Physica A 419 (2015) 183–195

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Physica A

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Comparing numerical integration schemes fortime-continuous car-following modelsMartin Treiber a,∗, Venkatesan Kanagaraj ba Technische Universität Dresden, Institute for Transport & Economics, Würzburger Str. 35, D-01187 Dresden, Germanyb Technion—Israel Institute of Technology, Haifa 32000, Israel

h i g h l i g h t s

• We propose novel performance metrics for numerical integration schemes.• For car-following models, the ballistic scheme is always superior to Euler’s scheme.• The standard RK4 scheme is only efficient for unperturbed single-lane traffic.• Heun’s scheme is generally the best for simple situations.• The ballistic scheme prevails for complex situations with stops and lane changes.

a r t i c l e i n f o

Article history:Received 23 June 2014Received in revised form 26 September2014Available online 16 October 2014

Keywords:Time-continuous car-following modelNumerical integrationEuler’s methodBallistic updateRunge–Kutta methodConsistency order

a b s t r a c t

When simulating trajectories by integrating time-continuous car-following models, stan-dard integration schemes such as the fourth-order Runge–Kutta method (RK4) are rarelyused while the simple Euler method is popular among researchers. We compare four ex-plicit methods both analytically and numerically: Euler’s method, ballistic update, Heun’smethod (trapezoidal rule), and the standard RK4. As performance metrics, we plot theglobal discretization error as a function of the numerical complexity. We tested the meth-ods on several time-continuous car-following models in several multi-vehicle simulationscenarios with and without discontinuities such as stops or a discontinuous behavior of anexternal leader. We find that the theoretical advantage of RK4 (consistency order 4) onlyplays a role if both the acceleration function of themodel and the trajectory of the leader aresufficiently often differentiable. Otherwise, we obtain lower (and often fractional) consis-tency orders. Although, to our knowledge, Heun’smethod has never been used for integrat-ing car-following models, it turns out to be the best scheme for many practical situations.The ballistic update always prevails over Euler’s method although both are of first order.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Time-continuous car-following models (or more precisely, their longitudinal dynamics components) prescribe the ac-celeration of individual cars as a function of the driver’s characteristic behavior and the surrounding traffic. Formally, theirmathematical formulation is equivalent to that of physical particles following Newtonian dynamics with the physical forces

∗ Corresponding author. Tel.: +49 351 463 36794; fax: +49 351 463 36809.E-mail address: treiber@vwi.tu-dresden.de (M. Treiber).URL: http://www.mtreiber.de (M. Treiber).

http://dx.doi.org/10.1016/j.physa.2014.09.0610378-4371/© 2014 Elsevier B.V. All rights reserved.

184 M. Treiber, V. Kanagaraj / Physica A 419 (2015) 183–195

replaced by ‘‘social forces’’ [1]. In contrast to car-following models formulated in discrete time (coupled maps) or fully dis-cretely (cellular automata), time-continuous car-followingmodelsmust be augmentedwith a numerical integrationmethodin all but the most trivial analytically solvable cases [2].

Mathematically speaking, time-continuous car-following models without explicit reaction time delay represent cou-pled ordinary differential equations (ODE). Some often investigated models of this class include the optimal-velocity model(OVM) of Ref. [3], derivatives such as the full-velocity difference model (FVDM) of Ref. [4], and the Intelligent-Driver Model(IDM) by Treiber et al. [5]. Early car-following models such as that by Gazis, Herman and Rothery (GHR, [6]) or the linearadaptive cruise control(ACC) model by Helly [7] also fall into this class.

Because of the complexity and possible event-oriented components of traffic flow scenarios, one generally assumes afixed common time step h and explicit numerical schemes to obtain trajectories from the model equations. In the generalliterature on numerical mathematics (see, e.g., Ref. [8]), the standard explicit numerical integration scheme for ODEs isthe fourth-order Runge–Kutta method (RK4). However, in the domain of microscopic traffic flow modeling, the use of thismethod is rarely stated (counterexamples include [9,10]). Instead, most authors apply simplermethods or do not specify thenumericalmethod at all. Commonly used schemes are the simple Eulermethod [11] or the ballistic update assuming constantaccelerations during one time step [2]. Notice that also the open-source traffic simulators SUMO [12] and AIMSUN [13] usesimple Euler update for the positions.

Many car-following models are formulated in discrete time by mapping the positions and speeds of the vehicles at timet to these variables at time t + h. Generally, such models are mathematically identical to a time-continuous car-followingmodel combined with a certain update procedure. For example, in the congested regime, Newell’s microscopic model [14]maps the position of vehicle i at time t + h to the position of the leading vehicle i− 1 at time t minus the (effective) vehiclelength. As shown in Ref. [2], this ismathematically equivalent to integrating theOVMby a simple Euler step of size hprovidedthe OVM’s optimal velocity function corresponds to a triangular fundamental diagram with a maximum density equal tothe inverted vehicle length and car-following time gaps equal to h. Similarly, a time-discrete model has been derived fromthe GHRmodel by using Euler update with h equal to the reaction time, and qualitatively different behavior has been foundcompared to integrating the original GHR model with the RK4 method [15]. In the context of car-following methods, theballistic method is particularly appealing since it allows to model reaction times without introducing explicit delays whichwould transform the ODEs of time-continuous car-followingmodels (such as all time-continuousmodels mentioned above)into delay-differential equations. It has been shown [16] that integration of the IDM by the ballistic methodwith time step his essentially equivalent to an explicit reaction time delay Tr = h/2 of the corresponding delay-differential equations (whichare, then, integrated by higher-order methods or very small time steps).

Nevertheless, it is often desired to approach the true solution of time-continuous car-following models as closely aspossible. A criterion for the quality of an integration scheme is its (local or global) consistency order stating how fast theapproximate numerical solution converges to the true solution when decreasing the time step h ([8], see Section 2 fordetails). However, for practical integration steps h, higher-order methods do not necessarily lead to lower discretizationerrors. Moreover, if the acceleration function of the model is not sufficiently smooth (differentiable) or the simulationscenario contains discontinuities such as stops, lane changes, or traffic lights, the actual consistency order of a givennumerical scheme is generally lower than its nominal order [8]. Finally, higher-order methods need several evaluations ofthemodel’s acceleration function per vehicle and per time stepwhile Euler’smethod and the ballistic scheme need only one.

This leads to the following question: ‘‘Does the higher numerical accuracy of higher-order schemes outweigh theirhigher numerical complexity in terms of computation time, for practical cases?’’ Specifically, we would like to know whichnumerical scheme has the lowest global discretization error for a given numerical complexity, and how this depends on themodel and the simulation scenario.

In this work, we profile four numerical methods, simple Euler, ballistic scheme, Heun’s rule or trapezoidal rule, and RK4,for three car-following models (OVM, FVDM, IDM) in several multi-vehicle simulation scenarios. We found that RK4 is, infact, superior if certain rather restrictive conditions for the differentiability of the acceleration function and the external data(in our case, the leader’s trajectory) are satisfied, and if a high numerical precision is required. In most practical situations,however, the ballistic scheme and the trapezoidal rule turn out to be the most efficient and robust methods, although thelatter is rarely used. Moreover, the ballistic update always prevails over simple Euler although both are of first order.

In the next two sections, we specify the integration schemes in the context of car-followingmodels and give an analyticalanalysis of the truncation errors. In Section 4,we describe the simulation tests, define the numerical complexity as ameasurefor the computational burden, and the discretization error in terms of a vector norm on the deviations of the trajectories. InSection 5, we present the simulations and results. Finally. Section 6 gives a discussion and an outlook.

2. Integration schemes for car-following models and their mathematical properties

2.1. Mathematical formulation

We start by writing the dynamics created by time-continuous car-following models without explicit reaction time as ageneral system of ordinary differential equations,

dy⃗dt

= f⃗ (y⃗, t). (1)

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The state vector y⃗ represents all positions and speeds, and f⃗ (.) characterizes the specific car-following model and possiblyexternal data such as an externally driven leading vehicle. Specifically,we consider a class of car-followingmodels defined by

dxidt

= vi, (2)

dvi

dt= a(si, vi, vi−1), (3)

where i = 1, . . . , n denote the index of a fixed number n of vehicles (the first vehicle has the lowest index), xi denotes theposition of the front bumper of vehicle i, vi its speed, and si = xi−1 − xi − li−1 the bumper-to-bumper gap where li−1 is thelength of the leading vehicle.

Amodel of this class is specified by the acceleration function a(s, v, vl). The simulation scenario is specified by the numbern of vehicles following each other, by the initial conditions xi(0) and vi(0) for all vehicles i, and by boundary conditionsprescribing the speed v1(t) and the acceleration a1(t) of the first vehicle i = 1. Specifically, we consider free-flow boundaryconditions [2],

dv1

dt= a1(v1, t), a1(v1, t) = afree(v1) = a(∞, v1, v1), (4)

and externally prescribed leader acceleration profiles

v1(t) = vext(t), a1(t) =dvext

dt. (5)

Periodic boundary conditions,

a1(v1, vn, x1, xn) = a(xn + Lroad − x1 − ln, v1, vn) (6)

would be possible as well, as would be external influences such as traffic lights. However, we do not allow entering andleaving vehicles at the upstreamanddownstreamends of the simulated road section (sources and/or sinks) since the ensuingtime dependent vehicle number n would violate the general form (1). Nevertheless, we do not expect that open boundaryconditions will influence our results in any significant way.

In order to cast the model equations (2) and (3) into the general form (1), we define the state vector as

y⃗ =

x⃗v⃗

= (x1, . . . , xn, v1, . . . , vn)

T . (7)

Then, the right-hand side of (1) becomes

f⃗ (y⃗, t) =

v⃗

a⃗(x⃗, v⃗, t)

=

v1...vn

a1(v1, t)a(x1 − x2 − l1, v2, v1)

...a(xn−1 − xn − ln−1, vn, vn−1)

. (8)

Notice that the external boundary condition (5) makes the right-hand side non-autonomous (the independent variable tappears explicitly) while, for free or periodic boundary conditions, the ODE is autonomous.

2.2. Convergence and consistency order

The quality of explicit numerical integration methods with respect to discretization errors is generally characterized byits consistency order p. A method has a local consistency order ploc > 0 within a region R of state variables y⃗ and times t ifit converges to the true solution y⃗(t) and if, for all integration time steps h > 0 and all {y⃗, t} ∈ R, the inequality

ϵloc ≡∥y⃗num(t + h) − y⃗(t + h)∥

h< Ahploc (9)

is satisfied for all h > 0 with a finite h-independent prefactor A. Here, ϵloc is the local truncation error, y⃗num(t + h) isthe numerical approximation for the true initial condition y⃗(t), ∥.∥ is some vector norm (e.g., the 1-norm which we willuse henceforth), and the consistency order is the highest positive value of ploc for which this inequality is satisfied. Thedenominator h ensures that the decreasing numerical errors are not a trivial consequence of decreasing time steps h.

More relevant for simulations of car-following models, however, is the global consistency order indicating how the cu-mulated discretization errors of a complete simulation run decrease with decreasing h. If the simulation starts at t = 0

186 M. Treiber, V. Kanagaraj / Physica A 419 (2015) 183–195

with given initial conditions y⃗(0) = y⃗0 and ends at t = T > 0 including m = T/h time steps, one may define the globalconsistency order by the highest value of pglob for which

ϵglob ≡ ∥y⃗num − y⃗∥ < Ahpglob (10)

is satisfied for all h > with a finite h-independent prefactor A. Here, ∥.∥ is a vector norm on all the components of y⃗ for alltime steps, see (31) or (32) below.

If the right-hand side f⃗ (y⃗, t) is Lipschitz continuous with respect to y⃗ in the whole region R covered by the trajectories,one can show [8] that (i) the true solution exists, (ii) it is unique, and (iii) convergent numerical methods have a uniqueconsistency order

ploc = pglob = p. (11)

The function f⃗ (y⃗, t) is Lipschitz continuous if it is differentiable with respect to the components of y⃗ nearly everywhere andif the gradients are bounded. We notice that, for car-following models, these are nontrivial requirements. For example, theyare not satisfied for any car-followingmodelwith a discontinuous acceleration function. In contrast, a non-differentiability atcertain points, typically introduced bymin- or max conditions, is allowed. Even for a perfectly smooth acceleration functionas that of the IDM, the Lipschitz condition is violated for gaps s → 0 (crashes) or diverging speeds or speed differences:in both cases, the IDM acceleration function and some derivatives thereof, and thus the Lipschitz constant, tend to infinity.While the latter can be excluded by the general dynamics of the model (v(t) ≤ v0 for all t > 0 if v(0) ≤ v0), the former canonly be verified a posteriori. In the following, we will base our investigations on the global discretization error (truncationerror) ϵglob as defined in (10).

Finally, we note that the prefactor A indicating the upper bound varies widely with the method and the problem at hand.Generally, A increases drastically with the order of themethod, if the situation includes abrupt changes of the state, e.g., stopand go traffic. These variations are of a high practical relevance since they imply that a higher consistency order does notnecessarily lead to a higher accuracy as we will show in Section 5.

2.3. The investigated integration methods

We investigate (i) the simple Euler method, (ii) the trapezoidal rule (Heun’s method), (iii) the standard fourth-orderRunge–Kuttamethod (RK4), and (iv) the ballistic update. For reference, themethods (i)–(iii) for integrating (1) are as follows:

Euler: k⃗1 = f⃗ (y⃗, t),

y⃗(t + h) = y⃗ + hk⃗1, (12)

trapezoidal: k⃗1 = f⃗ (y⃗, t), k⃗2 = f⃗ (y⃗ + hk⃗1, t + h),

y⃗(t + h) = y⃗ +h2

k⃗1 + k⃗2

, (13)

RK4: k⃗1 = f⃗y⃗, t

, k⃗2 = f⃗

y⃗ +

h2k⃗1, t +

h2

,

k⃗3 = f⃗y⃗ +

h2k⃗2, t +

h2

, k⃗4 = f⃗

y⃗ + hk⃗3, t + h

,

y⃗(t + h) = y⃗ +h6

k⃗1 + 2k⃗2 + 2k⃗3 + k⃗4

. (14)

The ballistic method can only be sensibly defined if the ODE (1) represents Newtonian-style acceleration equations for oneor several particles including time-continuous car-following models. The ballistic method assumes constant accelerationsduring one time step which will be taken as that at the beginning of this step:

y⃗(t + h) =

x⃗(t + h)v⃗(t + h)

=

x⃗v⃗

+ h

v⃗

a⃗(x⃗, v⃗)

+

12h2

a⃗(x⃗, v⃗)

0⃗

. (15)

This can be interpreted as a mixed first-order, second-order update consisting of an Euler update for the speeds, and atrapezoidal update for the positions. While the resulting order p = 1 is that of Euler’s method, it turns out that the prefactorA of Eq. (10) is significantly lower in nearly all situations. As for the Euler update, the acceleration function a⃗(x⃗, v⃗) needs onlyto be calculated once per update step while two and four calculations are necessary for the trapezoidal and RK4 updates,respectively. Since calculating the acceleration function represents the essential part of the numerical complexity, this givesa hint at the numerical efficiency of the ballistic update.

2.4. Adaptation of the methods for stopping vehicles

To be fully effective, the general numerical methods assume smoothness conditions on the model and the data that arerarely given when simulating car-following models. Regarding a common source of such discontinuities, namely vehicle

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stopping, we can nevertheless improve all methods in a systematic way by overriding the canonical formulation for such asituation. Due to the finite update times, all update formulas (12)–(15) will lead to negative speeds whenever a time stepincludes the stopping of vehicles. In this case, it would be better to estimate the stopping position directly. Specifically, wehave applied, for all methods, following heuristics:

• The special treatment is activated if, for a vehicle i, the speed of a predictor (intermediate step) or the final step of an in-tegration scheme is negative indicating that this vehicle has stopped at some time instant of this time step. It also impliesthat the acceleration ai(t) calculated at the beginning of this time step is negative.

• In case of activation, we override the originally calculated position by the old position plus the stopping distance for aconstant deceleration −ai(t),

xi(t + h̃) = xi(t) −v2i (t)

2ai(t)if vi(t) + h̃ai(t) < 0. (16)

Here, h̃ is either h or h/2 (the latter applies for the second and the third predictor of RK4).• Additionally, in case of activation, we reset the speed to zero.

This provision for stopping vehicles fits naturally to the ballistic approach but improves the other methods as well. For thehigher-order methods, it increases the maximum effective consistency order from p = 1 to 2 and decreases the absoluteerror, at least, if stopped vehicles are the only reason for non-smooth trajectories.

3. Analytic analysis of truncation errors

In order to get some insight into what can be expected by the numerical analysis of the subsequent sections, we willinvestigate explicit analytical expressions for the local truncation errors

δy⃗ = y⃗num(t + h) − y⃗(t + h) (17)

provided y⃗num(t) = y⃗(t) for the special case of a single vehicle (position x, speed v) following an externally driven leadingvehicle l, i.e., the time dependencies xl(t), vl(t) and al(t) = v̇l(t) (as well as the length ll) are externally prescribed.Furthermore, we assume smooth acceleration functions and smooth speed/acceleration profiles of the leader. Omitting theexternally fixed leading vehicle, the equations of motion (1) of the relevant following vehicle have the general form

y⃗ =

xv

, f⃗ =

v

a(xl − x − ll, v, vl)

. (18)

Introducing the short-hand notations

a = a(s(t), v(t), vl(t)), (19)

as =∂a(s, v, vl)

∂s, av =

∂a(s, v, vl)

∂v, avl =

∂a(s, v, vl)

∂vl, (20)

with all derivatives taken at time t (beginning of the time step), we obtain, after some calculations, the local truncationerrors of the two simplest methods as

Euler: δy⃗ =

δxδv

=

h2

2

a

as(vl − v) + ava + avlal

+ O(h3), (21)

ballistic: δy⃗ =

δxδv

=

h2

2

0

as(vl − v) + ava + avlal

+ O(h3). (22)

The analytical truncation errors of the trapezoid and RK4 schemes can also bewritten in thisway in terms of the accelerationfunction and derivatives thereof but they are too complex to be of use in this analysis.

The most common reason for non-smooth acceleration profiles is the stopping of a vehicle since, in many models, theacceleration function jumps from negative values for v > 0 to zero for v = 0 in order to prevent driving backwards. Assum-ing that, in the true solution, the stopping occurs after a fraction h∗ ∈ [0, h[ of a time step and setting any negative speedto zero in the numerical solution, we have a truncation error of zero for the speeds but a finite positional truncation errorwhich, for the uncorrected Euler and ballistic methods, is given by

Euler: δx = v(h − h∗) −12ah2

∗+ O(h3), (23)

ballistic: δx = v(h − h∗) −12a(h − h2

∗) + O(h3). (24)

Since, in this situation, h−h∗ = O(h) and the initial speed v = O(h) as well, the local truncation error of bothmethods is ofO(h2). Consequently, both schemes revert to a first-order method in this situation. The same is also true for the uncorrected

188 M. Treiber, V. Kanagaraj / Physica A 419 (2015) 183–195

trapezoid scheme and RK4. After adaptation of the schemes according to Section 2.4, we have, for the two above methods,the local truncation errors

Euler: δx = −12ah2

∗+ O(h3), (25)

ballistic: δx = O(h3). (26)As in the case of smooth accelerations, the positional update of the ballistic method is now of second order (for this specialcase, even the whole method is of second order). Similar considerations show that, in this situation, the corrected trapezoidand RK4 methods are second-order as well.

Several insights can be obtained from these expressions: first, the necessary time step depends on the car-followingmodel: the higher the absolute values of the model’s acceleration or of the derivatives thereof, the lower the required timestep for a given precision. For the Euler scheme, the relative error in the displacement of the vehicles is

ϵx =δxhv

= ha(s, v, vl)

2v+ O(h2), (27)

and that of the speed change

ϵx =δv

ha= h

as(vl − v) + ava + avlal2a

+ O(h2). (28)

Comparing the IDM and the OVM, we observe that, for typical parameterizations and typical scenarios like the city-scenarioto be discussed below, absolute values of the IDM acceleration are of the order of 1m/s2 while that of the OVM are of theorder of 10m/s2. Likewise, the right-hand side of (28) has typical absolute values of h times 0.5s−1 and h times 2s−1 for theIDM and OVM, respectively. Thus, for the same precision, the IDM time step can be about 4 times that of the OVM step.

Second, when analyzing the magnitude A of the truncation errors Ahp+1 of the pth-order methods (which, for smoothacceleration functions, can all can be expressed in terms of the acceleration function and derivatives thereof), they turn outto be of the order A ≈ A0/h

p0 where A0 and h0 are model-specific constants. For example, h0 ≈ 2s for the IDM in typical sce-

narios, and h0 ≈ 0.5s for the OVM. This means, at a time step h ≥ h0, the truncation error does not decrease systematicallywith the order of the method.

Third, due to its higher-order displacement truncation errors, the ballistic scheme is particularly efficient compared toEuler’s method if the simulation includes periods with slow or stopped vehicles or, equivalently, short gaps s. In such situ-ations, the partial derivative as of nearly all models is strongly elevated compared to free traffic. Thus, the local truncationerror of the displacements strongly increases the truncation error for the speed changes of the next step and, ultimately, theglobal discretization error.

Fourth, the proposed adaptation of the methods for stopped vehicles raises the order of the ballistic, trapezoid and RK4methods to p = 2 while it is p = 1 otherwise, regardless of the numerical method.

4. Methodology for assessing the integration schemes

In order to assess the numerical schemes, we need to specify the numerical complexity, the global discretization error,and the reference solution against which to calculate the global error. Since the true solution cannot be obtained for anynontrivial scenario (otherwise, there would be no need for simulation), setting up the reference (or, more precisely, anestimator for the reference with controlled errors), is a nontrivial task.

4.1. Numerical complexity

We define numerical complexity as the computation time to simulate a single vehicle on a single lane over a given sim-ulated time interval, e.g., 1s. The inverse of this quantity indicates the number of vehicles that can be simulated in real time.To determine the numerical complexity, we observe that nearly all of the computational burden consists in evaluating themodel’s acceleration function. Furthermore, a numerical method of nominal consistency order p needs at least p calcula-tions of the right-hand side f (.), i.e., of the acceleration function, and an efficient numerical method (including all thesepresented) needs exactly p such calculations. Hence, the numerical complexity is essentially proportional to the quantity

C =ph

(29)

denoting the number of evaluations of the acceleration function per vehicle and simulated unit time. The nominal consis-tency order is p = 1 for the Euler and ballistic updates, p = 2 for the trapezoidal rule, and p = 4 for RK4.

4.2. Reference solution

By virtue of the Lipschitz continuity of the chosen model and the boundedness of the dynamics, a unique exact globalsolution exists for our simulation scenarios. However, it cannot be calculated analytically for any but the most trivialsituations. We therefore calculate a ‘‘reference solution’’ y⃗ref(t) against which to compare the integration schemes by the

M. Treiber, V. Kanagaraj / Physica A 419 (2015) 183–195 189

RK4 scheme using a time step href = 10−4s which is smaller than the smallest time step of the actual investigations by afactor of 20. To test the validity of this reference, we repeat the calculation with h = 2href resulting in y⃗cmp(t) and calculatethe global error between these two solutions. Since one can show that the actual global consistency order pact of RK4 satisfiespact ≥ 1 for all the investigated scenarios, this provides an upper bound for the global error between the reference and theunknown true solution [8]:

∥y⃗ref − y⃗∥ ≤ ∥y⃗cmp − y⃗ref∥. (30)This controlled error of the reference solution ensures that all global discretization errors calculated in the following can bedetermined with uncertainties (i.e., second-order errors) of less than 1%.

4.3. Global discretization error

We have tested several global error norms, among them the 1-norms and 2-norms of the time series of location, speed,acceleration, and gap for one or more trajectories, and combinations thereof. Examples include the 1-norm of the speedtrajectory of the ith follower,

ϵi = ∥vnumi − vref

i ∥ =1m

mj=1

|vnumi (jh) − vref

i (jh)|, (31)

where m is the number of points of this trajectory, vnumi (jh) is the speed of the ith vehicle at time t = jh (after the jth time

step), and vrefi (jh) is the corresponding speed of the reference solution.

Another example is the 1-norm of the speeds of all trajectories,

ϵ = ∥v⃗num− v⃗∥ =

1nm

ni=1

mj=1

|vnumi (jh) − vref

i (jh)|, (32)

where v⃗(t) = (v1(t), . . . , vn(t))T.In agreement with the general mathematical result that all norms in finite-dimensional spaces are equivalent, we al-

ways obtained similar results, regardless of the quantity (speed or gap) or the chosen vehicle trajectory (or including alltrajectories), we will, henceforth, only consider ϵ10, i.e., the discretization errors of the speeds of the 10th vehicle (assumingsimulations with more than 10 followers).

5. Results

As main test cases for the simulations, we have used the city start–stop scenario as described in Chapter 11 of Ref. [2]:Initially, a queue of 20 identically parameterized cars is waiting behind a red traffic light. At t = 0, the traffic light turnsgreen and the queue of cars starts moving until a red traffic light 670m ahead stops the vehicle platoon again.

Most tests are performed with the IDM or variants thereof. The acceleration function of the original IDM reads [5]

aIDM(s, v, vl) = afree(v) − amax

s∗(v, vl)

s

2

, (33)

where the free-flow acceleration function afree(v) and the dynamic desired gap s∗(v, vl) are given by

afree(v) = amax

1 −

v

v0

4

, (34)

s∗(v, vl) = maxs0 + vT +

v(v − vl)

2√amaxb

, 0

. (35)

Furthermore, backwards motion is avoided by resetting the acceleration to zero if v = 0 and the original IDM accelerationis negative. The meaning and applied values of the five IDM parameters are listed in Table 1.

The IDM acceleration function (33) is smooth for speeds v > 0 and dynamic desired gaps s∗ > 0. While the latter isalways true in the presented simulations, the first condition is violated whenever there are some periods with at least onestopped vehicle.

In the theoretical ‘‘best case’’, both the acceleration function of the model and, if applicable, external data (such as exter-nally prescribed leading trajectories) are smooth, i.e., differentiable up to any order. This will produce smooth trajectoriesfor which all mathematical conditions for the theoretical consistency order are satisfied. We will start with this case beforewe simulate less ideal (and more realistic) scenarios by progressively reducing the degree of smoothness.

5.1. Smooth trajectories

Fig. 1 shows some characteristics of the resulting trajectories of the city start–stop scenario when parameterizing theIDM according to the second column of Table 1. Notice that, in this simulation, all cars move according to the IDM, i.e., there

190 M. Treiber, V. Kanagaraj / Physica A 419 (2015) 183–195

Table 1IDM parameters of the tests.

Parameter Standard set ‘‘creep-to-stop’’ set

Desired speed v0 15 m/s 15 m/sDesired time gap T 1 s 1 sMinimum gap s0 2 m 1 m

Maximum acceleration amax 1 m/s2 2 m/s2

Comfortable deceleration b 1.5 m/s2 1.5 m/s2

Fig. 1. Trajectories of the start–stop scenario: single-lane city traffic between two signalized intersections as simulated with the IDM with standardparameters.

is no externally prescribed leading trajectory. The red traffic light is simulated as a standing virtual vehicle of length zeroserving as a ‘‘virtual leader’’ for the first vehicle. When restricting the simulation time to 60s, no vehicle has stopped yet atthe end of the simulation time. Since, for nonzero speeds, the IDM acceleration function (33) and the resulting trajectoriesare smooth, this means that the conditions for the maximum theoretical consistency orders are satisfied, i.e., p = 1 forEuler’s and the ballistic methods, p = 2 for the trapezoidal rule, and p = 4 for RK4 method.

Fig. 2 shows the numerical accuracy of the 60s-simulation in the form of a log–log plot of the discretization errors as afunction of the numerical cost for all four integration methods presented in Section 2. As a measure for the numerical costC , we defined the number of calculations of the acceleration function per simulated vehicle and simulated second accordingto Eq. (29). As an error measure, we used the 1-norm ϵ10 of the speed differences of the 10th follower. Specifically, we haveassumed 16 different update times ranging from h = 0.002s to h = 2.4s. The reference trajectory was obtained by applyingthe RK4 method with an update time step h = 10−4s. We recorded the result every 2.4s which is the lowest commonmultiple of all update intervals h used for the test simulations. This is necessary to separate the discretization errors to beanalyzed from errors when interpolating data points and provides a common data basis for all test simulations.

We observe that the theoretical consistency orders are realized in the actual simulation, i.e., the asymptotic negativeslopes psim of the log–log plot are approximatively equal to p = 1 for the Euler and ballistic schemes, p = 2 for thetrapezoidal scheme, and p = 4 for RK4. Moreover, for h ≤ 0.5s (which corresponds to C ≥ 2p and includes the practicallyused intervals), the order of performance of the methods (from best to worst) is RK4, trapezoidal, ballistic, and Euler. Noticethat the ballistic scheme is always superior to Euler’s scheme (only about 30% of the error of the latter) although both havethe same consistency order p = 1.

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Fig. 2. Error norm of the speeds as a function of the numerical complexity for four numerical integration methods. The simulation is that of Fig. 1 for thesimulation interval [0s, 60s].

Fig. 3. Details of the acceleration profile during the stopping phase. Left: distinct stop (parameterization by the second column of Table 1); right: creepingstop (third column).

Fig. 4. Same as Fig. 2 but for a total simulation interval of 100s.

5.2. Discontinuous accelerations caused by stopped vehicles

We simulate the same city start–stop scenario as above with the same parameter set, but now for a simulation timeof tmax = 100s instead of 60s. As shown in Fig. 3, some vehicles have stopped with a discontinuous acceleration profileafter this prolonged time. For the higher-order methods, this means that the mathematical smoothness conditions for thetheoretical consistency order are no longer satisfied.

Does this have implications for the actual accuracy? Fig. 4 shows that the consistency orders of the Euler, ballistic, andtrapezoidal schemes essentially retain their theoretical values of p = 1, 1, and 2, respectively. Furthermore, the prefactors Aof Eq. (10) determining the absolute size of the discretization errors are essentially unchanged as well. In contrast, the errorsof the RK4 method significantly increase by a factor of about five. Nevertheless, RK4 remains the best method for practical

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Fig. 5. Discretization errors of the scenario of Fig. 4 when re-parameterizing the IDM by the 3rd column of Table 1 resulting in a creeping halt.

Fig. 6. Discretization errors when simulating the scenario of Fig. 4 with the optimal-velocity model (creeping halt).

update intervals. Remarkably, its simulated consistency order p ≈ 3.5 is only marginally below its theoretical value of 4. Atfirst sight, this is unexpected since, for mathematical reasons, the consistency order for a discontinuous acceleration profileshould not exceed p = 1, regardless of the method.

There are two reasons to explain these findings. Firstly, the discontinuity concerns just a single point, so the error con-tribution of lower consistency order is small and the asymptotic slope may not yet have been reached in the log–log plots.Secondly, we have taken special provisions to increase the accuracy of the stopping situation: Whenever a predictor or thefinal value of an integration step yields a negative speed, we override the normal algorithm by setting the speed to zero andthe position to the ballistically estimated stopping position which is calculated assuming a constant deceleration definedby the right-hand side of (8) for this step. This special-purpose procedure, which is described in Section 2.4 in more detail,increases the upper bound of the expected consistency order to 2 in all simulations that include stops but have smoothacceleration functions and data, otherwise.

Depending on the parameterization, it is possible that the simulated vehicles do not stop at a precisely defined time (asabove) but ‘‘creep to a halt’’ retaining a smooth acceleration profile at all times. An example parameterization for this behav-ior is given by the third columnof Table 1 resulting in the time series plotted in Fig. 3 right. Then, themathematical conditionsfor the full theoretical consistency order are satisfied again. In fact, the log–log plot of the discretization error vs. numericalcost (Fig. 5) shows little changes with respect to the first simulation without stops. Particularly, the slopes are consistentwith the theoretical expectation again. However, the RK4 method produced significantly larger errors compared to the 60ssimulation when simulating with comparatively large update time intervals. We also simulated this scenario with the OVMof Ref. [3] with typical parameters resulting in a creeping halt as well. Again, we found the expected theoretical consistencyorders but a higher prefactor, i.e., generally higher errors for all methods and all discretization time steps (see Fig. 6).

5.3. Non-smooth acceleration functions

Discontinuous acceleration profiles do not only result when vehicles stop but may also be generated by the model itselfif its acceleration function amic(s, v, vl) has regions in state space with kinks (non-differentiable points) or discontinuities,

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Fig. 7. Discretization errors of the start–stop scenario (tmax = 100s) for the IDM-Plus (left) and themodified IDM according to (37) (right). All IDM variantsare parameterized according to the second column of Table 1.

at least when the dynamics reaches these regions. We tested both cases by simulating the standard start–stop scenario(tmax = 100s) with two modifications of the IDM. As an example for an acceleration function with kinks, we simulated the‘‘IDM-Plus’’ proposed by Schakel et al. [17]. Its acceleration function reads

aIDM+(s, v, vl) = min

afree(v), amax

1 −

s∗(v, vl)

s

2

(36)

with the normal IDM expressions (34) and (35) for the free acceleration and the dynamic desired gap. The additional ‘‘min’’condition of the IDM-Plus1 leads to a kink in the acceleration profile when the remaining distance of the first vehicle to thered traffic light (modeled as a standing virtual vehicle of length zero) is approximatively sc = s0 + v0T + v2

0/(2√amaxb),

and also to (smaller) kinks of the accelerations of the following vehicles.Fig. 7 (left) shows the discretization errors for the IDM-Plus. We observe that both the error prefactors and the consis-

tency orders of the Euler, ballistic, and trapezoidal methods remains essentially unchanged while the consistency order ofRK4 reduces to about two. This is expected on theoretical grounds: a kink in the realized acceleration profile sets the upperlimit of the consistency order of any explicit method to pmax = 2 [8]. For practical simulation time intervals correspondingto a numerical cost of around 10(veh s)−1, however, we observe little difference with respect to discretization errors for asmooth acceleration profile.

To obtain amodelwith discontinuities in the acceleration functionwhich are present inmany othermodels (e.g., Ref. [14]),we modify the free-flow IDM acceleration function to

afree(v) =

a if v < v01 − v/v0 if v ≥ v0.

(37)

In this model, drivers reduce their acceleration abruptly from amax to zero once reaching their desired speed.Fig. 7 (right) shows that acceleration discontinuities greatly increase the discretization errors of the higher-order

methods. In line with theoretical expectations, all methods now have the consistency order p = 1. Regarding the absoluteerrors, Euler is worst while all other methods are essentially equivalent.

5.4. External data with discontinuous accelerations

Another source of discontinuities can be external system data, e.g., prescribed speed profiles of an external leader. Fig. 8shows a simulation where a platoon of vehicles follows a leader with an externally given discontinuous acceleration profilecorresponding to a speed profile with kinks. Assuming a model with a continuous acceleration function, this leads to anacceleration profile with kinks for the first follower, and to differentiable profiles for the further followers.

The log–log plot of the discretization errors (Fig. 8 bottom) reveals that the Euler, ballistic, and trapezoidal methods havetheir expected consistency orders and absolute errors while, surprisingly, RK4 has only consistency order p = 1. It seemsthat, at each abrupt behavioral change of the leader, the predictors of RK4 err to such an extent that the result is essentiallythat of the ballistic method.

5.5. Lane changes

The most severe source of discontinuities are active and passive lane changes, i.e., the considered vehicle changes itself,or another changing vehicle ‘‘cuts in’’ in front of it. From the mathematical point of view, this means that not only theaccelerations of the leader are discontinuous but the gaps and leading speeds as well. Since the two latter quantities are

1 The ‘‘min’’ function of the dynamic desired gap becomes only relevant in rare cases, none of which are included in the simulations.

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Fig. 8. Top: following a leader with a fixed speed profile (IDM, parameters as in the second column of Table 1); bottom: convergence diagram for the tenthfollower. The saturation at a global error of ≈e−12 is due to the finite time step of the leading vehicle’s profile.

exogenous variables of the model’s acceleration functions, the acceleration profile of the vehicle behind the lane-changingvehicle on the target lane is discontinuous as well reducing the consistency orders of all methods to p = 1.

This is, in fact, what we observe: Fig. 9 displays a simulation where the test vehicle encounters three cut-ins in front of it.All four integration methods have the empirical consistency order p = 1. Remarkably and unexpectedly, the absolute valueof the error for a given numerical cost is largest for RK4.

6. Discussion and conclusions

In this work, we have systematically investigated the global discretization error of several explicit numerical integrationschemes commonly used for simulating time-continuous car-followingmodels. To enable an equitable comparison betweensimple and higher-order methods, we have determined the errors as a function of the numerical complexity, i.e., the nor-malized computation time for simulating one vehicle over one time unit: a method is better if, for the same numericalcomplexity, its global errors are lower.

Generally, when integrating ODEs or systems thereof, the fourth-order Runge–Kutta scheme (RK4) is the de-facto stan-dard and other methods are rarely used.Why is this not the case for integrating car-followingmodels where Euler’s methodis most widespread? This contribution shows that, for typical traffic-related situations, the RK4 method is, in fact, not thebestmethod since the smoothness conditions to reach its theoretical consistency order p = 4 are rarely given. To investigatethe effect of violating these conditions, we simulated several scenarios and several models, from the ideal case to the mostsevere violations of smoothness:

(1) all trajectories remain smooth (sufficiently often differentiable) over the complete simulation time (Section 5.1),(2) the acceleration profile of the trajectories is continuous but not smooth due to the model’s acceleration function

(Section 5.3) or the external data (Section 5.4),(3) the acceleration profile is discontinuous due to stopped vehicles (Section 5.2) or by discontinuities in the model’s accel-

eration function which are reached by the system dynamics (Fig. 8),(4) the speed and gap profiles are discontinuous as a consequence of lane changes (Section 5.5).

We have found that RK4 and the trapezoidal scheme perform best if the trajectories are smooth or have, at most, kinks inthe acceleration.While, on a qualitative level, this is expected on theoretical grounds, we have provided quantitative resultswhich, to our knowledge, have not been given before. Moreover, with our novel treatment of stopping vehicles (Section 2.4),we can increase the order from one to two for this frequent case of acceleration discontinuities. In all other situations, how-ever, the consistency order of allmethods is restricted to one and the ballistic and trapezoidal schemes are of equal perfor-mance as RK4. Analytic consideration and numerical results show that, although of the same theoretical consistency order

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Fig. 9. Top: gap in front of the test vehicle. The three discontinuities represent three vehiclesmerging in front of this vehicle; bottom: convergence diagramfor the test vehicle.

p = 1 as Euler’smethod, the ballistic scheme is superior in all cases, typically requiring 30% of computation time for the sameaccuracy. Finally,when including lane changes, RK4 turned out to have theworst performance, evenworse than simple Euler.

In summary, we recommend the ballistic and trapezoidal methods as efficient and robust schemes for integrating car-following models. Although of the same theoretical consistency order p = 1 as Euler’s method, the ballistic scheme turnedout to be consistently better than Euler’s scheme.

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