Numerical Methods for Modern Traffic Flow Models. Alexander Kurganov

Transcription

1 Numerical Methods for Modern Traffic Flow Models Alexander Kurganov Tulane University Mathematics Department kurganov joint work with Pierre Degond, Université Paul Sabatier, Toulouse Anthony Polizzi, University of Michigan, Ann Arbor

2 Traffic Flow Models with Arrhenius Look-Ahead Dynamics [M.J. Lighthill, G.B. Whitham; 955] proposed the simplest deterministic continuum traffic flow model: u t + f(u) x = 0, f(u) = uv (u), V (u) = V m ( u) u(x, t): a relative car density (0 u ) V m const: the maximum possible speed V (u): dimensionless velocity function that depends only on u

3 [A. Sopasakis, M.A. Katsoulakis; 006] extended the Lighthill- Whitham model to a more realistic one by taking into account interactions with other vehicles ahead ( look-ahead rule). The resulting scalar conservation law with a global flux is: u t + F (u) x = 0, F (u) = V m u( u) exp( J u), where the contribution of short range interactions to the flux is: J w(x) := x+γ x J(y x)w(y) dy γ const > 0: look-ahead distance J: interaction potential 3

4 The simplest interaction potential is the constant one: Then J(r) = { /γ, 0 < r < γ, 0, otherwise. J u(x) := γ x+γ x u(y) dy Introducing the anti-derivative, U := global flux equation as: x u(ξ, t) dξ, we rewrite the F (u, U) = V m u( u)exp u t + F (u, U) x = 0 U(x + γ) U(x) γ 4

5 Finite-Volume Schemes -D System: u t + f(u) x = 0 ū n j x C j u(x, t n ) dx : cell averages over C j := (x j, x j+ ) This solution is approximated by a piecewise linear (conservative, second-order accurate, non-oscillatory) reconstruction: ũ n (x) = ū n j + sn j (x x j) for x C j 5

6 x j x j x j+ x j+ For example, s n j = minmod ( θūn j ūn j x, ūn j+ ūn j θūn j+, ūn j x x ) θ [, ], where the minmod function is defined as: minmod(z, z,...) := min j {z j }, if z j > 0 j, max j {z j }, if z j < 0 j, 0, otherwise. 6

7 Godunov-type upwind schemes are designed by integrating u t + f(u) x = 0 over the space-time control volumes [x j, x j+ ] [t n, t n+ ] t n+ t n x j x j / x j x j+/ x j+ 7

8 t n+ t n x j x j / x j x j+/ x j+ ū n+ j = ū n j x t n+ t n [ f(u(x j+, t)) f(u(x j, t)) ] dt In order to evaluate the flux integrals on the RHS, one has to (approximately) solve the generalized Riemann problem. This may be hard or even impossible... 8

9 Nessyahu-Tadmor Scheme The Nessyahu-Tadmor [H. Nessyahu, E. Tadmor; 990] scheme is a central Godunov-type scheme. It is designed by integrating u t + f(u) x = 0 over the different set of staggered space-time control volumes [x j, x j+ ] [t n, t n+ ] containing the Riemann fans t n+ t n x j x j+/ x j+ 9

10 t n+ t n x j x j+/ x j+ ū n+ j+ = x x j+ x j ũ n (x) dx x t n+ t n [ f(u(x j+, t)) f(u(x j, t)) ] dt Due to the finite speed of propagation, this can be reduced to: ū n+ j+ = ūn j + ūn j+ + x 8 (sn j sn j+ ) t [ ] f(u n+ x j+ ) f(un+ j ) 0

11 Values of u at t = t n+ expansion: u n+ j are approximated using the Taylor ũ n (x j ) + t u t(x j, t n ) ũ n (x) = ū n j + sn j (x x j) = ũ n (x j ) = ū n j u t (x j, t n ) = f x (ū n j ) The space derivatives f x are computed using the (minmod) limiter: ( f x (ū n j ) = minmod θ f(ūn j ) f(ūn j ), f(ūn j+ ) f(ūn j ), x x θ f(ūn j+ ) f(ūn j ) ) x

12 Nessyahu-Tadmor Scheme for a Traffic Flow Model with Arrhenius Look-Ahead Dynamics ū n j x C j u(x, t n ) dx : cell averages over C j := (x j, x j+ ) This solution is approximated by a piecewise linear (conservative, second-order accurate, non-oscillatory) reconstruction: ũ n (x) = ū n j + sn j (x x j) for x C j For example, s n j = minmod ( θūn j ūn j x, ūn j+ ūn j θūn j+, ūn j x x ) θ [, ]

13 The Nessyahu-Tadmor (NT) scheme is designed by integrating u t + F (u, U) x = 0 over the space-time control volumes [x j, x j+ ] [t n, t n+ ]: ū n+ j+ = x x j+ x j ũ n (x) dx x t n+ t n [ F (u(x j+, t), U(x j+, t)) ] F (u(x j, t), U(x j, t)) dt ū n+ j+ Due to the finite speed of propagation, this can be reduced to: = ūj + ū j+ + x 8 (sn j sn j+ ) t [ F (u n+ x j+, U n+ j+ ) F (un+ j, U n+ j ) ] 3

14 u and U at t = t n+ are approximated using the Taylor expansion: u n+ j ũ n (x j ) + t u t(x j, t n ) U n+ j Ũ n (x j ) + t U t(x j, t n ) ũ n (x) = ū n j + sn j (x x j) = u n j := ũn (x j ) = ū n j By integrating ũ n from x min to x we obtain the piecewise quadratic approximation of the antiderivative U: x Ũ n (x) = ũ n (ξ) dξ = x x min j j i=0 = x ū n i + ūn j (x x j i=0 In particular, ū n i + x x j ũ n (ξ) dξ ) + sn j (x x j )(x x j+ ), x C j U n j := Ũ n (x j ) = x j i=0 ū n i + x ūn j ( x) 8 s n j 4

15 u n+ j ũ n (x j ) + t u t(x j, t n ) U n+ j Ũ n (x j ) + t U t(x j, t n ) u t (x j, t n ) = F x (u n j, U n j ) () The space derivative F x is computed using the (minmod) limiter: ( F x (u n j, U j n ) = minmod θ F (ūn j, U j n) F (ūn j, U j n ), x F (ū n j+, U j+ n ) F (ūn j, U j n ), θ F (ūn j+, U j+ n ) F (ūn j, U j n) x x ) Finally, we integrate equation () with respect to x to obtain: U t (x j, t n ) = F (u n j, U n j ) 5

16 Numerical Examples Example Red Light Traffic u(x, 0) = {, 4 < x < 6 0, otherwise This corresponds to a situation in which the traffic light is located at x = 6 and it is turned from red to green at t = 0. First, fix γ = 6

17 x=/5 x=/0 0.6 ORDER ORDER REFERENCE 0.6 ORDER ORDER REFERENCE x=/0 x=/40 x=/ x=/0 x=/40 x=/

18 Impact of the look-ahead dynamics: global vs. non-global fluxes GLOBAL FLUX NON GLOBAL FLUX

19 We finally demonstrate the dependence of the computed solution on the look-ahead distance γ: as γ decreases, the drivers can see less and the original traffic jam dissipates more slowly when γ is large, the effect of the global flux is small 0.6 γ=0. γ=0. γ=0.5 γ= 0.6 γ=0 γ=5 γ= γ=

20 Formal limits of the global flux as γ and γ 0 The global factor exp ( U(x+γ) U(x) ) γ as γ assuming the total number of cars on the freeway is bounded. Therefore, F (u, U) f(u) = V m u( u) as γ that is, the global model reduces to the original, non-global one. ( U(x + γ) U(x) γ ) U x (x) = u(x) as γ 0. Therefore, F (u, U) f(u)e u = V m u( u)e u as γ 0 e u : slow down factor in the limiting low visibility case. 0

21 Example Traffic Jam on a Busy Freeway u(x, 0) = {, 6 < x < , otherwise Dispersive effect is much more prominent here since the waves are moving in the direction opposite to the vehicles velocity.

22 Even in the case of a large γ = 0, the global and non-global flux solutions are significantly different: t=0 t=6 t=3 t= t=0 t=6 t=3 t=

23 For a smaller γ = 5, the dispersive waves are getting together by time t = 0: t=0 t=6 t=3 t= For γ = 3, the larger waves catch the smaller ones much faster now and a dispersive wave package is formed at about t = 3: t=0 t=6 t=3 t=

24 When γ =, a dispersive wave package develops right away and is later transformed into one wave: t=0 t=6 t=3 t= When γ = 0.5, the dispersive effect can be observed at small times only: t=0 t=6 t=3 t=

25 For even smaller γ = 0., no dispersive effect can be observed: t=0 t=6 t=3 t= A certain smoothing effect can be observed though the solution seems to contain a sub-shock at the left edge of the wave: x=/00 x=/

26 Large and intermediate values of γ = dispersive effect Small values of γ = smoothing effect starts dominating The nature of these phenomena can be heuristically explained by expanding U(x + γ) in u t +F (u, U) x = 0, F (u, U) = V m u( u)exp ( U(x + γ) U(x) γ into the Taylor series about x: u t + [ ( V m (u u )e TS] x = 0, TS := u + γ ) u x + γ 6 u xx +... u t + V m ( 3u + u )e TS u x = V m (u u )e TS ( γ u xx + γ 6 u xxx +... Our conjecture: for small γ initially smooth solutions may remain smooth for all t. However, the viscosity coefficient in this case is small so that smooth solutions may develop sharp gradients. 6 ) )

27 Example 3 Numerical Breakdown Study To check our conjecture, we consider the smooth initial data: u(x, 0) = e (x 7) x=/800 γ= 5 γ=0. γ= 6 x=/600 x=/

28 For small γ = 0., the solution does not seem to break down at any stage of its evolution: t=0 t=4 t=8 t= For large γ = 0, the solution of the global model clearly breaks down in finite time like the solution of the non-global model: 8 6 γ=0 NON GLOBAL FLUX

29 Linear Interaction Potential A more realistic interaction potential is a linear one: J(r) = ( ) { r r, 0 < r < γ ϕ, ϕ(r) = γ 0, otherwise that is, In this case, J u(x, t) = x J(r) = ( ) γ r γ, 0 < r < γ 0, otherwise J(y x)u(y, t) dy = γ or after the integration by parts, J u(x, t) = γ γ x+γ x x+γ x ( y x γ U(y, t) dy U(x, t) ) u(y, t) dy. 9

30 It is easy to show that J u(x, t) u(x, t) as γ 0 This suggests that for small γ, the modified model is not expected to be much different from the original one. Indeed, when the visibility is very bad, all vehicles located within the interval (x, x + γ] are expected to have almost the same influence on the driver of the car located at x. However, when γ is not too small, the use of the linear interaction potential is motivated by the fact that the vehicles located further away from x (but still visible to the driver there) typically have smaller influence on the driver s behavior then the vehicles that are nearby. 30

31 Formula J u(x, t) = γ γ x+γ x U(y, t) dy U(x, t) can be rewritten as J u(x, t) = γ [ U(x + γ, t) U(x, t) γ U(x, t) ] where U denotes the antiderivative of U: U(x, t) := x U(ξ, t) dξ 3

32 The modified model is: F (u, U, U) = V m u( u) exp γ u t + F (u, U, U) x = 0 ( [ U(x + γ, t) U(x, t) γ U(x, t) ]) Compare with u t + F (u, U) x = 0 F (u, U) = V m u( u) exp ( U(x + γ) U(x) γ ) 3

33 Nessyahu-Tadmor Scheme for a Traffic Flow Model with Linear Interaction Potential u t + F (u, U, U) x = 0 ū n+ j+ = ūj + ū j+ + x 8 (sn j sn j+ ) λ F (u, U, U) F (u, U, U) (xj+, t n+ ) (xj, t n+ ) Therefore, to complete the description of the NT scheme we need to provide a recipe for the calculation of the intermediate point values {U(x j, t n+ )}. 33

34 From the Taylor expansion U(x j, t n+ ) U(x j, t n ) + t U t(x j, t n ) The approximate values of {U(x j, t n )} can be obtained by integrating the piecewise quadratic function j Ũ n = x ū n i + ūn j (x x j i=0 ) + sn j (x x j )(x x j+ ), x C j which results in a piecewise cubic approximation of U(x, t n ): Ũ n (x) = x x min j Ũ n (ξ) dξ = x wi n + wn j (x x j ) i=0 +ūn j (x x j )(x x j+ ) + sn j 6 (x x j )(x x j )(x x j+ ), x C j 34

35 Finally, integrating equation we obtain U t (x j, t n ) = F (u n j, U n j ) x U t (x j, t n ) = F (u(ξ, t n ), U(ξ, t n ), U(ξ, t n )) dξ x min which can be calculated by exact integration of the piecewise linear approximation of F. 35

36 Back to Example Traffic Jam on a Busy Freeway u(x, 0) = {, 6 < x < , otherwise 36

37 γ = 0 Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 37

38 γ = 5 Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 38

39 γ = 3 Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 39

40 γ = Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 40

41 γ = 0.5 Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 4

42 γ = 0. Constant interaction potential t=0 t=6 t=3 t= t=0 t=6 t=3 t= Linear interaction potential 4

43 Models of Formation and Evolution of Traffic Jams [H.J. Payne; 97,979] first second-order fluid dynamics models of traffic flow were proposed [C. Daganzo; 995] These models admit absurd results: negative car density backward car movement since cars in traffic have properties usual fluids do not have [A. Aw, M. Rascle; 000, 00] A new second-order model was derived from the microscopic Follow-the-Leader model 43

44 Aw-Rascle Model { ρt + (ρu) x = 0 (ρw) t + (ρwu) x = 0, w = u + p(ρ) ρ(x, t): a relative car density (0 ρ ) u > 0: dimensionless velocity w: drivers preferred velocity p(ρ): velocity offset (0 p(ρ) ) Example ([A. Aw, M. Rascle; 000, 00]) p(ρ) = ρ γ Not the best choice since the region [0, ρ ] [0, u ] is not invariant 44

45 { ρt + (ρu) x = 0 (ρw) t + (ρwu) x = 0, w = u + p(ρ) { t ρ + x (ρu) = 0 ( t + u x )(u + p(ρ)) = 0 ρ t + (ρu) x = 0 u t + ( u ) x = ρp (ρ)u x Compare with the pressureless gas dynamics (PGD) system: ρ t + (ρu) x = 0 u t + ( u )x = 0 ρ t + (ρu) x = 0 (ρu) t + (ρu ) x = 0 45

46 Example [F. Berthelin, P. Degond, M. Delitala, M. Rascle; 008] p(ρ) = ( ρ ρ ) γ for ρ ρ, γ > 0 ρ : maximal density Notice that p(ρ) as ρ ρ For the modified Aw-Rascle model the region [0, ρ ] [0, u ] is invariant 46

47 Modeling Traffic Jams [F. Berthelin, P. Degond, M. Delitala, M. Rascle; 008] t ρ ε + x (ρ ε u ε ) = 0 ( t + u ε x )(u ε + εp(ρ ε )) = 0, p(ρ ε ) = ( ρ ε ρ ) γ p 0 ρ ρ! No velocity offset unless ρ ρ! This is a very stiff problem for small ε 47

48 ε 0 For a nonsingular p(ρ) (e.g., p(ρ) = ρ γ ), { t ρ ε + x (ρ ε u ε ) = 0 ( t + u ε x )(u ε + εp(ρ ε )) = 0 εp(ρ ε ) 0 and ρ t + (ρu) x = 0 u t + ( u )x = 0 (P GD) For p(ρ) = ( ρ ρ ) γ, εp(ρ ε ) p(x, t) 0! p(x, t) may become positive and finite The formal limit of the Rescaled Modified Aw-Rascle Model is the Constrained PGD system: { t ρ ε + x (ρ ε u ε ) = 0 ( t + u ε x )(u ε + εp(ρ ε )) = 0 t ρ + x (ρu) = 0 ( t + u x )(u + p) = 0 0 ρ ρ, p 0, (ρ ρ) p = 0 48

49 Constrained PGD System t ρ + x (ρu) = 0 ( t + u x )(u + p) = 0 0 ρ ρ, p 0, (ρ ρ) p = 0! This system is still ill posed since the velocity offset p is undefined inside the clusters, that is, in the regions ρ = ρ! Rigorous passage to the ε 0 limit shows that u x = 0 inside the clusters When two cluster collide, they form a new cluster moving with the velocity of the cluster on the right. 49

50 Numerical Methods for the Constrained PGD System Method : Finite-Volume Upwind Scheme The idea: at each time step solve the PGD system and correct the velocities at the edges of clusters Assume that at some time level t the computed solution {ρ j (t), u j (t)} is available. Then it is evolved to t + t: { ρj (t) u j (t) } { ρ n (, t) ũ n (, t) } H ρ j+ (t) H u j+ (t) { ρj (t + t) u j (t + t) } d dt ( ρj (t) u j (t) ) H j+ (t) H = j (t), H x j+ (t) := H ρ j+ (t) H u j+ (t) 50

51 To incorporate the constraint, a special treatment of clusters is proposed. First, clusters are marked using I j ρ ρ x I j

52 Piecewise Linear Reconstruction ρ(t) = ρ j (t) + (ρ x ) j (x x j ) ũ(t) = u j (t) + (u x ) j (x x j ) for x C j Since the eigenvalues of the PGD are equal to u, they are nonnegative and thus we only need to compute the left-sided point values at each cell interface: ρ j+ (t) := ρ j (t) + x (ρ x) j u j+ (t) := u j (t) + x (u x) j Velocity Correction behind the Cluster: If I j (t) = 0 and I j+ (t) = and ρ j+ (t)u j+ (t) > ρ j+ 3 (t)u j+ (t) 3 then correct the velocity u j+ (t) := ρ j+ 3 (t)u j+ (t) 3 ρ j+ (t) 5

53 CFL Condition H ρ d dt ρ j+ j(t) = (t) H ρ j (t) x, H ρ j+ (t) = ρ j+ (t)u j+ (t) For the Forward Euler method, ( ρ n+ j = ρ n j λ ρ j+ u j+ provided either ρ j u j ) ρ ρ j+ u j+ ρ j u j 0 or λ ρ ρ n j ρ j u j ρ j+ u j+ Question: Will the time steps be too small? 53

54 Cluster Velocity Correction After the evolution step, the new solution {ρ j (t+ t), u j (t+ t)} and cluster markers {I j (t + t)} are available Assume that I j q = 0, I j q+ =... = I j =, I j+ = 0. Then If u j (t + t) > u j+ (t + t) and ρ j+ (t + t) > 0 set u j (t + t) := u j+ (t + t) If u j (t + t) > u j+ (t + t) and ρ j+ (t + t) = 0 set u j (t + t) := w w: preferred velocity 54

55 Example Riemann Initial Data t=0 t=0. t= Density 0.6 t=0 0.6 t= t= Velocity 55

56 Example Random Initial Velocity Initial density (left) and velocity (right) 56

57 Density at times t = 5, 50, 75, and 00 57

58 Method : Finite-Volume Cluster Propagation The approximate solution at t = t n is represented as a piecewise constant function, interpreted as a set of clusters of density ρ n j moving with the velocities u n j (CFL condition: λ max u n j ) j ρ ρ ρ ρ x 58 x

59 The evolved clusters are then projected onto the original grid ρ ρ ρ ρ x x 59

60 We have obtained the scheme for the PGD. Example PGD, Riemann Initial Data (same as in Example ) t=0 t=0. t= Density t= t= t= Velocity 60

61 Backward Mass Transfer ρ ρ ρ ρ x x 6

62 Example Riemann Initial Data t=0 t=0. t= Density t= t= t= Velocity 6

63 Example Random Initial Velocity Initial density (left) and velocity (right) 63

64 Density at times t = 70, 40, 0, and 80 64

65 Future Plans Second-order extension of the Finite-Volume Cluster Propagation method Extension to the case ρ = ρ (u) -D extension to the models of crowd dynamics 65