TTRAFFIC FLOW MODELS WITH LOOKING AHEAD-BEHINDDYNAMICS
YONGKI LEE † Abstract.
Motivated by the traffic flow model with Arrhenius look-ahead relaxationdynamics introduced in [A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66(2006), p. 921–944], this paper proposes a traffic flow model with look ahead relaxation-behind intensification by inserting look behind intensification dynamics to the flux. Finitetime shock formation conditions in the proposed model with various types of interac-tion potentials are identified. Several numerical experiments are performed in order todemonstrate the performance of the modified model. It is observed that, compare toother well-known traffic flow models, the model equipped with look ahead relaxation-behind intensification has both enhanced dispersive and smoothing effects. Introduction
In this paper, we are concerned with the shock formation phenomena - bounded solu-tions with unbounded derivatives - for a class of nonlocal conservation laws,(1.1) (cid:26) ∂ t u + ∂ x F ( u, ¯ u ) = 0 , t > , x ∈ R ,u (0 , x ) = u ( x ) , x ∈ R , where u is the unknown, F is a given smooth function, and ¯ u is given by(1.2) ¯ u ( t, x ) = ( K ∗ u )( t, x ) = (cid:90) R K ( x − y ) u ( t, y ) dy, where K to be chosen later. The nonlinear advection couples both local and nonlocalmechanism.This class of conservation laws, identified in [10], appears in several applications in-cluding traffic flows [20, 8], the collective motion of biological cells [2, 4], dispersive waterwaves [22, 6, 14], high-frequency waves in relaxing medium [7, 16] and the kinematicsedimentation model [9, 23, 1].There are some distinguished special cases of (1.1) with the kernel K : • A shallow water model proposed by Whitham [22] u t + 3 c h uu x + ¯ u x = 0 , corresponding to (1.1) with F ( u, ¯ u ) = c h u + ¯ u and K ( r ) = π exp( − π | r | / • The hyperbolic Keller-Segel model with logistic sensitivity [4](1.3) (cid:26) u t + [ u (1 − u ) ∂ x S ] x = 0 , − S xx + S = u, Mathematics Subject Classification.
Primary, 35L65; Secondary, 35L67.
Key words and phrases. nonlocal conservation law, shock formation, traffic flow, global flux. a r X i v : . [ m a t h . A P ] M a r YONGKI LEE † corresponding to (1.1) with F ( u, ¯ u ) = u (1 − u )¯ u and K ( r ) = ∂ r ( e −| r | / • A nonlocal dispersive equation modeling particle suspensions [18, 19, 23] u t + u x + (( K a ∗ u ) u ) x = 0 , corresponding to (1.1) with F ( u, ¯ u ) = u + ¯ uu , K a ( r ) = a − K ( r/a ) and(1.4) K ( r ) = (cid:26) / (3( r / − | r | < , , otherwise . Along with the above nonlocal models, the model that motivates the present work is thetraffic flow model with looking ahead relaxation, introduced by Sopasakis and Katsoulakis: • A traffic flow model with Arrhenius look-ahead dynamics [20](1.5) u t + [ u (1 − u ) e − ¯ u ] x = 0 , corresponding to (1.1) with F ( u, ¯ u ) = u (1 − u ) e − ¯ u and(1.6) K ( r ) = (cid:26) K /γ a , if − γ a ≤ r ≤ , , otherwise;Here, u ( t, x ) represents a vehicle density normalized in the interval [0 , γ a is a positiveconstant proportional to the look-ahead distance and K is a positive interaction strength.This model takes into account interactions of every vehicle with other vehicles aheadwithin the look ahead distance γ a .Some careful numerical study of the above traffic flow model is carried out in [8]. Inaddition to this, an improved interaction potential(1.7) K ( r ) := (cid:26) K γ a (1 + rγ a ) , if − γ a ≤ r ≤ , , otherwise , is introduced in [8]. This linear interaction potential is intended to take into account thefact that a vehicle’s speed is affected more by nearby vehicles than distant ones. In thecase of a good visibility(large γ a ), the numerical examples in [8] suggest that (1.5) withthe linear potential yields solutions that seem to better correspond to reality.Several finite time shock formation scenarios of solution to (1.5) with (1.6) were pre-sented in [11]. The authors in [10] identified threshold conditions for the finite time shockformation of (1.5) subject to two different potentials above. The sub-thresholds for finitetime shock formation conditions in [10] are consistent with the numerical result in [8].We set K = 1 through out this paper, since this parameter is not essential in ourblow-up analysis. Then we can rewrite nonlocal term ¯ u = K ∗ u associated with (1.6) and(1.7) as follows, respectively:(1.8) ¯ u ( t, x ) = 1 γ a (cid:90) x + γ a x u ( t, y ) dy, and(1.9) ¯ u ( t, x ) = 2 γ a (cid:90) x + γ a x (cid:18) x − yγ a (cid:19) u ( t, y ) dy. RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 3
In this paper, we extend (1.5) by considering a look behind intensification. That is, weconsider the traffic flow model with look ahead relaxation and look behind intensification :(1.10) (cid:26) ∂ t u + ∂ x ( u (1 − u ) e − ¯ u +˜ u ) = 0 , t > , x ∈ R ,u (0 , x ) = u ( x ) , x ∈ R . Here, ¯ u is given in (1.8) and (1.9). For the ˜ u = K b ∗ u , we shall consider constant andlinear interaction potentials. i.e.,(1.11) K b ( r ) = (cid:26) K /γ b , if 0 ≤ r ≤ γ b , , otherwise;and(1.12) K b ( r ) := (cid:26) K γ a (1 − rγ b ) , if 0 ≤ r ≤ γ b , , otherwise . Here, γ b is a nonnegative constant proportional to the the look-behind distance. By setting K = 1 again, we can rewrite the nonlocal term ˜ u associated with the above kernels asfollows, respectively:(1.13) ˜ u ( t, x ) := 1 γ b (cid:90) xx − γ b u ( t, y ) dy, and(1.14) ˜ u ( t, x ) = 2 γ b (cid:90) xx − γ b (cid:18) − x − yγ b (cid:19) u ( t, y ) dy. This look behind intensification model is intended to take into account the drivingbehavior of some drivers who actively react to vehicle distributions of one’s ahead andbehind. More precisely, considering the flux in (1.10), a vehicle’s velocity is determinedby (1 − u ) e − ¯ u +˜ u . Here, the local traffic density u at one’s location plays an major roles.In addition to this, averaged ahead and behind traffic densities of each driver adjust thevelocity via relaxation and intensification effects. The drivers equipped with this strategymonitor/compare the densities ahead and behind within assigned distances, and prefer tochoose accelerate(decelerate) when one has a relatively high density behind(ahead). It isnatural to assume that γ a ≥ γ b > YONGKI LEE † in a series of papers by Engelberg, Liu and Tadmor [5, 15, 21] for a class of Euler-Poissonequations.ii) We investigate performance of the proposed model (1.10) via numerical examples incomparison with (1.5) and the Lighthill - Whitham - Richards(LWR) [13, 17] model(1.15) ∂ t u + ∂ x ( u (1 − u )) = 0 . We are interested in both dispersive and smoothing effects of the proposed global flux.It is well known that the u x of the LWR model blows-up if there is a point such that u x ( t, x ) (cid:12)(cid:12) t =0 >
0. Indeed, the derivative of d = u x along the characteristic satisfies theRiccatti equation ˙ d = d that leads to the blow-up of d unless d (cid:12)(cid:12) t =0 ≤
0. For some circumstances, this result isunrealistic because no shock formation is observed in the free flow(i.e., sparse traffic).On the other hand, in [8], it is observed the global flux in look-ahead model (1.5)has some smoothing effect that seems to be able to prevent the shock formation. Butsimultaneously, the relaxation makes waves less dispersive. In our numerical examples inSection 3, it is observed that the proposed model (1.10) greatly improves waves’ dispersionand smoothing phenomena.Now, the finite time shock formation results are collectively stated as follows. In thetheorems, we assume that γ a ≥ γ b > Theorem 1.1. (Constant interaction potentials) Consider (1.10) with (1.8) and (1.13) .Suppose that u ∈ H and ≤ u ( x ) ≤ for all x ∈ R . If (1.16) sup x ∈ R [ u (cid:48) ( x )] > γ a + γ b γ a γ b (cid:18)
12 + √ (cid:115) − min (cid:26) − , γ a γ b γ a + γ b inf x ∈ R [ u (cid:48) ( x )] (cid:27)(cid:19) , then u x must blow-up at some finite time. Theorem 1.2. (Linear interaction potentials) Consider (1.10) with (1.9) and (1.14) .Suppose that u ∈ H and ≤ u ( x ) ≤ for all x ∈ R . If there is a point x ∈ R such that (1.17) u (cid:48) ( x ) > γ a + γ b γ a γ b (cid:18) (cid:115)
32 + (cid:0) γ a γ a + γ b ) (cid:1) (cid:19) , then u x must blow-up at some finite time. Regarding these results some remarks are in order.i) The condition (1.16) reflects some balance between sup x ∈ R [ u (cid:48) ( x )] and inf x ∈ R [ u (cid:48) ( x )].It seems the nonpositive term inf x ∈ R [ u (cid:48) ( x )] is more negative, then sup x ∈ R [ u (cid:48) ( x )] needs tobe large for the finite time shock formation. It can be interpreted that not only the cardensity behind the traffic jam but also the car density ahead of the traffic jam contributeto the formation of shock.ii) There is no direct comparison between (1.16) and look-ahead only model’s((1.5)-(1.8)) blow-up condition(1.18) sup x ∈ R [ u (cid:48) ( x )] > γ a (cid:18)
12 + √ (cid:115) − min (cid:26) − , γ a inf x ∈ R [ u (cid:48) ( x )] (cid:27)(cid:19) , RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 5 which is obtained in [10]. Roughly speaking, if inf x ∈ R [ u (cid:48) ( x )] is not too negative, thethreshold in (1.16) is higher than the one in (1.18). Also, for example, if γ a = 1 and γ b = 0 .
5, then the right hand side of (1.16) is bigger than that of (1.18) for any initialdata u ( x ). The same thing holds for majorities of γ a and γ b values. This is interestingbecause, even though we insert behind intensification ˜ u which in turn increases the wavesspeed, the blow-up thresholds actually increased. These blow-up results are consistentwith numerical experiments in Section 3.iii) In contrast to (1.16), the blow-up condition in (1.17) depends only on γ a , γ b andthe size of the initial slope. It is interesting to observe that a little difference in theinteraction kernels leads to different types of blow-up conditions; one involves two globalterms sup x ∈ R [ u (cid:48) ( x )] and inf x ∈ R [ u (cid:48) ( x )], the other involves local u (cid:48) ( x ) term only. Further-more, the blow-up threshold in (1.17) is higher than the one in (1.16) for not too negativeinf x ∈ R [ u (cid:48) ( x )]. Indeed, Example 3.4 in Section 3 shows that the model with linear interac-tion potentials has less steep wave(behind the traffic jam) than the model with constantpotentials for all time.We now summarize the main arguments in our proofs. We want identify some thresholdcondition for the shock formation of solutions to (1.10). The local existence and blow-upalternative results for (1.1) or (1.10) can be found in [10]. From Corollary 1 in [10], itsuffices to track the dynamics of u x . The idea is based on tracing M ( t ) := sup x ∈ R [ u x ( x, t )]and N ( t ) := inf x ∈ R [ u x ( x, t )]. The existence and differentiability(in almost everywheresense) of M ( t ) and N ( t ) are proved by Constantin and Escher [3], which we summarizein the following. Lemma 1.3. (Theorem 2.1 in [3] ) Let
T > and u ∈ C ([0 , T ]; H ) . Then for every t ∈ [0 , T ] there exists at least one point η ( t ) ∈ R with N ( t ) := inf x ∈ R [ u x ( t, x )] = u x ( t, η ( t )) , and the function N is almost everywhere differentiable on (0 , T ) with dNdt ( t ) = u tx ( t, η ( t )) a.e. on (0 , T ) . We also state a useful result, which is obtained in [12].
Lemma 1.4. (Lemma 3.3 in [12] ) Consider the following differential inequality, (1.19) dBdt ≥ a ( t )( B − b ( t ))( B − b ( t )) , B (0) = B , with a ( t ) > , b ( t ) ≤ b ( t ) and that a ( t ) , b ( t ) , b ( t ) are uniformly bounded.i) If B > max b , then B ( t ) will experience a finite time blow-up.ii) min { B , min b } ≤ B ( t ) , for t ≥ as long as B ( t ) remains finite on the time interval [0 , t ] . We remark that the above lemma remains valid even if (1.19) holds almost everywhere.We now conclude this section by outlining the rest of the paper. In Section 2, we proveTheorems 1.1 and 1.2. In Section 3, we demonstrate the performance of the proposedmodel via several numerical experiments.
YONGKI LEE † Proof of theorems
Proof of Theorem 1.1:
Let d := u x and apply ∂ x to the first equation of (1.10),˙ d := ( ∂ t + (1 − u ) e − ¯ u +˜ u ∂ x ) d = e − ¯ u +˜ u (cid:20) d + 2(1 − u )(¯ u x − ˜ u x ) d − u (1 − u ) (cid:8) ( − ¯ u x + ˜ u x ) + ( − ¯ u xx + ˜ u xx ) (cid:9)(cid:21) . (2.1)Define for t ∈ [0 , T ), M ( t ) := sup x ∈ R [ u x ( t, x )] = d ( t, ξ ( t )) ,N ( t ) := inf x ∈ R [ u x ( t, x )] = d ( t, η ( t )) . (2.2)Then, along ( t, ξ ( t )), using (1.8) and (1.13), we have − ¯ u xx + ˜ u xx = − γ a u x ( ξ + γ a ) + (cid:18) γ a + 1 γ b (cid:19) u x ( ξ ) − γ b u x ( ξ − γ b ) , ≤ (cid:18) γ a + 1 γ b (cid:19) ( M − N ) , (2.3)and (2.1) can be written as,˙ M = e − ¯ u +˜ u (cid:20) M + 2(1 − u )(¯ u x − ˜ u x ) M − u (1 − u ) (cid:8) ( − ¯ u x + ˜ u x ) + ( − ¯ u xx + ˜ u xx ) (cid:9)(cid:21) a.e. ≥ e − ¯ u +˜ u (cid:20) M + 2(1 − u )(¯ u x − ˜ u x ) M − u (1 − u ) (cid:26) ( − ¯ u x + ˜ u x ) + (cid:18) γ a + 1 γ b (cid:19) ( M − N ) (cid:27)(cid:21) a.e. (2.4)Along ( t, η ( t )), we have − ¯ u xx + ˜ u xx = − γ a u x ( η + γ a ) + (cid:18) γ a + 1 γ b (cid:19) N − γ b u x ( η − γ b ) ≤ , and (2.1) can be written as,˙ N = e − ¯ u +˜ u (cid:20) N + 2(1 − u )(¯ u x − ˜ u x ) N − u (1 − u ) (cid:8) ( − ¯ u x + ˜ u x ) + ( − ¯ u xx + ˜ u xx ) (cid:9)(cid:21) a.e. ≥ e − ¯ u +˜ u (cid:20) N + 2(1 − u )(¯ u x − ˜ u x ) N − u (1 − u )( − ¯ u x + ˜ u x ) (cid:21) a.e. (2.5)Now, we write the inequality in (2.5) as(2.6) ˙ N ≥ e − ¯ u +˜ u ( N − N )( N − N ) a.e. , where N ( u, ¯ u x , ˜ u x ) = − (1 − u )(¯ u x − ˜ u x ) − (cid:112) { (1 − u )(¯ u x − ˜ u x ) } + 2 u (1 − u )( − ¯ u x + ˜ u x ) RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 7 and N ( u, ¯ u x , ˜ u x ) = − (1 − u )(¯ u x − ˜ u x ) + (cid:112) { (1 − u )(¯ u x − ˜ u x ) } + 2 u (1 − u )( − ¯ u x + ˜ u x ) . We note that N ≤ ≤ N because 0 ≤ u ( t ) ≤
1. It can be shown later that N isuniformly bounded from below,(2.7) N ≥ − γ a + γ b γ a γ b . Applying Lemma 1.4 part (ii) to (2.6) with min ≤ u ≤ , | ω i |≤ γ N ( u, ω , ω ) = − γ a + γ b γ a γ b , weobtain N ( t ) ≥ min (cid:26) − γ a + γ b γ a γ b , N (0) (cid:27) =: (cid:18) γ a + γ b γ a γ b (cid:19) ˜ N . Substituting this lower bound into (2.4), we obtain˙ M ≥ e − ¯ u +˜ u (cid:18) M + (cid:26) − u )(¯ u x − ˜ u x ) − u (1 − u )( γ a + γ b ) γ a γ b (cid:27) M − u (1 − u )( − ¯ u x + ˜ u x ) + u (1 − u )( γ a + γ b ) ˜ N ( γ a γ b ) (cid:19) a.e. Rewriting of this inequality gives(2.8) ˙ M ≥ e − ¯ u +˜ u ( M − M )( M − M ) a.e. , where M ( ≥ M ) is given by M := −{ − u )(¯ u x − ˜ u x ) − u (1 − u )( γ a + γ b ) γ a γ b } (cid:113) { − u )(¯ u x − ˜ u x ) − u (1 − u )( γ a + γ b ) γ a γ b } + 8 u (1 − u )( − ¯ u x + ˜ u x ) − u (1 − u )( γ a + γ b ) ˜ N ( γ a γ b ) M has an uniform upper bound,(2.9) M ≤ γ a + γ b γ a γ b (cid:20)
12 + √ · (cid:113) − ˜ N (cid:21) . By Lemma 1.4 (i), if M (0) > γ a + γ b γ a γ b (cid:20)
12 + √ · (cid:113) − ˜ N (cid:21) , then M ( t ) will blow up a finite time. This is exactly the threshold condition as stated inTheorem 1.1.To complete our proof we still need to verify both claims (2.9) and (2.7).To verify (2.9), we set v := γ a γ b γ a + γ b (¯ u x − ˜ u x )= γ a γ b γ a + γ b (cid:26) γ a u ( x + γ a ) − (cid:0) γ a + γ b γ a γ b (cid:1) u ( x ) + 1 γ b u ( x − γ b ) (cid:27) . YONGKI LEE † From 0 ≤ u ( t ) ≤ − ≤ v ≤
1. If suffices to find upper bound for M overthe set Ω := { ( u, v ) ∈ R | ≤ u ≤ , − ≤ v ≤ } . In fact, M = −{ − u ) v − u (1 − u ) } + (cid:113) { − u ) v − u (1 − u ) } + 8 u (1 − u )( v − ˜ N )4( γ a γ b / ( γ a + γ b )) ≤ γ a + γ b γ a γ b (cid:2) (cid:113) − ˜ N ) (cid:3) . Here, we use max ( u,v ) ∈ Ω {− − u ) v + u (1 − u ) } = 2 which can be verified easily since theunderlying function is linear in v and quadratic in u . For the next one, max ( u,v ) ∈ Ω { u (1 − u )( v − ˜ N ) } = 2(1 − ˜ N ) is used, which is obtained from the upper bound u (1 − u ) ≤ / v defined above, we have Q := γ a γ b γ a + γ b N = − (1 − u ) v − (cid:112) { (1 − u ) v } + 2 u (1 − u ) v . By rearranging, Q = u (1 − u ) v − Q · (1 − u ) v ≤ u (1 − u ) v (cid:15)Q + (1 − u ) (cid:15) v , < (cid:15) < . (2.10)It follows that (1 − (cid:15) ) Q ≤ v (cid:15) { (1 − u ) + 2 (cid:15)u (1 − u ) }≤ (cid:15) , (2.11)where the maximum value is achieved at ∂ Ω. This gives Q ≤ (cid:15) (1 − (cid:15) ) . Since (cid:15) is arbitrary, we choose (cid:15) = to get Q ≤
1, hence Q ≥ −
1, which gives (2.7).This completes the proof of Theorem 1.1. (cid:3)
Proof of Theorem 1.2:
Let d := u x and apply ∂ x to the first equation of (1.10),˙ d := ( ∂ t + (1 − u ) e − ¯ u +˜ u ∂ x ) d = e − ¯ u +˜ u (cid:20) d + 2(1 − u )(¯ u x − ˜ u x ) d − u (1 − u ) (cid:8) ( − ¯ u x + ˜ u x ) + ( − ¯ u xx + ˜ u xx ) (cid:9)(cid:21) . (2.12)Using (1.9) and (1.14), we display the derivatives of the global terms(we omit the t − dependence).(2.13) ¯ u x = 2 γ a (cid:20) γ a (cid:90) x + γ a x u ( y ) dy − u ( x ) (cid:21) , ¯ u xx = 2 γ a (cid:20) γ a (cid:8) u ( x + γ a ) − u ( x ) (cid:9) − d ( x ) (cid:21) . RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 9
Also,(2.14) ˜ u x = 2 γ b (cid:20) − γ b (cid:90) xx − γ b u ( y ) dy + u ( x ) (cid:21) , ˜ u xx = 2 γ b (cid:20) − γ b (cid:8) u ( x ) − u ( x − γ b ) (cid:9) + d ( x ) (cid:21) . We reorder d ( x ) terms in ( − ¯ u xx + ˜ u xx ) and obtain˙ d = e − ¯ u +˜ u (cid:20) d + (cid:8) − u )(¯ u x − ˜ u x ) − u (1 − u )( 1 γ a + 1 γ b ) (cid:9) d − u (1 − u ) (cid:8) ( − ¯ u x + ˜ u x ) + ( 2 γ a − γ b ) u ( x ) + 2 γ b u ( x − γ b ) − γ a u ( x + γ a ) (cid:9)(cid:21) . (2.15)We should point out that in contrast to the model with constant potentials(see (2.3)), theabove equation does not have u x ( x + γ a ) or u x ( x − γ b )(see (2.3) ). Later this differenceleads to the structurally different blow up conditions.Rewriting of (2.15) gives(2.16) ˙ d = 2 e − ¯ u +˜ u ( d − D )( d − D ) , where D ( ≥ D ) is given by(2.17) D = − B + √ B + C , where B = (1 − u )(¯ u x − ˜ u x ) − u (1 − u )( 1 γ a + 1 γ b ) , and C = 2 u (1 − u ) (cid:26) ( − ¯ u x + ˜ u x ) + ( 2 γ a − γ b ) u ( x ) + 2 γ b u ( x − γ b ) − γ a u ( x + γ a ) (cid:27) . To find an upper bound of D , we set v := γ a γ b γ a + γ b ) (¯ u x − ˜ u x ) . Then, one can see that(2.18) | v | ≤ . Indeed, since v = γ a γ b γ a + γ b ) (cid:26) γ a (cid:90) x + γ a x u ( y ) dy − (cid:18) γ a + 2 γ b (cid:19) u ( x ) + 2 γ b (cid:90) xx − γ b u ( y ) dy (cid:27) , and 0 ≤ u ( x ) ≤ γ a γ b γ a + γ b ) D = 12 (cid:20) − (cid:8) (1 − u ) v − u (1 − u )2 (cid:9) + (cid:115)(cid:8) (1 − u ) v − u (1 − u )2 (cid:9) + (cid:0) γ a γ b γ a + γ b ) (cid:1) C (cid:21) , (2.19)where v and C are defined above. † As in the proof of Theorem 1.1 we letΩ := { ( u, v ) ∈ R | ≤ u ≤ , − ≤ v ≤ } . Then it holds γ a γ b γ a + γ b ) D ≤ (cid:20) (cid:114) (cid:0) γ a γ b γ a + γ b ) (cid:1) C (cid:21) . (2.20)Here, we use max ( u,v ) ∈ Ω { (1 − u ) v − u (1 − u ) / } = 1 which can be verified easily sincethe underlying function is linear in v and quadratic in u .To find an upper bound of (cid:0) γ a γ b γ a + γ b ) (cid:1) C , we expand and obtain (cid:0) γ a γ b γ a + γ b ) (cid:1) C = 2 u (1 − u ) (cid:20) v + (cid:0) γ a γ b γ a + γ b ) (cid:1) (cid:26) ( 2 γ a − γ b ) u ( x ) + 2 γ b u ( x − γ b ) − γ a u ( x + γ a ) (cid:27)(cid:21) ≤ (cid:20) (cid:0) γ a γ b γ a + γ b ) (cid:1) (cid:26) γ b (cid:27)(cid:21) . The last inequality holds because of (2.18), 0 ≤ u ( x ) ≤ u (1 − u ) ≤ / γ a − γ b ) < . Therefore, (2.20) leads to D ≤ γ a + γ b γ a γ b (cid:20) (cid:115)
32 + (cid:0) γ a γ a + γ b ) (cid:1) (cid:21) . Applying part (i) of Lemma 1.4, we complete the proof of Theorem 1.2. (cid:3) .3.
Numerical examples.
In this section, we demonstrate the performance of the proposed look ahead relaxation-behind intensification model (1.10), in comparison with look ahead relaxation only model(1.5) and LWR(non-global flux) model (1.15).In all numerical examples, Lax-Friedrichs numerical scheme is applied, and we takelarge enough computational domain so that no waves touch its boundary within the finalcomputational time. The Courant-Friedrichs-Lewy(CFL) numbers are chosen as 0 . .
25, and all solutions are computed on uniform grids with ∆ x = 1 /
100 or ∆ x = 1 / u ( t, x ) = K ∗ u , we used the trapezoidal method.We re-display the models of considerations here for readers’ convenience: • (LWR) Lighthill - Whitham - Richards model:(3.1) u t + [ u (1 − u )] x = 0 . • (Look-A) Look ahead relaxation model:(3.2) u t + [ u (1 − u ) e − ¯ u ] x = 0 . RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 11 • (Look-AB) Look-ahead relaxation-behind intensification model:(3.3) u t + [ u (1 − u ) e − ¯ u +˜ u ] x = 0 . Here, the global terms associated with constant interaction potentials((1.6) and (1.11))are given by(3.4) ¯ u ( t, x ) = 1 γ a (cid:90) x + γ a x u ( t, y ) dy, ˜ u ( t, x ) := 1 γ b (cid:90) xx − γ b u ( t, y ) dy. Also, the linear interaction potentials((1.7) and (1.12)) lead to(3.5)¯ u ( t, x ) = 2 γ a (cid:90) x + γ a x (cid:18) x − yγ a (cid:19) u ( t, y ) dy, ˜ u ( t, x ) = 2 γ b (cid:90) xx − γ b (cid:18) − x − yγ b (cid:19) u ( t, y ) dy. In all examples, we should point out that our look ahead-behind model is better in thesense of dispersive and smoothing effects.
Example 3.1. (two plateaus flow)
We consider equations LWR, Look-A and Look-ABwith constant interaction potentials(that is, (3.1), (3.2)-(3.4) and (3.3)-(3.4), respectively)with γ a = 1 and γ b = 0 . u (0 , x ) = 0 . . e − ( x +5) + 0 . e − ( x +3) . The situation corresponds to highly congested traffic( − < x <
2) is preceded by freetraffic and followed by less congested traffic. See Figure 1.In this example, on can clearly see the effects of the global fluxes. Look-A model’swaves(Red) lag behind that of the LWR waves(Blue). The waves in Look-AB modelhave faster pace than the ones in Look-A. This can be explained by the awareness oftraffic behind. While the waves with global fluxes develop no shock, the LWR wave(Blue)develop a shock discontinuity at t = 2. It is interesting to observe that the valley betweentwo initial plateaus is filled quickly in Look-AB model, while Look-A model has persistentvalley until t = 2 .
5. Overall, we should point out that Look-AB model has both enhanceddispersive and smoothing effects. † Figure 1.
Example 3.1: Time evolutions of the solutions equipped withLWR(Non-global flux), Look-Ahead flux and Ahead-Behind flux; γ a = 1and γ b = 0 . RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 13
Example 3.2. (red light traffic)
We consider equations LWR, Look-A and Look-AB withconstant interaction potentials(that is, (3.1), (3.2)-(3.4) and (3.3)-(3.4), respectively) with γ a = 1 and γ b = 0 . u (0 , x ) = (cid:26) . , x ∈ [ − , − , , otherwise . This initial condition corresponds to a situation in which the red traffic light is locatedat x = − Figure 2.
Example 3.2: Time evolutions of the solutions equipped withLWR(Non-global flux), Look-Ahead flux and Ahead-Behind flux; γ a = 1and γ b = 0 .
5, and zoom near the steepest gradients(lower right). † Example 3.3. (three plateaus; constant interaction potentials and linear interaction po-tentials) . We consider Look-AB with constant interaction potentials and linear interactionpotentials((3.3)-(3.5) and (3.3)-(3.4)), subject to the following initial data(3.8) u (0 , x ) = 0 . e − ( x +5) + 0 . e − ( x +2) + 0 . e − x , with ( γ a , γ b ) = (1 , . Figure 3.
Example 3.3: Time evolutions(left to right) of Look-AB withlinear interaction kernel(upper) and constant interaction kernel(middle),and LWR(lower).
Example 3.4. (steep plateau; constant interaction potentials and linear interaction po-tentials)
We consider Look-AB with constant interaction potentials and linear interaction
RAFFIC FLOWS WITH LOOKING AHEAD&BEHIND DYNAMICS 15 potentials((3.3)-(3.5) and (3.3)-(3.4)), subject to the following initial data(3.9) u (0 , x ) = 0 . e − x +2) , with ( γ a , γ b ) = (3 , . Figure 4.
Example 3.4 Time evolutions(left to right) of the solutionsequipped with; linear interaction kernel(magenta), constant interaction ker-nel(green) and LWR(blue), zoom in near the steepest slopes(lower right).
References [1] F. Betancourt, R. Burger, K. H. Karlsen and E. M. Tory. On nonlocal conservation laws modellingsedimentation.
Nonlinearity. , 24: 855–885, 2011.[2] M. Burger and Y. Dolak and C. Schmeiser. Asymptotic analysis of an advection-dominated chemo-taxis model in multiple spatial dimensions.
Commun. Math. Sci. , 6(1):1–28, 2008.[3] A. Constantin and J. Escher. Wave breaking for nonlinear nonlocal shallow water equations.
ActaMath. , 181: 229–243, 1998.[4] Y. Dolak and C. Schmeiser. The Keller-Segel model with logistic sensitivity function and smalldiffusivity.
SIAM J. Appl. Math. , 66: 286–308, 2005.[5] S. Engelberg, H. Liu and E. Tadmor Critical Thresholds in Euler-Poisson equations.
Indiana Univ.Math. J. , 50: 109–157, 2001.[6] D. D. Holm and A. N. W. Hone. A class of equations with peakon and pulson solutions (with aappendix by Braden H and Byatt-Smith).
J. Nonlinear Math. Phys , 12 Suppl. 1: 380–394, 2005. † [7] J. K. Hunter. Numerical solutions of some nonlinear dispersive wave equations. Lect. Appl. Math ,301–316, 1990.[8] A. Kurganov and A. Polizzi. Non-oscillatory central schemes for traffic flow models with Arrheniuslook-ahead dynamics.
Netw. Heterog. Media. , 4: 431–451, 2009.[9] G. Kynch. A theory of sedimentation.
Trans. Fraday Soc. , 48: 66–76, 1952.[10] Y. Lee and H. Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-aheaddynamics.
DCDS-A , 35(1): 323–3339, 2015.[11] D. Li, and T. Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics.
Netw.Heterog. Media. , 6: 681–694, 2011[12] T. Li, and H. Liu. Critical thresholds in hyperbolic relaxation systems.
J. Differential Equations ,247: 33–48, 2009[13] M. J. Lighthill and G. B. Whitham. On kinematic waves: II. A theory of traffic flow on long crowdedroads.
Proc. Roy. Soc., London, Ser. A , 229: 317–345, 1955[14] H. Liu. Wave breaking in a class of nonlocal dispersive wave equations.
Journal of Nonlinear MathPhys. , 13(3): 441–466, 2006.[15] H. Liu, E. Tadmor. Spectral dynamics of the velocity gradient field in restricted fluid flows.
Comm.Math. Phys. , 228: 435–466, 2002.[16] E. J. Parkes and V. O. Vakhneko. The calculation of multi-soliton solutions of the Vakhnenkoequation by the inverse scattering method.
Chaos Solitons Fractals. , 13: 1819–1826, 2002.[17] P. I. Richards. Shock waves on the highway.
Oper. Res , 4: 42–51, 1956[18] J. Rubinstein. Evolution equations for stratified dilute suspensions.
Phys. Fluids A. , 2(1): 3–6, 1990.[19] J. Rubinstein and J. B. Keller. Sedimentation of a dilute suspension.
Phys. Fluids A. , 1: 637–643,1989.[20] A. Sopasakis and M. Katsoulakis. Stochastic modeling and simulation of traffic flow: Asymmetricsingle exclusion process with Arrhenius look-ahead dynamics.
SIAM J. Appl. Math. , 66(3): 921–944,2006.[21] E. Tadmor and C. Tan Critical thresholds in flocking hydrodynamics with non-local alignment.
Philos Trans A Math Phys Eng Sc. , 372(2028), 2014.[22] G. B. Whitham. Linear and nonlinear waves.
John Wiley and Sons , 1974[23] K. Zumbrun On a nonlocal dispersive equation modeling particle suspensions.
Quart. Appl. Math. ,57: 573–600, 1999. † Department of Mathematical Sciences, Georgia Southern University, Statesboro,Georgia 30458
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