8.9 ρ / ρ j.8 f * (ρ) / (ρ j l / τ).6.4. ρ ρ c c...3.4.5 ρ / ρ j.6.7.8.9 Figure : The Kerner-Konhäuser model of speed-density and flow-density relations 9

5 ρ / ρ j 5 v / (l / τ) 5 q / (ρ j l / τ) 45 45 45 4 4 4 35 35 35 3 3 3 t / τ 5 5 5 5 5 5 5 5 5 4 6 4 6 4 6... 4.4 4.6 4.8.45.5.55.6 Figure 3: Solutions of the PW model by using the implicit method: N = and K = 5

.3 τ.3 4 τ.3 33 τ.3 5 τ.... ρ / ρ j........ 5 4.7 4.7 4.7 v (l / τ) 4.5 4.6 4.6 4.6 4.5 4.5 4.5 q / (ρ j l / τ).65.55.4.56.54.5.5.48.46.56.54.5.5.48.46.56.54.5.5.48.46 Figure 4: Comparison of solutions of the PW model with different methods: thinner solid lines for the implicit method thinner dashed lines for the fractional splitting method and thicker dashed lines for the explicit method

.3 τ.3 4 τ.3 33 τ.3 5 τ.... ρ / ρ j........ 5 4.7 4.7 4.7 v (l / τ) 4.5 4.6 4.6 4.6 4.5 4.5 4.5 q / (ρ j l / τ).65.55.4.56.54.5.5.48.46.56.54.5.5.48.46.56.54.5.5.48.46 Figure 5: Comparison of solutions of the LWR and PW models on a homogeneous roadway: thicker lines for the LWR model and thinner lines for the PW model

t / τ 5 45 4 35 3 5 ρ / ρ j.5.4.3. 5 45 4 35 3 5 v / (l / τ) 4 3 5 45 4 35 3 5 q / (ρ j l / τ).6.5.4.3.. 5 4 6 5 4 6 5 4 6 t / τ 5 5 5...9 5.8 3.9 3.8 3.7 3.6 3.5 5 5.75.7.65 4 6.7 4 6 3.4 3.3 4 6.6 Figure 6: The formation of a cluster for initial condition (34) 3

t = τ t =4 τ t 3 =33 τ t 4 =5 τ ρ / ρ j.5.5.5.5 5 5 5 5 4 4 4 4 v (l / τ) 3 3 3 3.5.5.5.5 q / (ρ j l / τ).5.5.5.5 Figure 7: Comparison of solutions of the LWR and PW models on an inhomogeneous ring road: Solid lines for the LWR model and dashed lines for the PW model 4