Comparison of Numerical Schemes for LWR Model under Heterogeneous Traffic Conditions

Abstract

First order macroscopic model like Lighthill-Whitham-Richards (LWR) has been extensively studied and applied for various homogeneous traffic problems. Applicability and adaptability of LWR models to various heterogeneous traffic conditions is still under exploration. Finding solutions for the macroscopic models using analytical methods is a complicated task, numerical approximations are used. The present study attempts to understand the suitability of different numerical schemes for a traffic conditions in detail. Various first order and second order numerical schemes were chosen for numerical integration. Derivation of the numerical scheme, several important issues like accuracy, stability and convergence of each scheme were discussed. Simulated variables like flow, density and speeds were compared with the original data collected from the two different urban arterials with and without bottlenecks in Delhi, India. The comparison of the results of various numerical schemes shows that Lax-Friedrichs and MacCormack schemes produced better results and more stable than the other schemes.

Full text

132 Period. Polytech. Transp. Eng. H. K. Gaddam, A. Chinthireddy, K. R. Rao
Comparison of Numerical Schemes
for LWR Model under Heterogeneous

Hari Krishna Gaddam1, Anjaneyulu Chinthireddy1,
K. Ramachandra Rao1*
Received 02 June 2015; accepted 15 October 2015
Abstract
First order macroscopic model like Lighthill-Whitham-
Richards (LWR) has been extensively studied and applied
for various homogeneous trafc problems. Applicability and
adaptability of LWR models to various heterogeneous trafc
conditions is still under exploration. Finding solutions for the
macroscopic models using analytical methods is a complicated
task, numerical approximations are used. The present study
attempts to understand the suitability of different numerical
schemes for a trafc conditions in detail. Various rst order
and second order numerical schemes were chosen for numeri-
cal integration. Derivation of the numerical scheme, several
important issues like accuracy, stability and convergence of
each scheme were discussed. Simulated variables like ow,
density and speeds were compared with the original data col-
lected from the two different urban arterials with and without
bottlenecks in Delhi, India. The comparison of the results of
various numerical schemes shows that Lax-Friedrichs and
MacCormack schemes produced better results and more stable
than the other schemes.
Keywords
LWR model, numerical schemes, v-k relationship
1 Introduction
Macroscopic trafc ow modelling represent how the behav-
iour of one characteristic of trafc (trafc ow, speed and density)
changes with respect to other trafc characteristic. Macroscopic
models mainly focus on overall trafc system features like con-
gestion, delay and queue formation but not on the behaviour of
a specic vehicle. The kind of problem may be extended from
estimation of macroscopic trafc ow variables to study of traf-
c operations or implementation of alternative route networks
based on travel time and delay; study of bottlenecks, Freeway
trafc ow problems and other. Selection of a model mainly
depends on simplicity, accuracy, efciency and extensibility.
Lighthill and Whitham model was the rst major step in
the macroscopic modelling with papers, ‘trafc ow on long
crowded roads’ and ‘ood movements in long rivers’ (Lighthill
and Whitham, 1955), Subsequently, Richards extended the
Lighthill and Whitham idea with the introduction of ‘shockwaves
on the highway’ in an identical approach (Richards, 1956). Thus
Lighthill-Whitham-Richards (LWR) model came into existence.
LWR model was successful in producing shockwaves and iden-
tifying trafc jams but has limitation in reproducing nonequilib-
rium trafc ow situations like stop and go pattern, queue dissi-
pation and hysteresis etc (Škrinjar et al., 2015). These drawbacks
have led to the formulation of higher order models.
Even though Payne type higher order models are good at
improving accuracy over rst order models but they are math-
ematically complicated and also it is difcult to understand and
solve analytically, whereas rst order models are easy to under-
stand, implement and analyse. This makes simulation of trafc
ow by using LWR model more attractive (Leo and Pretty, 1990).
Thus, the basic objective of this study is to check the applicabil-
ity of the LWR model to two different types of trafc situations.
One is with uniform lane conditions with free ow and second
one is with bottleneck situation with trafc ow is at capacity.
However the existing research is mainly focused on homog-
enous trafc ow condition and predominantly it is restricted to
cars only trafc (Bhavathrathan and Mallikarjuna, 2012; Mohan
and Gitakrishnan, 2013). The heterogeneous trafc environment
consists of different operational and performance characteristics
1 Department of Civil Engineering, IIT Delhi, Hauz Khas, New Delhi, India
* Corresponding author, e-mail: rrkalaga@civil.iitd.ac.in
44(3), pp. 132-140, 2016
DOI: 10.3311/PPtr.8297
Creative Commons Attribution b

P
P
Periodica Polytechnica
Transportation Engineering

133
Comparison of Numerical Schemes for LWR Models 2016 44 3
of vehicles and vehicles are categorised as motorized and non-
motorized vehicles. The Motorised vehicles comprises of cars,
busses, trucks, two-wheelers, LCVs, auto rickshaws and oth-
ers. Non-Motorised vehicles comprises of bicycles, cycle rick-
shaws and others. There is a high degree of variation in physical
and dynamical characteristics of vehicles ply on the road and
especially two wheelers have high manoeuvrability. Vehicles
on the road use same right of way. Complexity is increased by
presence of considerable amount of pedestrians, encroachment,
street parking and others. Altogether it makes modelling dif-
cult and thus there is a need for special methods to treat this
trafc environment (Khan and Maini, 1999). Few attempts were
made to incorporate heterogeneity by using higher order models
(Ajitha et al., 2014; Mallikarjuna and Bhavathrathan, 2014). It
is evident that a comparative study with rst order LWR is not
found. This study attempts to understand the suitability of LWR
model for heterogeneous trafc ow conditions.
In macroscopic model while simulating the trafc ow, it is
crucial to use proper numerical schemes. Macroscopic mod-
els are solved either by using analytical methods or numeri-
cal approximation schemes. Though analytical methods are
more accurate they are arduous to implement in all kinds of
trafc situations. On the other hand implementation of numeri-
cal schemes is easy. Partial differential equations can be
solved by using rst order and second order explicit numerical
schemes. It is always preferred to implement different numeri-
cal schemes and compare their simulation results (Helbing and
Treiber, 1999). Second objective of this study is to implement
different numerical schemes to trafc ow problems and nd
their accuracy in reproducing trafc behaviour.
Rest of the paper is organised as follows. The next section
explains the basic structure of the LWR model. Third section
describes about various numerical schemes adopted in the pre-
sent study, their accuracy and stability issues, fourth section
describes the data collection, while the fth section presents the
formulation of MATLAB® programming, sixth section presents
the results and discussion on the ndings in this study.
2 LWR model
In LWR model relationships among macroscopic variables
ow, density or concentration, position or location of entities
are modelled. The LWR model is a simple continuum model
which consists of conservation or continuity equation, funda-
mental ow equation, equation of state or equilibrium speed-
density relationship. Mathematical structure of the model is
given in Eqs. (1), (2) and (3).
Conservation of mass equation:
∂
∂+



∂
∂




=
k
t
dq
dk
k
x0
Fundamental equation:
qkv=∗
Equilibrium equation:
vvk
e
=
()
In the present study, numerical solutions are adopted over
analytical solutions as a solution method for LWR model. Finite
difference method, nite volume method and nite volume
methods are used for nding solutions for any kind of partial
differential equations. In this study nite difference numerical
schemes are used as solution methods.
3 Numerical solutions: Finite Difference Methods
(FDMs)
The LWR model, i.e. continuity equation coupled with
steady state equation needs to be solved numerically (Treiber
and Kesting, 2013). In many of the FDMs, derivatives are
replaced by appropriate nite difference approximations. Two
types of FDMs are implicit FDMs and explicit FDMs. In this
paper, explicit methods are used because they are useful for
varying boundary conditions according to realistic trafc ow
simulations. In explicit FDMs, derivatives are replaced by
backward, forward, central differences etc.
3.1 Explicit methods
3.1.1 First order numerical schemes
1. Upwind scheme
Kktxqk qk
i
n
i
n
i
n
i
n
+
()
−
()
=−
()()
−
()
()
1
1
∆∆
qk vk
i
n
i
n
i
n
()
=−
vv kk
i
n
fi
n
jam
n
=−
()()
()
1Greenshield
vv kk
i
n
fi
n
=−
()
()
()
exp0Underwood
2. Central difference scheme
kk txqkqk
i
n
i
n
i
n
i
n
+
()
−
()
+
()
=+
()()
−
()
()
1
11
05.∆∆
3. Downwind scheme
kktxqk qk
i
n
i
n
i
n
i
n
+
()
+
()
=+
()()
−
()
()
1
1
∆∆
4. Lax-Friedrichs scheme
kkk
txqkqk
i
n
i
n
i
n
i
n
i
n
+
()
+
()
−
()
−
()
+
()
=+
()
+
()()
−
(
1
11
11
05
05
.
.∆∆
))
()
5.
Leap-Frog scheme
kk txqkqk
i
n
i
n
i
n
i
n
+
()
−
()
−
()
+
()
=+
()()
−
()
()
11
11
∆∆
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(11)
(10)

134 Period. Polytech. Transp. Eng. H. K. Gaddam, A. Chinthireddy, K. R. Rao
3.1.2 Second order numerical schemes
1. Lax-Wendroff scheme
kkk
txqkqk
i
n
i
n
i
n
i
n
i
n
+
()
+
()
+
()
+
()
=+
()
−
()()
−
(
05
05
1
1
05
05
.
..
.∆∆
))
()
… Predictor
kktxqkqk
i
n
i
n
i
n
i
n+
()
+
()
+
()
−
()
+
()
=−
() ()
−
()
()
1
05
05
05
05
∆∆ .
.
.
.
… Corrector
2. MacCormack scheme
kk txqkqk
ii
n
i
n
i
n∗
+
()
=−
()()
−
()
()
∆∆ 1
… Predictor
kkktxqkq k
i
n
i
n
ii
i
+
()
∗∗
−
()
∗
=+
()
−
()()
−
()
()
1
1
05 05..∆∆
… Corrector
Eqs. (5), (6) and (7) are common for all other schemes.
3.2 Consistency and accuracy of numerical scheme
Consistency deals with the extent to which FDMs approxi-
mate the PDEs.
efAfN
h=
()
−
()
Truncationerror
Where
f (A) Analytical Solution and
f (N) Numerical Solution
The numerical scheme is said to be consistent if the trunca-
tion error (eh) satises
lim ,∆∆tx h
e
→=
00
The numerical scheme is said to be constituent of order (a,
b) accurate or simply of order (a, b) where
eo tx
h
ab
=
()
+
()
()
∆∆
3.3 Stability
A numerical scheme is said to be stable if truncation error,
round-off error etc. are not allowed to increase as the calculation
proceeds from one step to next step. CFL condition is a manda-
tory condition that should be satised for the stability of numer-
ical scheme. However CFL condition is necessary condition but
not a sufcient condition for the stability of numerical scheme.
3.3.1 Courant-Friedrichs-Lewy (CFL) condition
CC vtx
f
or Courant number
CFL≤
()
∆∆ 
CFL condition is that ∆x ⁄ ∆t should be greater than vf . Δx
and Δt are chosen in accordance with the CFL condition.
Table 1 Explicit nite difference methods (LeVeque, 1992)
Numerical scheme CFL Condition Stability
Upwind Satised if ∆t ≤ ∆x/ |a|Stable
Central difference Satised if ∆t ≤ ∆x/ |a|Unstable
Downwind Not satised Unstable
Lax-Friedrich Satised if ∆t ≤ ∆x/ |a|Stable
Leap-Frog Satised if ∆t ≤ ∆x/ |a|Stable
Lax-Wendroff Satised if ∆t ≤ ∆x/ |aStable
MacCormack Satised if ∆t ≤ ∆x/ |aStable
4 Data collection
The trafc data on Aruna Asaf Ali Marg (near IIT Delhi, N
Delhi, India) and Sri Aurobindo Marg (near INA metro station,
N Delhi, India) collected during March 2015. Three video cam-
eras are placed at entry point, near midpoint and exit point as
shown in Fig. 1 and Fig. 2. Spot speeds of vehicles collected
by using speed guns. Space mean speed is calculated from spot
speed data. Car, two-wheeler, three-wheeler and truck (or bus)
count for every 5 seconds manually extracted from trafc video
footage. Total hourly trafc volume at Aruna Asaf Ali Marg is
1533 veh/hr and Total hourly trafc volume at Sri Aurobindo
Marg is 3085 veh/hr. Trafc volume is converted from veh/hr to
pcus/hr using Static PCU values given in Table 2 (IRC, 1990).
Fig. 1 Aruna Asaf Ali Marg , New Delhi (India)
Fig. 2 Sri Aurobindo Marg, New Delhi (India)
Table 2 Recommended Static PCU values on urban roads.
By IRC-106 (1990)
Vehicle type
Equivalent PCU Factors for percentage
composition of Vehicle type in trafc stream
5% 10% and above
Two wheeler 0.5 0.75
Passenger car 1.0 1.0
Three wheeler 1.2 2
Bus 2.2 3.7
(12)
(12)
(13)
(14)
(17)
(18)
(18)
(16)

135
Comparison of Numerical Schemes for LWR Models 2016 44 3
4.1 Fitting v-k relationship
As shape of v-k graph control the performance of mathe-
matical model, attempts are made to get reliable v-k relation.
v-k graph is drawn by taking speed and density values at each
minute from three camera locations. The v-k plot for Arun Asaf
Ali Marg is shown in Fig. 3 and for Sri Aurobindo Marg in Fig.
4 and 5. Here the v-k relationships are summarized in Table
3. For the present study Greenshiled Linear relationship and
Underwood exponential relationship were taken into consid-
eration. Extensive review on macroscopic v-k relationships is
given in Wang et al. (2010).
Fig. 3 v-k graph for Aruna Asaf Ali Marg
Fig. 4 v-k graph for Sri Aurobindo Marg (Whole stretch)
Fig. 5 v-k graph for Sri Aurobindo Marg (Within the bottleneck)
5 Formulation of MATLAB® programming
The estimation of trafc ow variables using numerical approx-
imation is done in MATLAB® environment. The steps involved in
numerical simulation programme are discussed below.
Step1: Discretization: Solution for the continuum models
are estimated by discretizing the space (road) and time. ∆x and
∆t are spatial step and time step respectively. Macro variables
k, v, q are estimated at each of the nodes (j ∆x, n ∆t) with j, n ∈
{0, 1, 2 …..n}. Here Kn
j = K (j ∆x, n ∆t).
Step2: Initial conditions: Initial conditions on the road has
to be specied in the simulation program for evaluating traf-
c ow variables. Uniform trafc ow conditions are assumed
for Aruna Asaf Ali Marg trafc ow problem and Square wave
problem has chosen to specify the initial condition for Sri
Aurobindo Marg trafc ow problem. The values that chosen
for two different trafc situation are given below
For Aruna Asaf Ali Marg: K0(x) = 2 if 0 ≤ x ≤ 1600
Kx x
Kx
0
0
15 300 600
10
()
=≤≤
()
=





if
otherwise
For Sri Aurobindo Marg
The initial conditions affect the outcomes of the simulation
program only for short duration in open boundary case. This
is because errors due to specication of initial conditions are
propagated very quickly outside the simulation section (Helbing
and Treiber, 1999).
Table 3 Parameters for the numerical programming implementation
Location v-k Relationships Parameters
Aruna Asaf
Ali Marg
Linear Fit : V= 65.817(1-(K/152)) R² = 0.3432 Vf=65.817, Kj=152
Exponential Fit : V = 66.65exp-(K/125) R² = 0.3465 Vf=66.65, K0=125
Sri Aurobindo
Marg (whole
stretch)
Linear Fit: V = 53.47(1-(K/341)) R² = 0.5727 Vf=53.47, Kj=341
Exponential Fit: V = 57.673exp-(K/250) R² = 0.6236 Vf=57.673, K0=250
Sri Aurobindo
Marg (with in
the bottleneck)
Linear Fit: V = 40.509(1-(K/462.96)) R² = 0.7395 Vf=40.509, Kj=463
Exponential Fit: V = 43.099exp-(K/333.33) R² = 0.7773 Vf=43.099, K0=333

136 Period. Polytech. Transp. Eng. H. K. Gaddam, A. Chinthireddy, K. R. Rao
Boundary Conditions: Dirichlet (or xed) boundary condi-
tions are considered for the present problem. Dirichlet conditions
are empirically measured density values and continuously fed to
the simulation program. Dirichlet boundary conditions are speci-
ed by k (0, t), k (L, t) and are time dependent and they represent
boundary conditions at entry and exit point respectively.
Step 3: By specifying initial and boundary conditions for
the road, trafc ow variables q, k, v are estimated for the next
time step for the sections i = 2 to nx − 1(i = 1 and nx are
dened as source and sink cells for the program). If t = nal
time step then the program will be terminated.
5.1 For uniform lane condition
First trafc situation considered in this study is uniform lane
conditions at Aruna Asaf Ali Marg in New Delhi, trafc data
collected at this location shows free ow conditions and has lot
of uctuations due to presence of signals. Descriptions of the
parameters used in the modelling are given below.
Length of the road section under consideration is 1600 m.
Total length of the simulation time: 45 minutes (2700 sec)
∆x = 200 m, ∆t = 5 sec, initial conditions and boundary con-
ditions as specied in step 3 of Section 5.
v-k relationships developed for the entire section and param-
eter values used in the model Kjam , Vf , K0 are taken as men-
tioned in Table 3.
5.2 For lane drop condition
For this second situation, arterial road with lane drop (work
zone) i.e., Aurobindo Marg, New Delhi is chosen for trafc
data collection. Details about the road cross section and data
collection is discussed in section 4. On this arterial road trafc
ow is continuous and higher than the earlier mentioned uni-
form lane situation. Trafc behaviour is complicated and speed
reductions, high density variations can be expected at bottle-
neck if the trafc ow exceeds the capacity. Various numerical
schemes mentioned earlier are applied to check the efciency
in reproducing this situation. Descriptions of the parameters
used in the modelling mentioned below.
Length of the road section under consideration is 1000 m.
Length of the work zone is 200 m.
Total length of the simulation time: 35 minutes (2100 sec)
∆x=200 m, ∆t=5 sec, initial conditions and boundary condi-
tions as specied in Section 5.
Greenshield and Underwood steady state equations tted for
whole stretch and the work zone separately. Parameter values
Kjam, Vf, K0 are taken as mentioned in Table 3.
Various numerical schemes applied to these trafc condi-
tions for nding solution are explained in the previous section.
Outcomes and performance of the each scheme will be dis-
cussed in the next section.
6 Results and analysis
The performance of various numerical schemes studied in
estimating macroscopic ow variables by simulating two dif-
ferent eld situations. First order nite difference schemes like
upwind scheme, Lax-Friedrichs (LaxFr.) scheme and second
order nite difference schemes like Lax-Wendroff (LaxW.)
method, MacCormack (MacC.) scheme are used. We dropped
central difference, downwind and Leap Frog schemes because
of their instability while simulation. Trafc evolution cap-
tured for every 5 seconds in terms of density in pcu/km, speed
in km/h, and ow in pcu/h. Trafc evolution on Aruna Asaf
Ali Marg and Sri Aurobindo Marg are simulated by using
Greenshield and Underwood equilibrium speed-density rela-
tionships separately.
6.1 For Uniform Lane Condition
Figures 6 and 7 show the space–time evolution of the density
for different v-k relationship with different numerical schemes.
Fig. 6 Density evolution with respect to space and time for Greenshield
v-k relationship with Lax Friedrichs scheme
Fig. 7 Density evolution with respect to space and time
for Underwood V-K relationship with Laxwendroff scheme

137
Comparison of Numerical Schemes for LWR Models 2016 44 3
Density values are estimated for 5, 10, 15, 20, and 30 sec-
onds. Error in estimating density is very high for small time
step values and it gradually decreases with increasing time step
value. For Greenshield steady state equation, at lower time step
values MacCormack scheme works well whereas at higher time
steps values Lax-Friedrichs scheme and MacCormack scheme
give good results. For Underwood steady state equation there is
a clear distinction between upwind, Lax-Wendroff schemes and
other schemes. They prove accurate compared to other schemes.
The simulation results for both steady state equations shows that
Lax-Wendroff scheme with underwood steady state equation
yield good results for higher time step.
Mean Absolute Percentage Error (MAPE) values for various
schemes are shown in Table 4 and Table 5.
Table 4 MAPE values for various time steps for Aruna Asaf Ali Marg
Time (Sec) MAPE(%) for Greenshield
LaxFr. Upwind LaxW. MacC.
558.92 38.10 46.42 33.79
10 37.95 25.80 30.71 24.50
15 18.22 15.30 14.84 15.35
20 11.83 12.84 11.93 12.79
30 10.79 12.66 12.17 11.86
Table 5 MAPE values for various time steps for Aruna Asaf Ali Marg
Time (Sec) MAPE(%) for Underwood
LaxFr. Upwind LaxW. MacC.
560.83 41.86 51.84 46.11
10 39.39 29.50 35.28 31.17
15 19.27 15.83 16.84 18.48
20 12.98 11.15 11.50 14.17
30 12.30 9.49 8.91 12.30
Figure 8 and Fig. 9 gives a visual idea of variation of MAPE
versus different time step values. Efciency of all numerical
schemes is very low at lower time steps.
Fig. 8 Variation of MAPE value for various time steps using Greenshield v-k
relation
Fig. 9 Variation of MAPE value for various time steps using Underwood v-k
relation
Figure 10 and Fig. 11 shows the comparison between
observed and estimated density values for 30 sec time step.
Fig. 10 Density variation with respect to time step
using Lax –Friedrichs method
Fig. 11 Density variation with respect to time step
using Lax –Wendroff method
From this study it is understood that LWR model is unsuita-
ble for low trafc volume situations where interaction between
vehicles is less and steady state equation fails in capturing
speed uctuations around equilibrium ow.

138 Period. Polytech. Transp. Eng. H. K. Gaddam, A. Chinthireddy, K. R. Rao
6.2 For lane drop condition
Density evolution with respect to space and time is shown in
Fig. 12 and Fig. 13.
Fig. 12 Density evolution with respect to space and time
for Greenshield v-k relationship with Lax-Friedrichs scheme
Fig. 13 Density evolution with respect to space and time
for Underwood v-k relationship with MacCormack Scheme
High density region is observed before commencing of the
work zone that means trafc ow exceeded the capacity of the
work zone. From the numerical analysis it is observed that Lax-
Friedrichs and MacCormack schemes are outperforming other
methods. Table 6 and Table 7 shows MAPE values concerned
to various time steps.
Table 6 MAPE values obtained for various time steps
for Sri Aurobindo Marg
Time (Sec) MAPE(%) for Greenshield
LaxFr. Upwind LaxW. MacC.
524.84 25.61 22.22 25.97
10 17.32 24.29 18.46 19.89
15 10.95 24.76 17.30 15.20
20 8.05 23.75 16.36 11.31
30 6.81 26.13 17.92 8.19
Table 7 MAPE values obtained for various time steps
for Sri Aurobindo Marg
Time (Sec) MAPE(%) for Underwood
LaxFr. Upwind LaxW. MacC.
525.39 25.92 20.26 23.67
10 18.10 23.97 16.04 17.24
15 11.21 23.95 15.59 12.93
20 9.05 22.92 14.82 9.99
30 6.69 25.23 15.71 7.26
Lax Friedrichs and MacCormack schemes are very consist-
ent at all levels. Efciency of estimating macroscopic trafc
variables with Greenshield and Underwood steady state equa-
tion is very high in Lax Friedrichs case. Visual representation
of variation of MAPE values versus time step are shown in
Fig. 14 and Fig. 15.
Fig. 14 Variation of MAPE value for various time steps
using Greenshield v-k relation
Fig. 15 Variation of MAPE value for various time steps
using Underwood v-k relation
Figure 16 and Fig. 17 shows the comparison between
observed and estimated density values for 30 sec time step.

139
Comparison of Numerical Schemes for LWR Models 2016 44 3
Fig. 16 Density variation with respect to time step
using Lax –Friedrichs method
Fig. 17 Density variation with respect to time step
using Lax –Friedrichs method
MacCormack scheme is successful in preventing vehicles
moving backward from the jam region to the empty region
(Ngoduy et al., 2004). MacCormack scheme produces numeri-
cal dispersion, leads to oscillations in large density gradient
regions and this scheme is more sensitive to the nonlinear insta-
bilities compared to rst order schemes like Lax-Friedrichs and
upwind scheme. MacCormack scheme is efciently reproduce
trafc conditions in congested trafc situation. In bottleneck
situation trafc ow is continuous and there is interaction
among the vehicles. This kind of situation has profound inu-
ence on performance of rst order macroscopic models and
the results replicate the same. When compared to uniform lane
situation accuracy is improved in lane drop situation.
7 Discussion
The main aim of the present study was to examine the
performance of rst order macroscopic models in represent-
ing the trafc stream behaviour. For that two different trafc
situations have chosen to implement numerical simulation
approach. Based on stability, accuracy and convergence point
of view four numerical schemes have nalised and they are
upwind, Lax-Friedrichs, Lax-Wendroff and MacCormack
schemes. Numerical simulation was implemented for two
types of v-k relationships namely Greenshield and Underwood.
From this study it was found that efciency of the numerical
schemes at smaller time steps (5 sec) is very low in free ow
trafc behaviour observed at Aruna Asaf Ali Marg. At higher
time steps (30 sec) Lax-Wendroff scheme with Underwood
steady state relationship works well for Aruna Asaf Ali Marg.
Accuracy of the simple LWR model is affected in capturing
uctuations in trafc ow due to static fundamental relation-
ship used in the model. Whereas the efciency of numerical
schemes in lane drop situation was impressive. Interestingly
Lax-Friedrichs scheme estimates trafc ow variables more
accurately with both Greenshield and Underwood, followed
by MacCormack scheme. The trafc ow observed at Sri
Aurobindo Marg (lane drop situation) almost behaves like con-
tinuous uid ow that means there is a continuous interaction
among the vehicles. This could be one of the reasons behind the
success of LWR model and if one observes the v-k relationship
for Sri Aurobindo Marg there is a little uctuation of speeds
around the mean. Lax-Friedrichs scheme has been proved to be
stable approximation with different v-k relationships.
Selection of a macroscopic model for a particular problem
depends on its strength and weaknesses of the model. From the
literature it is understood that the rst order models fail in repro-
ducing non-equilibrium trafc phenomenon observed on the
road like stop and go pattern, queue pattern, formation of clus-
ters and hysteresis (Daganzo, 1995; Zhang, 1998). These draw
backs emerge from the fact that the steady state equation does
not allow the speed to uctuate around equilibrium speed. There
is only one speed for a given parameters. This problem also per-
sists in the two trafc conditions observed in the present study.
Accuracy of the model may improve by replacing steady state
equation with dynamic velocity equation. Numerical Stability
of the explicit nite difference schemes is depend on CFL con-
dition i.e., vf (∆t/∆x) ≤ 1 and ∆x, ∆t values chosen accordingly.
Accuracy of the scheme is greatly inuenced by CFL condition
and it is one of the drawbacks of the explicit nite difference
schemes. Implicit nite difference scheme can be used if stabil-
ity is the issue. This numerical study can be further extended to
represent trafc heterogeneity by incorporating the area occu-
pancy in place of linear density (Mallikarjuna and Rao, 2006).
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