A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway

Transcription

1 Manuscript A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway Peng Zhang, S.C. Wong and Chi-Wang Shu. Department of Civil Engineering, The University of Hong Kong, Hong Kong SAR, P.R. China. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, P.R.China. Division of Applied Mathematics, Brown University, Providence, RI 9, USA Abstract: As a new attempt to solve hyperbolic conservation laws with spatially varying flues, the weighted essentially non-oscillatory (WENO) method is applied to solve a multi-class traffic flow model for an inhomogeneous highway. The numerical scheme is based upon a modified equivalent system that is written in a standard hyperbolic conservation form. Numerical eamples, which include the difficult traffic signal control problem, are used to demonstrate the effectiveness of the WENO scheme. Keywords: non-strictly hyperbolic conservation laws; spatially varying flues; WENO reconstruction Introduction In this paper, we etend a multi-class Lighthill-Whitham-Richards traffic flow model [,,8] to deal with inhomogeneous road conditions. The variable road conditions are the number of lanes a() and the free flow (maimum) velocities {v l,f ()} m l= of m types of vehicles. Let l(,t) be the density per lane of the lth type, and let (,t) = l (,t) l= be the total density per lane. The velocity of the lth type of vehicles is a function of, which is denoted by v l (). Furthermore we assume that {v l } m l= are related by v l = b l ()v(), v () <, b l () = v l,f ()/v f, v f = ma ma (v l,f()). (.) l m The work described in this paper was jointly supported by grants from the Hong Kong Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7/E) and the National Natural Science Foundation of China (Project No. 4764). The research of the third author was supported by NSF grant DMS-745 and ARO grant W9NF Corresponding author. hhecwsc@hkucc.hku.hk.

2 Accordingly, the velocity differences between m vehicle types are reflected by the functions {b l ()} m l= and b l (). The model equations are acquired from the mass conservation of m types of vehicles, which read (a() l ) t + (a() l b l ()v()) =, l m. (.) We introduce the conservative solution variables u l = a() l, the vector u = (u,...,u m ) T, and the flu vector f = (f,...,f m ) T with f l = b l u l v(σu l /a). Accordingly, the model equations can be written as u t + f(u,θ()) =, (.) where the vector function θ() represents all inhomogeneous factors on the road, namely, θ() = (a(),b (),...,b m ()). In this traffic flow problem, each density l and the total density are bounded by a jam density jam, and thus u/a D, D = {u/a l, l =,...,m;σ m l= l jam }. (.4) Moreover, the function v() of (.) satisfies v() = v f, v( jam ) =. The study of this etended traffic flow system is significant both for practical application and theoretical interest. In real traffic, the drop or increase in traffic capacity that is reflected by θ() is frequent in many locations, such as on curves and slopes and near ramps and traffic accidents. In particular, by etension b l = b l (,t) can serve as a switch function in signal traffic or the like. However, it is very difficult to solve the eigen-polynomial of system (.) eplicitly. Moreover, the eistence of spatially varying flues poses significant difficulties for both analytical and numerical studies (see [-], [], [4-5], and [4-7] for relevant discussions). In this paper, some important features of the model are discussed under a modified equivalent system of (.), in which all of the components of θ are solution variables. Analytically, the hyperbolicity of the system is proven, and the wave-breaking patterns of the Riemann problem are predicted. We note that these descriptions are mostly based on the relevant studies in [4] and [8]. The maimum absolute value of all of the eigenvalues is estimated, which is an essential parameter in the proposed numerical schemes. We note that these eigenvalues cannot be eplicitly solved (see Section for this discussion). In developing the numerical schemes, it may not be efficient to apply the standard methods (e.g., TVD, RKDG, and WENO schemes) to system (.) directly due to its spatially varying flues. It

3 should be mentioned that a wave propagation method was recently developed that generated good numerical results in solving elastic waves in heterogeneous media, which constitute a system with spatially varying flues []. Nevertheless, this method is limited to problems in which an eigenvalue has a fied sign for all involved u and θ(), and requires that the eigenvalues can be solved eplicitly. Therefore, this method is not suitable for our problem, in which one of the eigenvalues changes sign, and all are implicit. In [4-7], Zhang et al. developed a numerical scheme to deal with spatially varying flues for the scalar case, which can also be etended to the vector case. With a modified equivalent system of (.), a component-wise WENO scheme that applies the La-Friedrichs numerical flu in the finite volume method and the flu splitting in the finite difference method are discussed in Section. As the modification gives rise to a standard hyperbolic conservation form, the scheme is theoretically sound, and thus gives good results. Although the Riemann problem of the system generates very complicated wave structures, the numerical results are in good agreement with the claimed wave patterns (Section 4.). For sharp changes of b l from their maima to minima, which describes the sharp braking of vehicles, the resultant strong discontinuity at the interface is well captured (Section 4.). We note that this is the first time that the WENO scheme has been applied to hyperbolic conservation laws with spatially varying flues. Similar applications to other problems may be possible, which is addressed in Section 5 with several concluding remarks. Hyperbolicity and wave structure of the model equations To study the hyperbolicity and solution structure of the system, we rewrite (.) in the standard conservation form, which means that we add the identity θ t = to (.). For convenience, we treat θ as a scalar in the following discussions, but the results are equally applicable to the vector case. Accordingly, (.) and the added identity can be viewed as the following equivalent (m+) (m+) system: U t + F =, (.) where U = (u,θ) T and F = (f(u,θ),) T. We note that the construction of equivalent forms of the conservation laws with spatially varying flues is widely applied, for eample in [,,5,5]. According to this standard conservation form, we study the hyperbolicity of the system. We write the Jacobian F U and the matri for solving the eigen-pairs as follows: f θ F U = f u, F U λi m+ = f u λi m, (.) λ where = (,...,) has m components, f θ = (f θ,...,f mθ ) T and I m is the unit matri. Let f θ f u λi m = P m (λ),

4 and then the eigen-polynomial of F U is F U λi m+ = λp m (λ). (.). Hyperbolicity of the model equations For the discussed system, it is not difficult to obtain [8] P m (λ) = Q(λ) m (v l λ), Q(λ) = + l= l= l v l v l λ. (.4) For u/a D, where D is the open domain that corresponds to D of (.4), we suppose that {v l } m l= are distinct, namely, v <... < v m, (.5) then it is easy to verify that, by (.4), sgn(p m (v l )) = ( ) l, sgn(p m (v + l= v l l )) =. (.6) By the intermediate value theorem, (.6) suggests m distinct real eigenvalues {λ} m l= of the Jacobian f u, which are separated by m velocities as follows: v + l= l v l < λ < v < λ <... < v l < λ l < v l <... < v m < λ m < v m. (.7) For u/a D, m real eigenvalues of f u are also ensured, and some inequalities of (.7) change to equalities. The hyperbolicity of this latter case is rather complicated, but to a degree can be viewed as the limiting case of the former (see [8] for more details). For simplicity, this case is ecluded from this section and from Section.. By (.), therefore, we conclude that the Jacobian F u has m + real eigenvalues {λ l } m l= and that λ =, and thus that (.) is a hyperbolic system. Furthermore, by (.7) it is obvious that these m + eigenvalues are distinct if and only if λ λ =, in which case (.) is strictly hyperbolic. As is shown in the following, it is possible to have λ = λ =, in which case system (.) is generally non-strictly hyperbolic. Substituting λ with λ = in (.4) and with all v l being given by (.), we have Q() = q () v(), sgn(p m()) = sgn(q ()), (.8) where q() = v(). Usually, q() is supposed to be strictly concave [,,8], q () <. Assume that q( ) is the maimum of q(), such that q ( ) =, q () > for <, q () < for >, 4

5 where is the critical density, as it is called in many traffic flow models. Accordingly, (.8) and (.6) give the following conclusion: λ >, if <, =, if =, <, if >. This is again derived by the intermediate value theorem. Suppose that (r T l,z l) T is an eigenvector that corresponds to λ l, where r l is a m vector and z l is a scalar, then we have λ = λ l : (F U λ l I m+ ) r l z l f θ = f u λ l I m λ l r l z l = (f u λ l I m )r l + z l f θ λ l z l (.9) =. This indicates that we can take z l to be zero and r l to be the eigenvector of f u that corresponds to λ l. Suppose that ( r T, z) T is an eigenvector that corresponds to λ =. Similarly, we have λ = : (F U λi m+ ) r z = f u f θ r z which implies that r is the solution of the algebraic equations = f u r + zf θ =, f u r = zf θ. (.) Denote by r = (r,...,r m ) a non-singular matri of right-eigenvectors of f u. The matri of the right-eigenvectors of F U then reads R r r. z If λ, then f u is non-singular. We choose z in (.), and thus r is uniquely determined as r = zf u f θ. In this case R is non-singular and R = r zf u f θ, R = z r r f u f θ / z If λ =, then f u is singular. Consequently, R must be singular. This is obvious for z =. For z, if Eq.(.) has a solution to r, then we would read rank(f u, zf θ ) = rank(f u ) = rank(f u λ I m ) = m, for which an identity is required. This is generally unlikely under the assumption that f θ. For the discussed traffic flow problem, the required identity can be arranged as l= f l, or a v l l=. b l b l l v v a, for λ =, or =. However, it is obvious that this condition does not hold for the traffic flow model. In summary, system (.) is strictly hyperbolic for λ, but non-strictly hyperbolic for λ =. 5

6 . Wave structure of the model equations We consider the Riemann problem u, if <, u = u m+, if >, θ L, if <, θ = θ R, if >, (.) where the epression for θ means that a() and b l () are discontinuous of at the interface, with θ L = (a L,b L,...,bL m ), and θr = (a R,b R,...,bR m ). The Riemann problem was solved eactly for a general scalar form of (.) (m = ) in [4]. For the case m =, λ and λ are eplicit, but the solution is very complicated. For m >, no λ l can be solved eplicitly in the discussed traffic flow problem, and thus it is very difficult to derive the analytical results [8]. However, some predictions of the wave structure are possible, because the wave breaking near the interface would be similar to that for the scalar case that is discussed in [4]. Moreover, the monotone changes across or within the waves other than the interface are well understood for the system in which θ() is constant [8], and these conclusions are still applicable in each of the regions D + = {(,t) >, t > } and D = {(,t) <, t > }. The claimed wave structure in this section serves as a comparison for the numerical results in Section. We call a wave that corresponds to the λ l - (or λ-) characteristic field the λ l - (or λ-) wave. In accordance with (.7), these {λ l } m l= -waves must be in the region D+ = {(,t) >, t > }, and the λ-wave can be easily verified as a contact at the interface =. See [-,8-] for detailed accounts of the basic properties of hyperbolic waves. Let u L = u(,t ) and u R = u(,t + ), which are, respectively, the left and right solution states that are adjacent to the interface, and we then have, by the Rankine-Hugoniot jump condition, f(u L,θ L ) = f(u R,θ R ). (.) As indicated in [4], the two states u L and u R should be also connected by the λ l characteristics. This argument implies that the propagation of the λ -characteristics may change the angle across the interface, and should satisfy λ (u L,θ L )λ (u R,θ R ). (.) Here, the equality means that λ (u L,θ L ) or λ (u R,θ R ) coincides with the contact λ =, in which case system (.) must be non-strictly hyperbolic. We have, by (.), a L b L l L l v(l ) = a R b R l R l v(r ), l =,...,m. Let α l = a L b L l /(ar b R l ) and α l = /α l, and the aforementioned equation is then equivalent to R l v(r ) = α l L l v(l ), l; R v( R ) = v( L ) α l L l, (.4) l= 6

7 where the second equation is obtained by the summation of the first over l. Similarly, we obtain L l v(l ) = α l R l v(r ), l; L v( L ) = v( R ) α l R l. (.5) In the discussed traffic flow problem, it is reasonable to assume that l= if α k > for some k, then α l l, (.6) which indicates a non-increase or non-decrease (α l or α l ) in traffic capacity for all types of vehicles. Otherwise, the problem would be unrealistic, because the capacity drops for some vehicle types but increases for others in the same location. Whether system (.) is strictly hyperbolic will be subject to (.), because θ() is only changeable across the interface. Let u be the left solution state of the λ -wave (Figs. and ), and we then have the following description. t λ wave λ wave t λ wave u L=u R u u λ wave L u u= u R u (b) Fig. Wave patterns that correspond to a strictly hyperbolic system (.): the λ -characteristics pass through the interface from left to right; (b) the λ -characteristics pass through the interface from right to left. () Suppose that λ (u L,θ L )λ (u R,θ R ) >, then system (.) is strictly hyperbolic and we would epect m + waves. In addition to the λ-contact at the interface, only one λ -wave is needed before the λ -wave. This λ -wave is either in D + or D for the consideration of the following three cases. If λ (u,θ L ) > and λ (u,θ R ), then it is natural to have u L = u and λ (u L,θ L ) >. As λ (u R,θ R ) >, it is inferred by (.9) that R <. Such a unique R and the components of u R are solvable by (.4) if and only if v( L ) α l L l < v( ) = q( ). (.7) l= 7

8 In this case we can say that the propagation of λ (u,θ L ) is able to pass through the interface and then change to λ (u R,θ R ), in which state u R forms the λ -wave with u in D +. Fig. shows this wave pattern, where a wave is represented simply by a radial. (b) If λ (u,θ L ) and λ (u,θ R ) <, then similarly we have u R = u and λ (u R,θ R ) <, and thus λ (u L,θ L ) < and L >. By (.5), L and { L l }m l= are uniquely solvable if and only if v( R ) α l R l < v( ) = q( ). (.8) l= In this case, we can say that the propagation of λ (u,θ R ) is able to pass through the interface and then change to λ (u L,θ L ), in which state u L forms the λ -wave with u in D. This wave pattern is shown in Fig.(b). (c) If λ (u,θ L ) > and λ (u,θ R ) <, then we have either u L = u or u R = u. Accordingly, the wave pattern is similar to that of () or (b), as is shown in Figs. and (b). We note that at least one of Eqs. (.4) (with u L = u ) and (.5) (with u R = u ) is solvable, because at least one of conditions (.7) (with u L = u ) and (.8) (with u R = u ) must hold under assumption (.6). λ wave u u L t = R λ rarefaction u wave λ λ rarefaction u L = t u R λ wave u λ wave (b) Fig. Wave patterns that correspond to a non-strictly hyperbolic system (.): the λ -characteristics of u = u are reflected from the interface, forming an etra λ -wave in D ; (b) the λ -characteristics of u = u are reflected from the interface, forming an etra λ -wave in D +. () Suppose that λ(u L,θ L )λ(u R,θ R ) =, then system (.) is non-strictly hyperbolic and we predict a total of m+ waves in the following. For all of the cases under consideration, we need two λ -waves that are separated by the interface. This wave structure is also characterized by a λ -rarefaction, which is either in D or in D +, and is distinguished by two wave patterns that are shown in Figs. and (b), respectively. 8

9 If λ (u,θ L ) >, λ (u,θ R ), and (.7) is not satisfied for u L = u so that (.4) is unsolvable (ecept possibly for R = ), then by the assumption we can only choose λ (u R,θ R ) =, namely R =. Moreover, by (.4) and (.5), we have v( L ) α l L l = v( ) = q( ), L v( L ) = v( ) α l R l. (.9) l= l= Comparing the first equation of (.9) with (.7), we can see that a maimal principle is applied. The second equation of (.9) is solvable, because it is implied that α l (α l ) for all l, and thus v( )Σ m l= α l R l v( ) = q( ). Otherwise, by (.6), we have α l for all l, and (.7) will be satisfied by u L = u. However, this contradicts our assumption. In this case, L is chosen such that λ(u L,θ L ), namely L. Therefore, the propagation of λ (u,θ L ) is unable to pass through the interface, and reflects backward so that u and u L form a λ -wave in D. Meanwhile, u R and u form another λ -wave that can be identified as a rarefaction (see Proposition.), because R = >. This wave pattern is shown in Fig.. (b) If λ (u,θ L ), λ (u,θ R ) <, and (.8) is not satisfied for u R = u so that (.5) is unsolvable (ecept possibly for L = ), then we have λ(u L,θ L ) = or L =. This case is parallel to (), and the wave structure is shown in Fig.(b), where two predicted λ -waves are also separated by the interface. In contrast to, here it is the λ -wave in D that is identified as a rarefaction (also see Proposition.), because > L =. (c) If λ (u,θ L ) < and λ (u,θ R ) >, which suggests that λ (u,θ) changes sign across the interface, then we have λ (u R,θ R ) = for α l. In this case, the wave structure is similar to that of (), and is also shown in Fig.. For α l, we have λ (u L,θ L ) =, and the wave pattern is similar to that of ()(b), as shown in Fig.(b). We have the trivial case of this for λ (u,θ L ) = or λ (u,θ R ) =. For the scalar case m =, in which the λ l -waves disappear for l > and u = u m+ becomes known initially, the wave patterns as shown in Figs. and are identical to those that were solved eactly and proved to be unique in [4]. Furthermore, the conclusions for all of the λ l -waves in [8] can be directly applied in this Riemann problem, because each of the waves is either in D or in D +, where θ() is constant. In general, all of these λ l -characteristic fields are genuinely nonlinear, and thus they correspond to either shocks or rarefactions. Let u and u + be the left and right states of a certain λ l -wave, respectively, and, supposing that the function v() is concave, namely, v (), then the following two propositions in [8] may be cited. Proposition. A λ l -shock is characterized by the following: < +, k < + k for k l and k > + k for k < l. 9

10 Proposition. A λ l -rarefaction is characterized by the following: > +, k > + k for k l and k < + k for k < l. Moreover, all of the functions (θ) and k(θ) are monotone in the rarefaction fan, where θ = /t. It was also indicated in [8] that the two propositions should be applicable to a generally given function v(). Numerical schemes For the numerical approimation of the model equations, we follow the basic ideas that are stated in Section. We first apply the modified equivalent system (.), which is written in two parts as follows: u t + f(u,θ) =, θ t + o =, (.) where the flu o =. We note that θ is a solution vector for temporal evolution in the computation. Let ˆf and ô be the two numerical flu functions that correspond to the flues f and o, respectively, of (.). Then, a standard conservative scheme of (.) reads: du i dt + ( ˆf i+/ ˆf i / ) =, dθ i dt + (ô i+/ ô i / ) =. (.) In the following, the numerical flues ˆf i+/ and ô i+/ are reconstructed by the WENO method through the La-Friedrichs flu splitting method. As the characteristic decomposition of the system is impossible due to the implicitness of all of the λ l, or the singularity of the matri R that contains the right eigenvectors (see discussions of (.7)-(.)), the component-wise WENO reconstruction is adopted. For a detailed account of the WENO scheme, see [9,6,7] or [,8], which are more relevant to the discussed model.. Component-wise finite volume (FV) WENO scheme The fifth-order accurate WENO FV scheme applies the cell averages {(u j,θ j )} i+ j=i to reconstruct (u i+/,θ i+/ ), which are cell boundary values of i+/ on the left-hand side. With {(u j,θ j )} i+ j=i, (u + i+/,θ+ i+/ ) are similarly constructed, and are cell boundary values of i+/ on the right-hand side. Thus, we use the La-Friedrichs numerical flues as follows: ˆf i+/ = (f(u i+/ ) + f(u+ i+/ ) α(u+ i+/ u i+/ ))

11 ô i+/ = α(θ+ i+/ θ i+/ ). (.) Eqs. (.) and (.) constitute a complete semi-discretized scheme.. Component-wise finite difference (FD) WENO scheme In this scheme, we first use the La-Friedrichs flu splitting as follows: f + = (f(u,θ) + αu), f = (f(u,θ) αu); o + = αθ, o = αθ, (.4) where (u,θ) = (u j,θ j ), j = i,...,j +. With (f + j,o+ j ) = (f+ (u j,θ j ),o + (θ j )) v j for j = i,...,i +, we proceed with the WENO reconstruction in the cell I i to obtain v i+/, and then set ( ˆf + i+/,ô+ j+/ ) = v i+/. Similarly, with (f j,o j ) = (f (u j,θ j ),o (θ j )) v j for j = i,...,i+, we proceed with the WENO reconstruction in the cell I i+ to obtain v + i+/, and then set ( ˆf i+/,ô j+/ ) = v+ i+/. The numerical flues are thus given by ˆf i+/ = ˆf + i+/ + ˆf i+/, ô i+/ = ô + i+/ + ô i+/. (.5) Eqs. (.5) and (.) give the scheme. In (.) and (.4), α ma ma λ l(u,θ). As all of the λ l are implicit, a slightly larger estimation (u,θ) l m of α is made according to (.7): α = ma (u,θ) ma( v + l= l v l, v m ). The maimum is taken over the relevant region of (u,θ) = (u j,θ j ) for all j. Finally, we apply the third-order accurate TVD Runge-Kutta time discretization [7], for which the semi-discrete scheme (.) is written as the ODEs u t = L(u,θ). For the numerical stability of the described WENO scheme, considerable numerical eperience suggests the following CFL condition [6]: (n) t(n) α CFL, where CFL can be taken as.6. Note that t (n) and α (n) are corresponding values at time level n.

12 4 Numerical eamples In Section 4., numerical tests are conducted to determine the waves that are described in Section.. The traffic that approaches the stop line of a traffic signal is simulated in Section 4.. These eamples are computed using the component-wise FD WENO scheme. We note that the component-wise FV WENO scheme generates similar results. For a clear observation of these waves, the velocities of (.) are set to be linear [8,], v() = v f ( jam ). In all of the illustrations, the densities l and are scaled by jam so that l,, and the spatial and temporal lengths L and T of the computational domain (,L) (,T) are scaled to unity. These dimensionless variables are also used wherever they are not followed by a unit. 4. Resolution of the waves compared to the analytical results The initial data in this section are given by the Riemann problem (.), with the translation of = to = somewhere in (,L). Furthermore, we set m =, which is small enough for a clear observation of the wave breaking. For simplicity, b l () are set to be constant for all l, namely, b () =.5, b () =.75, b () =. (4.) The other parameters are L = 8m, T = 4s, = m, and t (n) =.6 /α (n), where T is the simulation time. Figs.- show the numerical results, which are designed to reproduce all of the analytical wave patterns in Section., and to confirm Propositions. and.. In each figure, we observe the waves as follows. First, we count the wave number. Four (m + ) waves are observed in Figs.-6 that correspond to the wave patterns that are described in (), (b), and (c) and illustrated in Figs. and (b). In these cases, the system is strictly hyperbolic. In Figs.7-, there are five (m + ) waves, which correspond to the wave patterns that are described in (), (b), and (c) and illustrated in Figs. and (b). In these cases, the system must be non-strictly hyperbolic.

13 ..5 λ -rarefaction.5..5 λ -shock λ-shock l λ -contact (b) Fig. Wave pattern () with =., a L =, a R =, u /a L = (.,.,.) T, and u m= /a R = (.5,.5,.) T as compared with Fig.. density l ; (b) total density λ -contact. l λ-shock λ -shock λ -shock (b) Fig.4 Wave pattern ()(b) with =.5, a L =, a R =, u /a L = (.,.,.) T, and u m+ /a R = (.,,.5) T as compared with Fig.(b). density l ; (b) total density...6 λ -shock..5 λ-shock λ -shock l.4.. λ -contact (b) Fig.5 Wave pattern ()(c) with =.5, a L =, a R =, u /a L = (.,.5,.5) T, and u m+ /a R = (.,.,.) T as compared with Fig.. density l ; (b) total density.

14 λ -contact λ -rarefaction l..7.6 λ -shock λ-shock (b) Fig.6 Wave pattern ()(c) with =.4, a L =, a R =, u /a L = (.,.5,.) T, and u m+ /a R = (.,.5,.5) T as compared with Fig.(b). density l ; (b) total density λ -contact..7 l λ-shock λ-rarefaction λ -shock λ -shock (b) Fig.7 Wave pattern () with =., a L =, a R =, u /a L = (.,.5,.5) T, and u m+ /a R = (.5,.5,.) T as compared with Fig.. density l ; (b) total density. Secondly, we identify all of the λ l -waves in each figure according to Propositions. and.. We first observe the monotone change of the total density that is shown in the right part of the figure, such that each of these waves is identified to be either a shock or a rarefaction. Accordingly, as shown in the left part of the figure, the monotone changes for all density l across or in the same wave are observed and confirmed to be in accordance with Proposition. (for a shock) or Proposition. (for a rarefaction). Thirdly, as shown in Figs.7(b)-(b), we note that R = =.5 (in Figs.7(b) and 9(b)) or L = =.5 (in Figs.8(b) and (b)) for the λ -rarefaction. These are the analytical results of the wave breaking across the interface ( λ-contact). In summary, we claim that the numerical results that are given by the developed scheme are completely in accordance with the analytical properties that are predicted in Section.. 4

15 ..7.6 λ -rarefaction λ -shock..5 λ -shock l.4 λ -contact.. λ-shock (b) Fig.8 Wave pattern ()(b) with =.5, a L =, a R =, u /a L = (.,.5,.5) T, and u m+ /a R = (.5,.,.5) T as compared with Fig.(b). density l ; (b) total density λ -contact l.7 λ -shock λ-rarefaction λ-shock λ-rarefaction (b) Fig.9 Wave pattern ()(c) with =.4, a L =, a R =, u /a L = (.,.,.) T, and u m+ /a R = (.5,.5,.5) T as compared with Fig.. density l ; (b) total density. In this observation, we also note that λ (u,θ R ) and λ (u m+,θ R ) (m = ) share the same sign in all of the eamples. This can be seen in Figs.(b)-(b)(by (.9)) and should be true in general. Therefore, some of the predictions in Section. can also be made conveniently for the substitution of λ (u,θ R ) with λ (u m+,θ R ). 5

16 λ-rarefaction.5 l λ -shock.4 λ -contact λ -rarefaction. λ-shock (b) Fig. Wave pattern ()(c) with =.5, a L =, a R =, u /a L = (.4,.,.) T, and u m+ /a R = (.,,.) T as compared with Fig.(b). density l ; (b) total density. 4. Traffic signal control We etend the developed scheme to an application in which the functions of θ are also temporal. This is very common in traffic problems, and signal control is a typical eample of the etension. At the stop line, the signal display turns from green to red (at t = s) and holds, for eample, for seconds, and then turns back to green for another seconds. To reflect this regular change of traffic signal, which is near =.5 on a road with a section length L = m and a constant a(), we suppose that all b l (m = ) are given by (.6) for the green signal, but are zero near =.5 and for the red light, as is described by the following: (,,), if.4 < <.6, and < t 6[t/6] s, (b,b,b ) = (.5,.75, ). otherwise. For the initial condition: The numerical result is shown in Fig.. (4.) u(,)/a = (.5,.5,.) T. (4.) l (b) 6

17 .5.5 t (c) Fig. Traffic states during a signal cycle that is controlled by (4.). The initial data are given by (4.) and the other parameters are =.5m and t(n) =. /α(n). Densities of all classes at t = s; (b) total density at t = s; (c) change of total density for the simulation time T = 6s. We also set (,,,, ), if.4 < <.6, and < t 6[t/6] s, (b, b, b, b4, b5 ) = (.6,.7,.8,.9, ). otherwise, (4.4) and u(, )/a = (.5,.5,.,.5,.5)T. (4.5) The numerical result for this case is shown in Fig t Fig. Change of total density for the simulation time T = 6s during a signal cycle that is controlled by (4.4). The initial data are given by (4.5) and the other parameters are =.5m and t(n) =. /α(n). 7

18 5 Conclusion With the modified system, this numerical test of the multi-class traffic flow model on an inhomogeneous highway shows the robustness of the WENO reconstruction when combined with the La-Friedrichs flu splitting or La-Friedrichs numerical flu. Based on this standard hyperbolic conservation system (despite it being non-strictly hyperbolic), the La-Friedrichs flu gives correct numerical viscosity for the convergence of the numerical solutions to physically relevant solutions, whereas the WENO reconstruction reduces the surplus numerical viscosity to achieve a high level of resolution of the claimed waves. This again indicates the robustness of the WENO reconstruction, especially when it is combined with the La-Friedrichs flu to solve many compleities in application problems. The application of the WENO scheme in this paper can be etended to solve other hyperbolic conservation laws with spatially varying flues, such as elastic waves in heterogeneous media. It would also be interesting to investigate the use of other well-known numerical schemes, such as the RKDG scheme [-5]. References [] P. Baiti and H. K. Jenssen, Well-posedness for a class of conservation laws with L data, Journal of Differential Equations 4 (997), [] D. B. Bale, R. J. LeVeque, S. Mitran, and A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flu functions, SIAM Journal of Scientific Computing 4 (), [] B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Mathematics of Computation 5 (989), [4] B. Cockburn and C. W. Shu, The Runge-Kutta local projection P-discontinuous Galerkin methods for scalar conservation laws. Modelisation Mathematique et Analyse Numerique 5 (99), 7-6. [5] B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approimation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 697, Springer, 998, pp

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