1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4...........................

11 2 5

1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4................................. 3 2.5............................... 4 2.6............................. 5 3 5 3.1............................. 5 3.2..................... 6 3.3....................... 8 3.4 1............................... 9 4 10 5 12 6 Lax Friedrichs 13 6.1...................................... 13 6.2................................ 14 6.3.................................... 15 7 15 7.1................................ 15 7.2.................................. 16 7.3........................... 17 8 20 24 i

Lax Friedrichs ii

11 3 1 ( ) ( ) 2 3 3 20[km/h] 9[m] 100[km/h] 10[km/h] 3) 4 5 6 Lax Friedrichs 7 Lax Friedrichs 2 2.1 t x x 0 (t) dx 0 (t)/dt d 2 x 0 (t)/dt 2 x i (t)(i = 1, 2, 3, ) 2 v i v i = dx i (t)/dt N N v i (t)(i = 1, 2, 3,, N) 1 1 v(x, t) 1

x x x x 1 2 3 4 Fig. 2.1 x 1 t 1 v(x 1, t 1 ) x 2 v(x 2, t 1 ) x 1 t 2 v(x 1, t 2 ) x i t j x i (t j ) v(x i, t j ) v i (t j ) v(x i (t j ), t j ) = v i (t j ) (2.1) v(x, t) x, t 1 2 2 2.2 q q q(x, t) 1 ρ L d ρ = 1 L + d (2.2) L d Fig. 2.2 2

2.3 3 ( ) v 0 ρ 0 τ v 0 τ v 0 τ ρ 0 v 0 τ τ ρ 0 v 0 τ q q = ρ 0 v 0 (2.3) ( ) = ( ) ( ) x, t q(x, t) = ρ(x, t)v(x, t) (2.4) 2.4 a b Fig. 2.3 Fig.2.3 x = a, x = b N N = b a ρ(x, t)dx (2.5) x = a, x = b ( q(a, t), q(b, t)) dn/dt x = a x = b q(x, t) 3

dn dt = q(a, t) q(b, t) (2.6) (2.5), (2.6) d b ρ(x, t)dx = q(a, t) q(b, t) (2.7) dt a x = a, x = b (2.7) x = a, x = b x y b b x t x a ρ(y, t)dy = q(a, t) q(x, t) (2.8) x ρ(x, t) = q(x, t) (2.9) t x (2.4) ρ t + (ρv) = 0 (2.10) x 2.5 Lighthill Whitham v = v(ρ) (2.11) 1) (2.11) (2.10) ρ t + (ρv(ρ)) = 0 (2.12) x (2.10) ( ) v max v ρ = v (ρ) 0 (2.13) ρ max v(ρ max ) = 0 (2.14) 4

2.6 2.5 q = ρv(ρ) (2.15) 2 0 1. (ρ = 0) 2. (v = 0 ρ = ρ max ) Fig.2.4 q 0 max Fig. 2.4 3 3.1 (2.10) 2.5 (2.2) L d 3) 5

( ) + ( ) = ( ) Table 3.1 3) [km/h] [m] [m] [m] 20 6 3 9 30 8 6 14 40 11 11 22 50 14 18 32 60 17 27 44 70 19 39 58 80 22 54 76 90 25 68 93 100 28 84 112 Table 3.1 3.2 Fig.3.1 v = Aρ + v max (3.1) A = v max ρ max (3.2) (3.1) (3.2) (2.2) v = v max ρ max 1 L + d + v max (3.3) ρ = ρ max d = 0 (3.3) 6

v v max 0 max Fig. 3.1 v = v maxl d + L + v max (3.4) (3.4) Table 3.1 Fig. 3.2 d 0.12 0.1 data L=0.004 L=0.005 L=0.006 L=0.007 0.08 0.06 0 0 20 40 60 80 100 v Fig. 3.2 L ( ) + ( ) [km] v max = 100 [km/h] 7

3.3 1 f(x) = f 0 + x 1 x 0 x x 0 (f 1 f 0 ) (3.5) (0 v 20) (20 < v 30) (30 < v 40) (90 < v 100) (3.6) d = 0.00045v (0 v 20) d = 0.0005v 0.001 (20 < v 30) d = 0.0008v 0.010 (30 < v 40) d = 0.0010v 0.018 (40 < v 50) d = 0.0012v 8 (50 < v 60) d = 0.0014v 0 (60 < v 70) d = 0.0018v 0.068 (70 < v 80) d = 0.0017v 0.060 (80 < v 90) d = 0.0019v 0.078 (90 < v 100) (3.6) (3.6) (2.2) Fig. 3.3 3.2 80 70 L=0.004 L=0.005 L=0.006 L=0.007 60 50 40 30 20 10 0 20 30 40 50 60 70 80 90 100 v Fig. 3.3 8

3.4 1 Table. 3.1 Fig. 3.4 1 d 0.12 data 0.1 0.08 0.06 0 0 20 40 60 80 100 v Fig. 3.4 1 v = k ρ p + q (3.7) 100 3 k = 3830, p = 30, q = 17 (3.7),(2.2) d d = v q k + p(v q) L (3.8) L = 0.005[km] k, p, q d = v + 17 0.005 (3.9) 3830 30(v + 17) Table 3.2 30 [km/h], 40[km/h], 50[km/h] 20[km/h], 60[km/h], 70[km/h] 80[km/h] 9

[km/h] [km] [km] [km] 0 0 0.00012 20 0.009 0.0085-0.0094 0.01166 30 0.014 0.0135-0.0144 0.0144 40 2 15-24 18 50 2 15-24 18 60 4 35-44 57 70 0.058 0.0575-0.0584 0.0663 80 0.076 0.0755-0.0764 0.1004 90 0.093 0.0925-0.0934 0.1676 100 0.112 0.1115-0.1124 0.3606 Table 3.2 4 4) t 0 = 1.0[s] Table 4.1 µ [km/h] 20 30 40 50 60 70 80 90 100 0.60 0.59 0.58 0.55 0.53 0.50 0.47 0.47 0.47 Table 4.1 µ = 0.53 Table 3.1 t 0 v d v 2 v 2 0 = 2ad (v 0 : a : ) (4.1) d = v2 2a ( ) d = t 0 v v2 2a (4.2) (4.3) 10

ma = µmg (4.4) a a = µg (4.5) (2.2),(4.3),(4.5) 1 2µg v2 + t 0 v 1 ρ + L = 0 (4.6) 2 ( v = t 0 + t 2 0 2 ) µg ( 1 ρ + L) / 1 µg = µ 2 g 2 t 0 2 + 2µg( 1 ρ L) µgt 0 (v > 0) (4.7) t 0 = 1.0(s), µ = 0.53, g = 9.8(m/s 2 ), L = 5.0(m) (3.6) (4.7) Fig.4.1 (4.7) 100[km/h] [m/s] 27.8[m/s] 2 (4.7) v [m/s] 30 data keisan 25 20 15 10 5 0-5 0 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Fig. 4.1 11

5 (Fig. 5.1) N v mgsin F mgcos mg Fig. 5.1 ma = F mg sin θ (5.1) F = µn = µmg cos θ (5.2) a a = g(µ cos θ + sin θ) (5.3) N F mgsin mgcos v mg Fig. 5.2 (Fig. 5.2) ma = F + mg sin θ F = µn = µmg cos θ a = g(µ cos θ sin θ) (5.4) 12

(5.3) θ x θ(x) ( θ(x) > 0 θ(x) < 0) (4.5) a = g(µ cos θ(x) + sin θ(x)) (5.5) (4.7) µ µ cos θ(x) + sin θ(x) (5.6) v(ρ, x) = (µ cos θ(x) + sin θ(x)) 2 g 2 t 0 2 + 2g(µ cos θ(x) + sin θ(x))( 1 ρ L) (2.10) (µ cos θ(x) + sin θ(x))gt 0 (5.7) ρ t + q(ρ, x) x = 0 (5.8) q(ρ, x) = ρv(ρ, x) (5.9) 6.1 6 Lax Friedrichs t x t+ P4 P P 2 1 3 x- x P x+ t Fig. 6.1 Lax Friedrichs 13

ρ t + q(ρ, x) x = 0 (6.1) { ρ(x, 0) = ρ0 (x) (0 < x < M) ρ x (0, t) = ρ x (M, t) = 0 (t > 0) (6.2) Lax Friedrichs (6.1) 2 q ρ q(ρ(x, t), x) = x ρ x + q x x x = q ρ ρ x + q x (6.3) (6.1) ρ t + q ρ ρ x + q x = 0 (6.4) Lax Friedrichs 1 ρ t ρ(p 4 ) ρ(p 1 ) t ( ρ(p 1 ) = ρ(p ) 3) + ρ(p 2 ) 2 (Fig. 6.1) q(ρ) x (6.4) ρ(p 4 ) (ρ(p 2 ) + ρ(p 3 ))/2 t ρ(p 4 ) = ρ(p 3) + ρ(p 2 ) 2 + q(ρ(p 3), x + x) q(ρ(p 2 ), x x) 2 x (6.5) = 0 (6.6) t 2 x {q(ρ(p 3), x + x) q(ρ(p 2 ), x x)} (6.7) 6.2 0 M t x x x P P P P 0 2 4 6 t P-1 P1 P3 P x 5 P7 Fig. 6.2 14

2 Lax Friedrichs 1 0 x = 0 x P 1 P 1 P 0 P 1 P 1 M x = M x P 7 P 5 P 6 P 5 P 5 6.3 Courant Friedrichs Lewy (CFL) CFL q ρ (ρ, x) t x 1 (6.8) q ρ v ρ (ρ, x) = (ρv(ρ, x)) ρ = v(ρ, x) + ρv ρ (ρ, x) µg (6.9) = ρ 2 µ 2 g 2 t 2 0 + 2µg(1/ρ L) q ρ = µ 2 g 2 t 2 0 + 2µg( 1 ρ L) µgt µg 0 ρ µ 2 g 2 t 2 0 + 2µg(1/ρ L) (6.10) 7 7.1 x M 1000[m] x 1000 ( x = 1) ρ(x, 0) ρ(x, 0) = [ /m] (7.1) 50[km/h] 60[s] Fig. 7.1 15

3 2 5 4 3 2 1 9 8 1 9 0 100 200 300 400 500 600 700 x 800 900 1000 0 10 20 40 30 t 50 60 8 7 (a) 3 (b) 2 Fig. 7.1 7.2 (x = 500) 5 (8 % ) ( 10 % 3) 4) ) 5 x=0 x=500 x=1000 Fig. 7.2 Fig. 7.3 (x = 500) 5 (8 % ) 16

05 95 05 95 9 85 8 75 7 65 9 85 8 0 100 200 300 400 500 600 700 x 800 900 1000 0 10 20 40 30 t 50 60 75 7 65 (a) 3 (b) 2 Fig. 7.3 5 x=0 x=500 x=1000 Fig. 7.4 Fig. 7.5 7.3 ρ(x, 0) ρ(x, 0) = 0.01 sin θ + [ /m] (7.2) (2.2) Table 3.1 40 [km/h] ρ 60 [km/h] ρ 40 60 [km/h] 17

4 35 3 4 35 3 25 2 15 1 05 95 25 2 15 1 0 100 200 300 400 500 600 700 800 x 900 1000 0 10 20 40 30 t 50 60 05 95 (a) 3 (b) 2 Fig. 7.5 Fig. 7.6 8 6 8 6 4 2 8 6 4 2 0 100 200 300 400 500 600 700 x 800 900 1000 0 10 20 40 30 t 50 60 4 2 8 6 4 2 (a) 3 (b) 2 Fig. 7.6 Fig. 7.7, 7.8 Fig. 7.9 60 18

8 6 8 6 4 2 8 6 4 2 0 100 200 300 400 500 600 700 x 800 900 1000 0 10 20 40 30 t 50 60 4 2 8 6 4 2 (a) 3 (b) 2 Fig. 7.7 8 8 6 6 4 4 2 2 8 8 6 6 4 4 2 2 (a) (b) 20 8 8 6 6 4 4 2 2 8 8 6 6 4 4 2 2 (c) 40 (d) 60 Fig. 7.8 19

8 6 4 2 8 6 4 2 Fig. 7.9 60 Fig. 7.10, 7.11 60 Fig. 7.12 5 5 5 5 5 0 100 200 300 400 500 600 700 x 800 900 1000 0 10 20 40 30 t 50 60 5 (a) 3 (b) 2 Fig. 7.10 8 20

5 5 5 5 5 5 (a) (b) 20 5 5 5 5 5 5 (c) 40 (d) 60 Fig. 7.11 21

5 5 5 Fig. 7.12 60 Lax Friedrichs ( ) Fig. 8.1 A B 22

B A Fig. 8.1 23

[1] R.Harberman : MATHMATICAL MODEL : TRAFFIC FLOW (,, 1981) [2], :, (, 1969), pp63-65 [3] :, pp98-99 [4] :, pp14-15 24