GANIT J. Bagladesh Math. Soc. (ISSN 66-3694 3 ( 43-5 A FINITE DIFFERENCE SCHEME FOR A FLUID DYNAMIC TRAFFIC FLOW MODEL APPENDED WITH TWO-POINT BOUNDARY CONDITION M. O. Ga, M. M. Hossa ad L. S. Adallah Departmet of Mathematcs Jahagragar Uversty, Savar, Dhaka, Bagladesh e-mal: adallahls@gmal.com, osmaga@yahoo.com Receved 9.. Accepted 6.7. ABSTRACT A flud dyamc traffc flow model wth a lear velocty-desty closure relato s cosdered. The model reads as a quas-lear frst order hyperbolc partal dfferetal equato (PDE ad order to corporate tal ad boudary data the PDE s treated as a tal boudary value problem (IBVP. The dervato of a frst order explct fte dfferece scheme of the IBVP for two-pot boudary codto s preseted whch s aalogous to the well kow Lax-Fredrchs scheme. The Lax-Fredrchs scheme for our model s ot straght-forward to mplemet ad oe eeds to employ a smultaeous physcal costrat ad stablty codto. Therefore, a mathematcal aalyss s preseted order to establsh the physcal costrat ad stablty codto of the scheme. The fte dfferece scheme s mplemeted ad the graphcal presetato of umercal features of error estmato ad rate of covergece s produced. Numercal smulato results verfy some well uderstood qualtatve behavor of traffc flow.. Itroducto The problem of computer smulato techques of the traffc flow models has become a mportat area the feld of umercal soluto methods. May research groups are volved dealg wth the problem wth dfferet kds of traffc models (lke fluddyamc models, ketc models, mcroscopc models etc. for several decades. E.g. I [], the author shows that f the kematcs wave model of freeway traffc flow ts geeral form s approxmated by a partcular type of fte dfferece equato, the fte dfferece results coverge to the kematcs wave soluto despte the exstece of shocks the latter. Errors are show to be approxmately proportoal to the mesh spacg wth a coeffcet of proportoalty that depeds o the wave speed, o ts rate of chage wth desty, ad o the slope ad curvature of the tal desty profle. The asymptotc errors are smaller tha those of Lax's frst-order, cetered dfferece method whch s also coverget. I [], the author develops a fte dfferece scheme for a prevously reported o-equlbrum traffc flow model. Ths scheme s a exteso of Goduov's scheme to systems. It utlzes the solutos of a seres of Rema problems at cell boudares to costruct approxmate solutos of the o-equlbrum traffc flow model uder geeral tal codtos. Moreover, the Rema solutos at both left (upstream ad rght (dowstream boudares of a hghway allow the specfcato of correct boudary codtos usg state varables (e.g., desty ad/or speed rather tha
44 Ga, Hossa ad Adallah fluxes. Prelmary umercal results dcate that the fte dfferece scheme correctly computes etropy-satsfyg weak solutos of the orgal model. I [3], the authors cosder a mathematcal model for flud dyamc flows o etworks whch s based o coservato laws. The approxmato of scalar coservato laws alog arcs s carred out by usg coservatve methods, such as the classcal Goduov scheme ad the more recet dscrete veloctes ketc schemes wth the use of sutable boudary codtos at juctos. Rema problems are solved by meas of a smulato algorthm whch processes each jucto. The above dscusso gves us the motvato to study ad vestgate effcet fte dfferece scheme for the traffc flow smulato. Wth ths eds, ths artcle, secto, we cosder a flud-dyamc traffc flow model whch has bee developed frst by Lghthll ad Whtham (955 ad Rchard (956 shortly called LWR model ad preset the basc features of the model based o Haberma (977 [4], Klar [5]. The model descrbes traffc pheomea resultg from teracto of may vehcles by dscussg the fudametal traffc varables lke desty, velocty ad flow. I partcular, a lear velocty-desty closure relato yelds a quadratc flux-desty relato leads to formulate a frst order o- lear partal dfferetal equato. The exact soluto of the o-lear PDE as a Cauchy problem s preseted. However, order to corporate tal ad boudary data, the o-lear frst order partal dfferetal s appeded by tal ad boudary value ad formulates a tal boudary value problem (IBVP. Certaly, a umercal method s eeded for the umercal mplemetato of the IBVP practcal stuato ad t s completely uavodable to use umercal method to solve real traffc flow problem. Therefore, based o the study of geeral fte dfferece method for frst order o-lear PDE from Leveque 99 [6], we preset a frst order explct fte dfferece scheme, whch s aalogous to the well-kow Lax-Fredrch scheme, for our cosdered traffc flow model as a IBVP wth two-sded boudary codto. The scheme s ot straght forward to mplemet ad oe eeds to employ a smultaeous physcal costrat ad stablty codto. Thus, secto 4, we perform mathematcal aalyss ad establsh the physcal costrats ad stablty codto. We develop ecessary computer programmg code for the mplemetato of the umercal scheme ad perform umercal smulatos order to verfy some qualtatve traffc flow behavor for varous traffc parameters. I secto 5, we preset the umercal features of error estmato ad rate of covergece. We also preset umercal smulatos results order to verfy some well-uderstood qualtatve traffc flow behavor.. A Traffc Model based o a lear velocty-desty fucto The flud-dyamc traffc flow model s used to study traffc flow by collectve varables such as traffc flow rate (flux q ( x, t, traffc speed v ( x, t ad traffc desty ( x, t, all + of whch are fuctos of space, x R ad tme, t R. The well-kow LWR model ([4], [5], [7] based o the prcple of mass coservato reads as
A Fte Dfferece Scheme for a Flud Dyamc Traffc Flow Model 45 q + = Isertg a lear velocty-desty closure relatoshp the flux q takes the form v ( ( = v q (3 ( = v ( =. v ad ( leads to formulate a olear frst order hyperbolc partal dfferetal equato (PDE of the form + v ( = The graph of the o-lear flux fucto gve by equato (3 s kow as Fudametal Dagram as sketched below. q ( (4 q Fgure : Fudametal Dagram of Traffc Flow 3. Exact soluto of the o-lear PDE by the method of characterstcs The olear PDE (4 ca be solved f we kow the traffc desty at a gve tal tme,.e., f we kow the traffc desty at a gve tal tme we ca predct the traffc desty for all future tme t t, prcple. The we have to solve a tal value problem (IVP of the form t
46 Ga, Hossa ad Adallah The exact soluto of the IVP (5 s gve by + v ( ( t, x = ( x = ( t, x = (, x = ( x = x v ( t (6 However, realty t s very complcated to approxmate the tal desty ( of the x (5 Cauchy problem as a fucto of t from gve tal data. Therefore, there s a demad of some effcet umercal methods for solvg the IVP (5. Nevertheless, cosderg a smple form of tal value, the aalytc soluto (6 ca be used to perform a error estmato of the umercal scheme. 4. Lax-Fredrchs scheme for the umercal soluto of the IBVP I ths secto we preset the dervato of the fte dfferece scheme for our model wth lear desty-velocty relato appeded wth tal ad two pot boudary codto. The frst order scheme s aalogous to the well-kow Lax-Fredrchs scheme [6]. We establsh the physcal costrats ad stablty codto for our verso of Lax- Fredrchs scheme. For ths, we cosder the traffc model (4 as a IBVP wth two-pot boudares as below. (IBVP q( + =, t t T, a x b wth I.C. ( t, x = ( x; a x b ad B.C. (t, a = a ( t; t t T (t, b = b ( t; t t T where, by equato (3, ( =. q v. As we cosder that the cars are rug oly the postve x-drecto, so the speed must be postve,.e. the rage of, q ( = v( (8 (7 Dscretzg the tme dervatve the IBVP (7 at ay dscrete pot =, L, M ; =, L, N ; by the forward dfferece formula ( t, x for
A Fte Dfferece Scheme for a Flud Dyamc Traffc Flow Model 47 + ( t, x t q ad the spatal dervatve by the cetral dfferece formula q t, x q + q x ( Isertg (9 ad ( (7, the dscrete verso of the o-lear PDE formulates the frst order fte dfferece scheme of the form where [ q( q( ] + t = +, =, L, M ; =, L, N ( x (9 ( ( q ( = v ( I the fte dfferece scheme, the tal data for all =, L, M ; s the dscrete versos of the gve tal value ( ad the boudary data ad for x all =, L, N are the dscrete versos of the gve boudary values a (x ad b (x respectvely. Ufortuately, despte the qute atural dervato of the scheme (, t suffers from severe stablty problems ad s useless practce. But f we replaces by ( + + the the ustable scheme ( becomes stable provded t x s suffcetly small, as t wll be proved the ext secto s proposto. The ( takes the form [ q( q( ] + t = ( + + + ; =, L, M ; =, L, N (3 x Ths dfferece equato s kow as Lax-Fredrchs scheme. Now we study the stablty codto of ths scheme for our model whch, ( q( = v (. 5. Stablty ad Physcal Costrats codtos The mplemetato of Lax-Fredrchs scheme s ot straght forward. Sce the car s dq movg oe drecto, so the characterstc speed must be postve. dt a b
48 Ga, Hossa ad Adallah.e. q ( = v ( (4 q ( v Proposto: The stablty ad physcal costrat codto of the Lax-Fredrchs scheme (3 s guarateed by the smultaeous codtos respectvely v t ad = k ( x, k. x Proof: Rewrtg the o-lear PDE (7 as + q ( =, the Lax-Fredrchs scheme (3 takes the form + t where λ : = q ( x [ ] t = ( + + ( x q + (5 = ( + + λ! + λ+ (6 + = [( + λ + ( λ + ] ; (7 whch s weghted arthmetc mea or smply weghted mea of ad for λ. The equato (6 mples that f λ, the ew soluto s a covex combato of the two prevous solutos. That the soluto at ew tme-step ( + at a spatal-ode, s a average of the solutos at the prevous tme-step at the spatal-odes ad +. Ths meas that the extreme value of the ew soluto s the average of the extreme values of the prevous two solutos at the two cosecutve odes. Therefore, the ew soluto cotuously depedet of the tal value, =, L, M ad the Lax- Fredrchs scheme s stable for + t λ : = q ( (8 x The by the codto (5, the stablty codto (8 ca be guarateed by v t γ : = (9 x
A Fte Dfferece Scheme for a Flud Dyamc Traffc Flow Model 49 Thus wheever oe employs the stablty codto (9, the physcal costrats codto (4 ca be guarateed mmedately by choosg = k ( x, k. ( 6. Numercal results ad dscusso We mplemet the Lax-Fredrchs scheme by developg a computer programmg code ad perform umercal smulato as descrbed below. Error Estmato of the Numercal Scheme I order to perform error estmato, we cosder the exact soluto (6 wth tal codto ( x = x, we have ( t, x = x = ( x v ( t ( t, x ( x v = ( tv t / / We prescrbe the correspodg two-sded boudary values by the equatos ad ( t a ( t b x v t / a = ( t, xa = ( vt / ( t, x x v t / b = b = (3 vt / For the above tal ad boudary codtos wth =.67 (.km/sec =6. km/hour, satsfyg the physcal costrat codto ( v = ( x =5/km 5 the spatal doma [5 km, km], we perform the umercal expermet for 4 mutes t =. tme steps for a hghway of 5 km spatal grd pots wth step sze x = meters =.5 whch guaratees the stablty codto (9 γ =. 668 <. We compute the relatve error -orm defed by: L e : e h = (4 e for all tme where e s the exact soluto (6 ad h s the umercal soluto computed by the Lax-Fredrchs scheme. Fgure shows the relatve errors for Lax-Fredrchs scheme, the relatve errors rema below.4 whch s qute acceptable. It s obseved that the error s decreasg wth respect to the smaller descretzato parameters t ad x, whch shows a very good feature of covergece of the Lax- Fredrchs scheme.
5 Ga, Hossa ad Adallah Error L Norm 4 x -5 3.5 3.5.5 δt=., δx=.5 δt=.5, δx=.5 δt=.333, δx=.833 δt=.5, δx=.65 δt=., δx=.5.5 5 5 5 Tme Secod Fgure : Covergece of Lax-Fredrchs Scheme Now we cosder the tal desty usg se fucto ad perform umercal computato the spatal doma [, ] km. Fgure 3(a shows the tal desty ad the desty after sx mutes. 3(b shows the propagatg traffc waves at two m, four m ad sx m for the Lax-Fredrchs Scheme. 55 5 tal 6mutes 45 Desty of Car [vehcles/.km] 4 35 3 5 5 5 3 4 5 6 7 8 9 Hghway Km
A Fte Dfferece Scheme for a Flud Dyamc Traffc Flow Model 5 Desty of Car [vehcles/.km] 4 35 3 5 m 4m 6m 4 6 8 Hghway Km Fgure 3(a: Ital Desty Profle Fgure 3(b: Tme evoluto of desty profle Fgure 4(a ad 4(b shows the desty u ( t, x profles ad velocty v( t, x profles for three dfferet tmes ad oe ca read from the two fgures that the desty ad velocty are matag the egatve relato, as gve by equato (, throughout the computatoal process as expected. 45 4 m m 3m.58.56.54 m m 3m Decty of car/.km 35 3 5 5 3 4 5 6 7 8 9 Hghway Km Fgure 4(a: Desty profles Velocty of Car.km/sec].5.5.48.46.44.4.4 3 4 5 6 7 8 9 Hghway Km Fgure 4(b: Velocty profles
5 Ga, Hossa ad Adallah Velocty[.km/sec].65.6.55.5.45.4.35 Flux of Traffc.8.6.4..8.6.4. Fudametal Dagram of Traffc Flow.3 5 5 5 3 35 4 45 5 55 Desty /.km 3 4 5 6 7 8 Desty /.km Fgure 5(a: Traffc velocty as a fucto of desty Fgure 5(b: Traffc Flux as a fucto of desty Fally, Fgure 5(a presets the computed car velocty as a fucto of desty ad Fgure 5(b shows the computed flux (traffc flow as a fucto of desty. Fgure 5(b verfes qualtatve behavor, the well-kow Fudametal dagram as Fgure. Cocluso From the umercal results we observed that our verso of Lax-Fredrchs scheme for the cosdered traffc flow model s adequate for traffc flow smulato. The tme-step the establshed stablty codto ad physcal costrats codto s ot stff ad ths resulted computatoal effcecy of the scheme. The computatoal results showed the accuracy up to fve decmal places ad a good rate of covergece. The umercal smulato results verfed some well-kow qualtatve traffc flow behavor. The scheme ca be exteded for mult-lae traffc flow smulatos whch we left for future work. REFERENCES. Carlos F. Dagazo. A fte dfferece approxmato of the kematc wave model of traffc flow, Trasportato Research Part B: Methodologcal Volume 9, Issue 4, (Elsever, p.6-76, 995.. H. M. Zhag. A fte dfferece approxmato of a o-equlbrum traffc flow model, Trasportato Research Part B: Methodologcal Volume 35, Issue 4, (Elsever, p. 337-365,. 3. Gabrella Brett, Roberto Natal, Beedetto Pccol, A Flud-Dyamc Traffc Model o Road Networks, Comput Methods Eg., CIMNE, Barceloa, Spa 7. 4. Rchard Haberma, Mathematcal models, Pretce-Hall, Ic., 977. 5. Axel Klar, Rehart D. Kuhe ad Ramurd Wegeer, Mathematcal models for Vehcular Traffc, Preprt, Dept. of Math., TU of Kaserlauter, Germay, 996. 6. Radall J. LeVeque, Numercal Methods for Coservato Laws, secod Edto, 99, Sprger. 7. Clve L. Dym, Prcples of Mathematcal Modelg, Academc press, 4. 8. L S Adallah, Shajb Al, M O Ga, M K Padt & J Akhter A Fte Dfferece Scheme for a Traffc Flow Model based o a Lear Velocty-Desty Fucto JU J. of Sc. Vol 3, No., 9.
A Fte Dfferece Scheme for a Flud Dyamc Traffc Flow Model 53