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QIQI WANG: So today, we're going
to be continuing finite volume.

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00:00:35,600 --> 00:00:38,410
We started on finite
volume on Monday.

12
00:00:38,410 --> 00:00:41,010
And just a very quick
recap, finite volume

13
00:00:41,010 --> 00:00:43,630
doesn't solve all the
differential equations,

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00:00:43,630 --> 00:00:46,270
but it solves a class of
differential equations

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00:00:46,270 --> 00:00:51,420
that we care most, that
is conservation laws.

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00:00:51,420 --> 00:00:55,430
Conservation laws are
differential equations that

17
00:00:55,430 --> 00:00:57,820
can be returned in this form.

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00:00:57,820 --> 00:01:00,640
And today, I'm just going
to be more specific.

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00:01:00,640 --> 00:01:07,440
Let's say, rho is the density
of a conservative quantity.

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00:01:07,440 --> 00:01:15,470
The time derivative of rho
plus the divergence of a flux,

21
00:01:15,470 --> 00:01:18,530
which is a function
of rho, is equal to 0.

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00:01:18,530 --> 00:01:22,560
So this is a general
conservation law

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00:01:22,560 --> 00:01:25,650
when there is no [INAUDIBLE],
when there is no generation

24
00:01:25,650 --> 00:01:31,575
or it's appearing off any
material within the control

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00:01:31,575 --> 00:01:32,860
volume.

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00:01:32,860 --> 00:01:35,980
Everything that increases has
to come from outside the control

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00:01:35,980 --> 00:01:36,540
volume.

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00:01:36,540 --> 00:01:42,460
Anything that decreases has to
go out of the control volume.

29
00:01:42,460 --> 00:01:49,460
And this is the
differential form.

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00:01:49,460 --> 00:01:51,910
And in one dimension
of cos, the divergence

31
00:01:51,910 --> 00:01:54,210
is just spatial derivative.

32
00:01:54,210 --> 00:01:58,080
It's just a partial
x of F of rho.

33
00:01:58,080 --> 00:01:59,720
And today we have
going to be solving

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00:01:59,720 --> 00:02:03,650
one of the equations like this.

35
00:02:03,650 --> 00:02:07,210
And in order to
apply finite volume,

36
00:02:07,210 --> 00:02:11,690
we need to make it
an integral form.

37
00:02:11,690 --> 00:02:18,220
We have to really
integrate this in space.

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00:02:18,220 --> 00:02:26,220
And instead of a
divergence operating space,

39
00:02:26,220 --> 00:02:31,831
we get this-- d dt of an
integral over any control

40
00:02:31,831 --> 00:02:32,330
volume.

41
00:02:32,330 --> 00:02:35,280
Let me describe it as omega.

42
00:02:35,280 --> 00:02:38,630
Omega is a control volume.

43
00:02:38,630 --> 00:02:43,110
So we are basically integrating
this whole thing in omega.

44
00:02:43,110 --> 00:02:44,930
And this is the
first thing we get.

45
00:02:44,930 --> 00:02:47,783
And the second thing
we get is that now we

46
00:02:47,783 --> 00:02:50,788
are doing this in impossibly
multiple dimensions,

47
00:02:50,788 --> 00:02:53,650
like last week [INAUDIBLE].

48
00:02:53,650 --> 00:02:58,862
Now if you integrate the
divergence of something--

49
00:02:58,862 --> 00:03:01,658
you're integrating a
divergence of a vector

50
00:03:01,658 --> 00:03:04,210
field-- in a control
volume, what do you get?

51
00:03:04,210 --> 00:03:07,010


52
00:03:07,010 --> 00:03:10,120
A listing something
like the integral

53
00:03:10,120 --> 00:03:12,670
of a divergence of a
vector in a control volume

54
00:03:12,670 --> 00:03:14,023
equal to something [INAUDIBLE].

55
00:03:14,023 --> 00:03:24,166


56
00:03:24,166 --> 00:03:25,607
Something can
relate a divergence.

57
00:03:25,607 --> 00:03:29,086


58
00:03:29,086 --> 00:03:30,760
[INAUDIBLE] the divergence 0.

59
00:03:30,760 --> 00:03:34,536


60
00:03:34,536 --> 00:03:39,290
[INAUDIBLE] is so cool.

61
00:03:39,290 --> 00:03:42,925
There are two theorems
in method popular that I

62
00:03:42,925 --> 00:03:45,660
can see that are
very important, both

63
00:03:45,660 --> 00:03:51,470
relating a spatial derivative
to a spatial integral.

64
00:03:51,470 --> 00:03:56,650
And the divergence theorem
is the more fundamental one.

65
00:03:56,650 --> 00:03:58,070
I can feel a more
fundamental one,

66
00:03:58,070 --> 00:04:04,060
because it basically tells
you the spatial integral

67
00:04:04,060 --> 00:04:09,880
of a divergence can be expressed
with something that doesn't

68
00:04:09,880 --> 00:04:12,530
involve derivative at all.

69
00:04:12,530 --> 00:04:17,140
It can be represented as the
integral over the surface

70
00:04:17,140 --> 00:04:18,920
of the control volume.

71
00:04:18,920 --> 00:04:25,030
The surface normal starts
with just the flux vector

72
00:04:25,030 --> 00:04:27,710
is equal to 0.

73
00:04:27,710 --> 00:04:32,480
So divergence theorem says
that the integral of this

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00:04:32,480 --> 00:04:36,582
is equal to this--
divergence theorem.

75
00:04:36,582 --> 00:04:39,110


76
00:04:39,110 --> 00:04:42,740
So what you see is that
this is the 2D or 3D version

77
00:04:42,740 --> 00:04:47,390
of the integral form of
the conservation law.

78
00:04:47,390 --> 00:04:50,630
What we are saying is
that now, again, this

79
00:04:50,630 --> 00:04:55,658
is the total amount of mass,
or stuff, within the control

80
00:04:55,658 --> 00:04:56,336
volume.

81
00:04:56,336 --> 00:04:58,400
And today we're going
to be doing half.

82
00:04:58,400 --> 00:05:00,210
We see that the
total [INAUDIBLE]

83
00:05:00,210 --> 00:05:02,600
half within a control volume.

84
00:05:02,600 --> 00:05:09,230
And this is the flux through
anywhere in the surface.

85
00:05:09,230 --> 00:05:12,090
So there is a normal.

86
00:05:12,090 --> 00:05:16,540
And this normal is also at
normal of the control volume.

87
00:05:16,540 --> 00:05:18,776
So n points towards the outset.

88
00:05:18,776 --> 00:05:23,855
So this F of rho is the
flux towards the outset.

89
00:05:23,855 --> 00:05:30,157
That makes sense, because
if F points outward--

90
00:05:30,157 --> 00:05:33,510
if the flux is towards
the outside-- then

91
00:05:33,510 --> 00:05:36,880
this term would be
positive, because F

92
00:05:36,880 --> 00:05:38,240
is going to be aligned with n.

93
00:05:38,240 --> 00:05:39,890
This term is positive.

94
00:05:39,890 --> 00:05:42,260
And the time derivative
will be negative

95
00:05:42,260 --> 00:05:46,340
because things are going out,
the stuff should decrease.

96
00:05:46,340 --> 00:05:49,468
If asked if the flux
vector points inward, then

97
00:05:49,468 --> 00:05:54,040
it is going opposite to the
outward point of normal,

98
00:05:54,040 --> 00:05:55,264
opposite to n.

99
00:05:55,264 --> 00:05:58,420
And this integral
would be negative.

100
00:05:58,420 --> 00:06:01,120
So if this thing is negative,
then we have to add to 0

101
00:06:01,120 --> 00:06:04,490
so this can have some positive,
which means the amount of stuff

102
00:06:04,490 --> 00:06:08,218
is going to increase
as a function of time.

103
00:06:08,218 --> 00:06:09,600
Is it clear?

104
00:06:09,600 --> 00:06:13,450
This is now generalizing
that in 2D or 3D.

105
00:06:13,450 --> 00:06:16,910
And of course, we can
still write that in 1D.

106
00:06:16,910 --> 00:06:20,710
In the one dimensional
special case,

107
00:06:20,710 --> 00:06:24,980
what is this partial omega?

108
00:06:24,980 --> 00:06:27,860
Partial omega is the
boundary of omega.

109
00:06:27,860 --> 00:06:31,152
What is the boundary of the
one dimensional control volume?

110
00:06:31,152 --> 00:06:34,100


111
00:06:34,100 --> 00:06:38,680
The point, or two points--
two points, right?

112
00:06:38,680 --> 00:06:45,500
And the two points, I have R and
L. And outward pointing normal

113
00:06:45,500 --> 00:06:49,170
is going to be just a 1 at
the right hand side at R,

114
00:06:49,170 --> 00:06:51,830
is going to be minus
1 at L. So I'm just

115
00:06:51,830 --> 00:06:59,040
going to be saying this is F
of rho at R minus F of rho at L

116
00:06:59,040 --> 00:07:00,688
is equal to 0.

117
00:07:00,688 --> 00:07:05,960
Or as we said on Monday, the
time derivative of total stuff

118
00:07:05,960 --> 00:07:10,942
is equal to the stuff coming in
from the left minus the stuff

119
00:07:10,942 --> 00:07:14,344
coming out from the right, if
you moved both of this stuff

120
00:07:14,344 --> 00:07:18,850
[INAUDIBLE] side by side.

121
00:07:18,850 --> 00:07:23,690
So this is the kind of
differential equations

122
00:07:23,690 --> 00:07:29,630
we are solving, and the
scheme is finite volume.

123
00:07:29,630 --> 00:07:37,120
And finite volume basically
follows exactly this formula.

124
00:07:37,120 --> 00:07:40,300
Instead of integral
over control volume,

125
00:07:40,300 --> 00:07:44,760
we are storing the average
of the control volume, which

126
00:07:44,760 --> 00:07:47,650
is simply the integral divided
by the size of the control

127
00:07:47,650 --> 00:07:50,030
volume.

128
00:07:50,030 --> 00:07:53,870
It could modify the cell average
with the size of the cell,

129
00:07:53,870 --> 00:07:56,360
with the integral.

130
00:07:56,360 --> 00:07:59,420
And in order to think of
the DDP of the cell average,

131
00:07:59,420 --> 00:08:06,025
you just divide this also by
the cell volume, or cell size.

132
00:08:06,025 --> 00:08:08,662
So you get the
evolution equation

133
00:08:08,662 --> 00:08:14,010
that allows you to control
using finite volume.

134
00:08:14,010 --> 00:08:19,030
Now this equation is not
approximate This is exact.

135
00:08:19,030 --> 00:08:23,290
What we need to approximate
in finite volume is this flux.

136
00:08:23,290 --> 00:08:27,230
This flux is a flux evaluated
after the cell boundaries

137
00:08:27,230 --> 00:08:29,428
or cell [INAUDIBLE].

138
00:08:29,428 --> 00:08:32,260
And the finite volume
is [INAUDIBLE].

139
00:08:32,260 --> 00:08:33,924
You only have the cell average.

140
00:08:33,924 --> 00:08:38,324
So you need to approximate the
interface value from the cell

141
00:08:38,324 --> 00:08:38,990
average.

142
00:08:38,990 --> 00:08:42,030
And that is the approximation
we have been talking about.

143
00:08:42,030 --> 00:08:45,460


144
00:08:45,460 --> 00:08:47,400
So that's kind of a review.

145
00:08:47,400 --> 00:08:49,810
The first new thing
we're going to talk about

146
00:08:49,810 --> 00:08:53,960
is, actually, why do
we use finite volume.

147
00:08:53,960 --> 00:08:59,670
We use finite volume because
it captures shock waves,

148
00:08:59,670 --> 00:09:03,360
because if there is
discontinuity in the solution,

149
00:09:03,360 --> 00:09:05,440
finite difference doesn't work.

150
00:09:05,440 --> 00:09:08,530
I'm going to show
you with a-- what?

151
00:09:08,530 --> 00:09:10,790
Cancel.

152
00:09:10,790 --> 00:09:17,522
I'm still testing around
with the code you wrote.

153
00:09:17,522 --> 00:09:20,261
But let me go to
today's directory,

154
00:09:20,261 --> 00:09:21,760
and I'm going to
show you two demos.

155
00:09:21,760 --> 00:09:24,290


156
00:09:24,290 --> 00:09:29,490
I'm going to show you
a simulate-- so first,

157
00:09:29,490 --> 00:09:31,920
I'm going to tell you
what this is doing.

158
00:09:31,920 --> 00:09:36,730
So I have two functions,
one is simulate fv,

159
00:09:36,730 --> 00:09:39,420
and one is simulate fd.

160
00:09:39,420 --> 00:09:44,850
So these two functions
are identical,

161
00:09:44,850 --> 00:09:48,510
except for this single letter.

162
00:09:48,510 --> 00:09:50,276
One is fd, one's fv.

163
00:09:50,276 --> 00:09:51,900
But otherwise, I'm
just doing the same.

164
00:09:51,900 --> 00:09:54,300
I'm using [INAUDIBLE]
45 to integrate in time.

165
00:09:54,300 --> 00:09:56,980


166
00:09:56,980 --> 00:09:59,550
Conditions on that
I'm [INAUDIBLE]

167
00:09:59,550 --> 00:10:01,900
having space, using
either finite volume

168
00:10:01,900 --> 00:10:03,920
or finite difference.

169
00:10:03,920 --> 00:10:07,870
I see there is a question
on what this ''at'' do.

170
00:10:07,870 --> 00:10:12,860
This "at" is basically
take a function I wrote,

171
00:10:12,860 --> 00:10:19,550
which is either du
dp fv or du dp fd

172
00:10:19,550 --> 00:10:24,340
and feed that into the ODE 45.

173
00:10:24,340 --> 00:10:28,489
Anytime you use ODE 45 you
need this "at" to make it work.

174
00:10:28,489 --> 00:10:30,655
Now the difference between
finding fd and finding fv

175
00:10:30,655 --> 00:10:34,444
is that in this we're using
the standard finite difference

176
00:10:34,444 --> 00:10:37,369
to solve the differential
form of the Burger's equation.

177
00:10:37,369 --> 00:10:41,536


178
00:10:41,536 --> 00:10:47,869
So du dp is equal to
minus u times du dx.

179
00:10:47,869 --> 00:10:48,855
This is du dx.

180
00:10:48,855 --> 00:10:50,827
Now you should be
familiar with this.

181
00:10:50,827 --> 00:10:55,530
I'm taking u minus a
specific version of u--

182
00:10:55,530 --> 00:11:00,270
this is assuming [INAUDIBLE]
conditions-- divided by dx.

183
00:11:00,270 --> 00:11:08,400
So this is a [INAUDIBLE] space
on a difference of du dx.

184
00:11:08,400 --> 00:11:10,570
So here I am solving
the differential form

185
00:11:10,570 --> 00:11:13,620
of the equation of
the finite difference.

186
00:11:13,620 --> 00:11:18,790
In finite volume-- let's review
what we do in finite volume.

187
00:11:18,790 --> 00:11:20,950
du dx [INAUDIBLE] flux.

188
00:11:20,950 --> 00:11:23,386
Now, this is not using
the differential form,

189
00:11:23,386 --> 00:11:25,102
but using the [INAUDIBLE]
form because we

190
00:11:25,102 --> 00:11:26,670
are computing the flux.

191
00:11:26,670 --> 00:11:31,520
And here, I'm assuming
the opposite direction

192
00:11:31,520 --> 00:11:34,410
is always the left.

193
00:11:34,410 --> 00:11:37,230
Here, I'm assuming like I
have an initial condition that

194
00:11:37,230 --> 00:11:41,648
is opposite, and it's
easy to illustrate

195
00:11:41,648 --> 00:11:43,620
using a case like that.

196
00:11:43,620 --> 00:11:47,010
So the wind always come from
the left, u is always positive.

197
00:11:47,010 --> 00:11:59,080
That means when [INAUDIBLE]
it is a minus [INAUDIBLE].

198
00:11:59,080 --> 00:12:04,825
And I'm doing du dt equal to
flux at the left minus flux

199
00:12:04,825 --> 00:12:06,290
at the right divided by dx.

200
00:12:06,290 --> 00:12:07,990
So this is finite volume.

201
00:12:07,990 --> 00:12:12,070
So let's see, what is the key
difference between the results

202
00:12:12,070 --> 00:12:13,790
of these two schemes.

203
00:12:13,790 --> 00:12:16,737
And in either one, I'm starting
with something positive,

204
00:12:16,737 --> 00:12:20,535
so sine x plus 1 over 2
because I'm assuming also

205
00:12:20,535 --> 00:12:22,170
in finite difference
and finite volume,

206
00:12:22,170 --> 00:12:23,622
the wind comes from the left.

207
00:12:23,622 --> 00:12:25,080
So that I'm doing
[? backwarding ?]

208
00:12:25,080 --> 00:12:30,614
space is a finite
difference or finite volume.

209
00:12:30,614 --> 00:12:32,506
So I have a positive solution.

210
00:12:32,506 --> 00:12:35,390
I told them to see how it does.

211
00:12:35,390 --> 00:12:37,912
So I'm going to [INAUDIBLE]
on a different sort,

212
00:12:37,912 --> 00:12:39,370
and somebody new
will tell me, what

213
00:12:39,370 --> 00:12:42,520
is wrong with a finite
difference solution.

214
00:12:42,520 --> 00:12:43,570
Watch carefully.

215
00:12:43,570 --> 00:12:44,750
I'm going to start.

216
00:12:44,750 --> 00:12:48,390
You need to tell me
what's going wrong.

217
00:12:48,390 --> 00:12:50,670
So it looks like any
Burger's equation.

218
00:12:50,670 --> 00:12:52,236
I mean, we have a shock wave.

219
00:12:52,236 --> 00:12:54,110
Things are going.

220
00:12:54,110 --> 00:12:58,994
So what's wrong with this?

221
00:12:58,994 --> 00:13:01,410
Is anything wrong?

222
00:13:01,410 --> 00:13:04,100
Solution [INAUDIBLE].

223
00:13:04,100 --> 00:13:04,870
Let me run again.

224
00:13:04,870 --> 00:13:13,952


225
00:13:13,952 --> 00:13:15,300
Anything looks wrong?

226
00:13:15,300 --> 00:13:18,910


227
00:13:18,910 --> 00:13:21,994
Many different circles it'd
be tracing, but it's not.

228
00:13:21,994 --> 00:13:22,946
AUDIENCE: [INAUDIBLE]

229
00:13:22,946 --> 00:13:27,699


230
00:13:27,699 --> 00:13:28,990
QIQI WANG: You have sharp eyes.

231
00:13:28,990 --> 00:13:32,110
Now, let's look at the
finite volume solution.

232
00:13:32,110 --> 00:13:34,977
Exactly the same, but
using finite volume.

233
00:13:34,977 --> 00:13:38,260


234
00:13:38,260 --> 00:13:39,320
What's the difference?

235
00:13:39,320 --> 00:13:42,380


236
00:13:42,380 --> 00:13:45,900
This [INAUDIBLE] the shock
wave is correct in this case.

237
00:13:45,900 --> 00:13:48,800
While the shock just
[INAUDIBLE] it forms,

238
00:13:48,800 --> 00:13:51,760
the finite difference
just stands there.

239
00:13:51,760 --> 00:13:55,170
The speed of the shock
wave is finite difference,

240
00:13:55,170 --> 00:13:58,310
is wrong in finite difference.

241
00:13:58,310 --> 00:14:03,700
So by using the differential
form of the equation

242
00:14:03,700 --> 00:14:09,410
and disregarding the
conservative form,

243
00:14:09,410 --> 00:14:13,913
you are not guaranteed to have
the correct speed of the shock

244
00:14:13,913 --> 00:14:14,918
wave.

245
00:14:14,918 --> 00:14:18,540
It actually solves the
other part correctly.

246
00:14:18,540 --> 00:14:20,940
The only part that it
doesn't capture correctly

247
00:14:20,940 --> 00:14:24,460
is how the shock wave
is supposed to behave.

248
00:14:24,460 --> 00:14:27,670
And if we use finite
volume, because we

249
00:14:27,670 --> 00:14:31,190
are looking at the conservative
form of the differential

250
00:14:31,190 --> 00:14:36,920
equation of this form,
the other integral form.

251
00:14:36,920 --> 00:14:42,460


252
00:14:42,460 --> 00:14:46,980
This form of [INAUDIBLE]
is to have shock waves.

253
00:14:46,980 --> 00:14:52,450
Solve this one, and you take the
divergence of the [INAUDIBLE]

254
00:14:52,450 --> 00:14:57,460
that is discontinuous, for it
acts derivative of something

255
00:14:57,460 --> 00:14:59,010
that is discontinuous.

256
00:14:59,010 --> 00:15:03,760
That is no longer a
correct thing to do.

257
00:15:03,760 --> 00:15:06,522
And therefore, you
get wrong behavior

258
00:15:06,522 --> 00:15:09,360
when there is discontinuity
like shock waves.

259
00:15:09,360 --> 00:15:13,710


260
00:15:13,710 --> 00:15:14,360
Any questions?

261
00:15:14,360 --> 00:15:20,180


262
00:15:20,180 --> 00:15:25,726
Oh, and by the way,
another thing is this.

263
00:15:25,726 --> 00:15:30,290


264
00:15:30,290 --> 00:15:31,531
I should let you look.

265
00:15:31,531 --> 00:15:33,030
In addition to the
shock wave, there

266
00:15:33,030 --> 00:15:36,506
is another thing that is not
right in the finite difference.

267
00:15:36,506 --> 00:15:39,390


268
00:15:39,390 --> 00:15:42,281
Lets look again at
the finite volume.

269
00:15:42,281 --> 00:15:45,150
The Burger's equation's
a conservation law,

270
00:15:45,150 --> 00:15:48,522
which means that [INAUDIBLE]
under the curve should be

271
00:15:48,522 --> 00:15:50,960
[INAUDIBLE].

272
00:15:50,960 --> 00:15:54,540
The integral of the solution
u, of course, the whole domain,

273
00:15:54,540 --> 00:15:59,650
should be the same, because
anything that comes out of here

274
00:15:59,650 --> 00:16:00,880
should come back into here.

275
00:16:00,880 --> 00:16:03,380
It's like when you step out
of the door, you come back.

276
00:16:03,380 --> 00:16:05,170
You're not disappearing.

277
00:16:05,170 --> 00:16:10,341
But if you look at
the finite difference,

278
00:16:10,341 --> 00:16:14,130
at this point, it [INAUDIBLE].

279
00:16:14,130 --> 00:16:17,580
Things just disappear
out of nowhere.

280
00:16:17,580 --> 00:16:22,060
The solution is not conserved.

281
00:16:22,060 --> 00:16:26,540
So two things that finite
difference is missing.

282
00:16:26,540 --> 00:16:30,420
One is, it doesn't have
to behave as a shock wave.

283
00:16:30,420 --> 00:16:36,430
Two, it is the solution
is not conservative.

284
00:16:36,430 --> 00:16:39,252
Any question on why
we do finite volume?

285
00:16:39,252 --> 00:16:46,020


286
00:16:46,020 --> 00:16:48,100
So this is why we
do finite volume.

287
00:16:48,100 --> 00:16:53,660
And the next question
is another question,

288
00:16:53,660 --> 00:16:58,710
probably, you have when you are
going through the last lecture

289
00:16:58,710 --> 00:17:02,110
and reading through the
material is, why the hell

290
00:17:02,110 --> 00:17:04,092
do we do upwind?

291
00:17:04,092 --> 00:17:07,839
Why do we look at
the upwind direction?

292
00:17:07,839 --> 00:17:11,200
So the upwind of flux is this.

293
00:17:11,200 --> 00:17:14,690
So we look at a few volumes.

294
00:17:14,690 --> 00:17:17,990
k minus 1, k plus 1.

295
00:17:17,990 --> 00:17:22,750
The finite volume, we store the
average value in the volumes.

296
00:17:22,750 --> 00:17:25,680
So we have k minus
1/2 and k plus 1/2

297
00:17:25,680 --> 00:17:31,430
as the volume interfaces
or the cell interfaces.

298
00:17:31,430 --> 00:17:37,930
And the upwind flux is basically
saying F at k minus 1/2

299
00:17:37,930 --> 00:17:48,740
is either the F of rho at
k minus 1 or F of rho at k,

300
00:17:48,740 --> 00:17:58,224
depending on where the wind
comes from, depending on if,

301
00:17:58,224 --> 00:18:08,100
should say-- let me use C
as the speed of the wind.

302
00:18:08,100 --> 00:18:11,040
So if c is greater
than zero, the winds

303
00:18:11,040 --> 00:18:15,710
have positive speed,
which means the wind goes

304
00:18:15,710 --> 00:18:16,807
towards the right.

305
00:18:16,807 --> 00:18:18,755
And then you look at
the upwind direction,

306
00:18:18,755 --> 00:18:23,280
you should be looking at
when you are computing

307
00:18:23,280 --> 00:18:27,186
the upwind value of the
plus [INAUDIBLE] minus 2,

308
00:18:27,186 --> 00:18:30,994
you should look towards
the left, [INAUDIBLE].

309
00:18:30,994 --> 00:18:40,470
So when k is greater--
so when C is less than 0,

310
00:18:40,470 --> 00:18:44,824
that means the speed of
the wing is negative,

311
00:18:44,824 --> 00:18:46,490
opposite direction
is towards the right.

312
00:18:46,490 --> 00:18:52,322
The winds are [INAUDIBLE]
opposite direction for this,

313
00:18:52,322 --> 00:18:53,780
you should be
looking at the right

314
00:18:53,780 --> 00:18:58,480
and use the k value instead.

315
00:18:58,480 --> 00:19:00,830
And here, c is what?

316
00:19:00,830 --> 00:19:05,250
c is this.

317
00:19:05,250 --> 00:19:09,550
c is the c that appears in this.

318
00:19:09,550 --> 00:19:16,430


319
00:19:16,430 --> 00:19:22,440
c is the value that appears
in the differential form

320
00:19:22,440 --> 00:19:26,580
of the equation, over here.

321
00:19:26,580 --> 00:19:29,560
So as you can
imagine, if we have

322
00:19:29,560 --> 00:19:34,000
[INAUDIBLE] like the [INAUDIBLE]
vacuum equation we have,

323
00:19:34,000 --> 00:19:37,480
this c exactly determines
whether your solution

324
00:19:37,480 --> 00:19:40,302
moves towards the left or
moves towards the right.

325
00:19:40,302 --> 00:19:43,770
So if this c is positive, the
solution will go to the right

326
00:19:43,770 --> 00:19:45,792
and winds go towards the right.

327
00:19:45,792 --> 00:19:49,830
If c is negative, it goes
the opposite direction.

328
00:19:49,830 --> 00:19:52,320
But in general, in
a conservation law,

329
00:19:52,320 --> 00:19:56,400
how do we convert
our conservation law,

330
00:19:56,400 --> 00:20:01,910
which is partial F rho partial
x equal to 0, into to that form?

331
00:20:01,910 --> 00:20:05,263
But if we convert this
into this one, what is c?

332
00:20:05,263 --> 00:20:09,430


333
00:20:09,430 --> 00:20:15,080
Can we always convert
the conservative form

334
00:20:15,080 --> 00:20:19,730
into this, like a vacuum
scheme looks like, leaving

335
00:20:19,730 --> 00:20:23,523
a vacuum looking like form?

336
00:20:23,523 --> 00:20:25,020
AUDIENCE: Just take
the derivative.

337
00:20:25,020 --> 00:20:27,440
QIQI WANG: I just take
the derivative, exactly.

338
00:20:27,440 --> 00:20:31,750
If I take c equal to
partial-- not partial.

339
00:20:31,750 --> 00:20:34,480
I do too many pd's.

340
00:20:34,480 --> 00:20:41,670
dF d rho, then I get my C.
In the Burger's equations,

341
00:20:41,670 --> 00:20:47,330
c is [INAUDIBLE] rho because F
is equal to half of rho square,

342
00:20:47,330 --> 00:20:48,590
right?

343
00:20:48,590 --> 00:20:51,445
So when I take derivative,
I just get [INAUDIBLE].

344
00:20:51,445 --> 00:20:53,970
As you see if you're looking
at all the equations,

345
00:20:53,970 --> 00:20:57,530
you'll see what we
did [INAUDIBLE].

346
00:20:57,530 --> 00:20:59,368
And by the way,
you may be thinking

347
00:20:59,368 --> 00:21:01,758
of why did I use these here.

348
00:21:01,758 --> 00:21:07,210
Because if you look at a gas
dynamic equation, the C you get

349
00:21:07,210 --> 00:21:11,340
is going to be the
speed of sound.

350
00:21:11,340 --> 00:21:13,370
OK.

351
00:21:13,370 --> 00:21:13,870
OK.

352
00:21:13,870 --> 00:21:18,140
Why do we need do upwind?

353
00:21:18,140 --> 00:21:21,280
I mean, the real
mathematical reason

354
00:21:21,280 --> 00:21:24,750
is something I'm going to
talk about [INAUDIBLE].

355
00:21:24,750 --> 00:21:29,160
But, like, I think Dr.
[? Vikram ?] [? Dag ?] is going

356
00:21:29,160 --> 00:21:31,590
to be giving the lecture
right after the midterm.

357
00:21:31,590 --> 00:21:33,710
But he's going to
talking about stability.

358
00:21:33,710 --> 00:21:35,630
And this is for stability.

359
00:21:35,630 --> 00:21:43,010
I'm just going to
demonstrate it empirically

360
00:21:43,010 --> 00:21:49,780
in the shared folder
we had-- not this one--

361
00:21:49,780 --> 00:21:51,460
we had from last lecture.

362
00:21:51,460 --> 00:21:52,250
OK.

363
00:21:52,250 --> 00:21:54,500
So I had a driver here.

364
00:21:54,500 --> 00:21:57,232
And I have a bunch
of people's schemes.

365
00:21:57,232 --> 00:21:59,420
Who's scheme is
working by the way?

366
00:21:59,420 --> 00:22:00,629
Who knows?

367
00:22:00,629 --> 00:22:01,920
Which one of them should I use?

368
00:22:01,920 --> 00:22:04,920


369
00:22:04,920 --> 00:22:09,610
Who kind of knows more or
less your script is working?

370
00:22:09,610 --> 00:22:10,140
OK.

371
00:22:10,140 --> 00:22:10,639
Yours?

372
00:22:10,639 --> 00:22:11,778
OK.

373
00:22:11,778 --> 00:22:12,385
[INAUDIBLE]

374
00:22:12,385 --> 00:22:13,260
AUDIENCE: [INAUDIBLE]

375
00:22:13,260 --> 00:22:16,718


376
00:22:16,718 --> 00:22:17,570
QIQI WANG: OK.

377
00:22:17,570 --> 00:22:18,070
All right.

378
00:22:18,070 --> 00:22:25,790
So I'm going to
change this to-- OK.

379
00:22:25,790 --> 00:22:27,200
Thanks.

380
00:22:27,200 --> 00:22:28,650
So I can run it here.

381
00:22:28,650 --> 00:22:31,290


382
00:22:31,290 --> 00:22:34,527
And I get a solution.

383
00:22:34,527 --> 00:22:36,860
I mean, you probably didn't
treat the boundary condition

384
00:22:36,860 --> 00:22:38,760
correctly, but that's fine.

385
00:22:38,760 --> 00:22:40,550
So there is something
on the boundary

386
00:22:40,550 --> 00:22:45,420
that is not working correctly.

387
00:22:45,420 --> 00:22:50,073
I think that you might have set
the [INAUDIBLE] here to be 0.

388
00:22:50,073 --> 00:22:51,636
So you just set the
[INAUDIBLE] here

389
00:22:51,636 --> 00:22:54,765
to be 0 instead of
[INAUDIBLE] the [INAUDIBLE]

390
00:22:54,765 --> 00:22:57,980
blowing with the [INAUDIBLE].

391
00:22:57,980 --> 00:23:02,450
Because I'm having a [INAUDIBLE]
find the word, so [INAUDIBLE],

392
00:23:02,450 --> 00:23:03,450
right?

393
00:23:03,450 --> 00:23:04,950
[INAUDIBLE]

394
00:23:04,950 --> 00:23:07,540
But that's fine.

395
00:23:07,540 --> 00:23:09,710
OK.

396
00:23:09,710 --> 00:23:15,480
So I think it should be
relatively easy to modify.

397
00:23:15,480 --> 00:23:19,190
F-- yeah, yeah, yeah.

398
00:23:19,190 --> 00:23:19,690
OK.

399
00:23:19,690 --> 00:23:22,000
So I just need to
treat the last one.

400
00:23:22,000 --> 00:23:28,620
So F_k_plus_half(and), right?

401
00:23:28,620 --> 00:23:33,130
The last one is equal to-- OK.

402
00:23:33,130 --> 00:23:38,060
I will just cheat
here, F_bad_k(and).

403
00:23:38,060 --> 00:23:39,700
Because I know my
solution is positive,

404
00:23:39,700 --> 00:23:42,710
so I just take the left one.

405
00:23:42,710 --> 00:23:46,740


406
00:23:46,740 --> 00:23:47,240
OK.

407
00:23:47,240 --> 00:23:49,820
So here is k_plus_half.

408
00:23:49,820 --> 00:23:51,570
AUDIENCE: That should
have taken care of--

409
00:23:51,570 --> 00:23:54,189
QIQI WANG: Oh, that should
have taken care of that.

410
00:23:54,189 --> 00:23:55,064
AUDIENCE: [INAUDIBLE]

411
00:23:55,064 --> 00:24:08,631


412
00:24:08,631 --> 00:24:09,506
AUDIENCE: [INAUDIBLE]

413
00:24:09,506 --> 00:24:12,312


414
00:24:12,312 --> 00:24:13,020
QIQI WANG: Sorry?

415
00:24:13,020 --> 00:24:16,520


416
00:24:16,520 --> 00:24:20,702
k_plus_half of i minus--
that is end, right?

417
00:24:20,702 --> 00:24:21,327
AUDIENCE: Yeah.

418
00:24:21,327 --> 00:24:24,670
That's [INAUDIBLE].

419
00:24:24,670 --> 00:24:30,050
QIQI WANG: I think you
shouldn't be-- F_bar_k, right?

420
00:24:30,050 --> 00:24:32,650


421
00:24:32,650 --> 00:24:33,500
OK.

422
00:24:33,500 --> 00:24:43,794
And else it is equal
to F_bar_k(1), right?

423
00:24:43,794 --> 00:24:46,601


424
00:24:46,601 --> 00:24:47,100
OK.

425
00:24:47,100 --> 00:24:48,340
So let's see if that works.

426
00:24:48,340 --> 00:24:51,450


427
00:24:51,450 --> 00:24:52,110
Oh, that works.

428
00:24:52,110 --> 00:24:53,350
OK, nice.

429
00:24:53,350 --> 00:24:53,910
OK.

430
00:24:53,910 --> 00:24:55,750
So we have this
scheme over here.

431
00:24:55,750 --> 00:24:57,860
We have an upwinding scheme.

432
00:24:57,860 --> 00:25:01,620
So as we did in
finite [INAUDIBLE],

433
00:25:01,620 --> 00:25:06,530
will know that backward in space
or forward in space anything

434
00:25:06,530 --> 00:25:09,380
you [INAUDIBLE] have a
one-sided [INAUDIBLE],

435
00:25:09,380 --> 00:25:12,990
you get first order accuracy.

436
00:25:12,990 --> 00:25:15,250
But if you take
essential difference

437
00:25:15,250 --> 00:25:18,380
or take the average
of left and right,

438
00:25:18,380 --> 00:25:20,410
you get secondary accuracy.

439
00:25:20,410 --> 00:25:23,050
So why don't we use that?

440
00:25:23,050 --> 00:25:28,020
I'm going to change your
code to do exactly that.

441
00:25:28,020 --> 00:25:31,590
I'm going to instead of doing
all this conditional thing,

442
00:25:31,590 --> 00:25:42,180
I'm just going to do this
is equal to half of F_bar_k

443
00:25:42,180 --> 00:25:43,940
and F_bar_k plus 1.

444
00:25:43,940 --> 00:25:48,040


445
00:25:48,040 --> 00:25:49,960
This thing's divided by 2.

446
00:25:49,960 --> 00:25:53,100
So I'm not going to be
using your conditions.

447
00:25:53,100 --> 00:25:58,280
I'm just going to do k_plus_half
equal to actually k_plus_half,

448
00:25:58,280 --> 00:26:02,550
k_plus k_plus over 2.

449
00:26:02,550 --> 00:26:07,500
And instead of doing
this condition--

450
00:26:07,500 --> 00:26:11,940
apologize first for going to
make the [INAUDIBLE] unstable.

451
00:26:11,940 --> 00:26:24,000
But k_plus_half(and) is going to
be F_bar_k(and) plus F_bar_k(1)

452
00:26:24,000 --> 00:26:25,200
divide by 2.

453
00:26:25,200 --> 00:26:30,690
So now if I do this, I always
set the interface value

454
00:26:30,690 --> 00:26:33,810
to be the average between
the two [INAUDIBLE].

455
00:26:33,810 --> 00:26:34,675
What do I get?

456
00:26:34,675 --> 00:26:38,240


457
00:26:38,240 --> 00:26:39,860
Now, it looks like correct.

458
00:26:39,860 --> 00:26:40,360
Oh.

459
00:26:40,360 --> 00:26:43,162


460
00:26:43,162 --> 00:26:44,650
All right.

461
00:26:44,650 --> 00:26:45,920
So what we see here?

462
00:26:45,920 --> 00:26:49,130
We see here in the
beginning, it looks fine.

463
00:26:49,130 --> 00:26:54,690
Let's look more carefully
what actually happens.

464
00:26:54,690 --> 00:26:59,285
Let's set dt equal to 0.001
to be more-- let's see

465
00:26:59,285 --> 00:27:00,130
what happens.

466
00:27:00,130 --> 00:27:00,770
OK.

467
00:27:00,770 --> 00:27:04,040
Usually, the evolution-
let me [INAUDIBLE]

468
00:27:04,040 --> 00:27:05,290
it, so it doesn't [INAUDIBLE].

469
00:27:05,290 --> 00:27:08,100


470
00:27:08,100 --> 00:27:13,620
[INAUDIBLE] I'm going
to do ylim(0,2),

471
00:27:13,620 --> 00:27:16,530
so that doesn't [INAUDIBLE].

472
00:27:16,530 --> 00:27:17,980
OK.

473
00:27:17,980 --> 00:27:21,270
In the beginning, we see
it is behaving correctly.

474
00:27:21,270 --> 00:27:23,710
And I am probably getting
secondary accuracy, right?

475
00:27:23,710 --> 00:27:26,345


476
00:27:26,345 --> 00:27:29,810
But as soon as the
[INAUDIBLE] is formed,

477
00:27:29,810 --> 00:27:33,100
something bad is
going to happen.

478
00:27:33,100 --> 00:27:34,040
Now, [INAUDIBLE].

479
00:27:34,040 --> 00:27:36,838


480
00:27:36,838 --> 00:27:38,326
All right.

481
00:27:38,326 --> 00:27:38,826
[INAUDIBLE]

482
00:27:38,826 --> 00:27:44,293


483
00:27:44,293 --> 00:27:46,300
So, yeah.

484
00:27:46,300 --> 00:27:49,214
So we've got these calculations.

485
00:27:49,214 --> 00:27:53,211
And it looks pretty much
like in [INAUDIBLE] analysis

486
00:27:53,211 --> 00:27:59,156
you get this [INAUDIBLE]
phenomenon, like right or wrong

487
00:27:59,156 --> 00:28:01,700
[INAUDIBLE].

488
00:28:01,700 --> 00:28:11,390
So this is actually why we need
to take an upwinding scheme,

489
00:28:11,390 --> 00:28:15,600
especially if you have a
nonlinear differential equation

490
00:28:15,600 --> 00:28:17,439
with potentially shock waves.

491
00:28:17,439 --> 00:28:22,260


492
00:28:22,260 --> 00:28:24,150
I mean, if you go
to math class, I

493
00:28:24,150 --> 00:28:28,400
can probably talk more
about shock waves being

494
00:28:28,400 --> 00:28:33,100
a dissipated phenomenon
and a second order scheme,

495
00:28:33,100 --> 00:28:35,773
like essential difference that
has no numerical dissipation.

496
00:28:35,773 --> 00:28:40,350
And you need the numerical
dissipation [INAUDIBLE].

497
00:28:40,350 --> 00:28:42,550
I'm just going to
demonstrate numerically.

498
00:28:42,550 --> 00:28:45,970
This type of thing is going to
happen if we don't do upwinding

499
00:28:45,970 --> 00:28:47,155
and you have shock wave.

500
00:28:47,155 --> 00:28:49,750


501
00:28:49,750 --> 00:28:52,020
So for something that
doesn't have shock wave

502
00:28:52,020 --> 00:28:56,890
it's probably fine as long as
you use an appropriate time

503
00:28:56,890 --> 00:28:58,650
integrator.

504
00:28:58,650 --> 00:28:59,365
OK.

505
00:28:59,365 --> 00:29:04,342
So today-- I mean, this
is why upwinding-- we

506
00:29:04,342 --> 00:29:06,092
are going to be doing
something like this.

507
00:29:06,092 --> 00:29:08,576


508
00:29:08,576 --> 00:29:09,950
So I tried to
stimulate distance.

509
00:29:09,950 --> 00:29:13,976


510
00:29:13,976 --> 00:29:16,880
This is a lot of parts, right?

511
00:29:16,880 --> 00:29:19,598
And if you do a simulation
of individual parts,

512
00:29:19,598 --> 00:29:21,005
yeah, you can do this.

513
00:29:21,005 --> 00:29:24,260
But [INAUDIBLE] this
[INAUDIBLE] it really

514
00:29:24,260 --> 00:29:26,390
looks like-- I mean,
each [INAUDIBLE] more

515
00:29:26,390 --> 00:29:29,086
looks more or less
like a molecule.

516
00:29:29,086 --> 00:29:33,810
So you really get a continuum
of cars or [INAUDIBLE].

517
00:29:33,810 --> 00:29:38,000
And one of the ideas is to
instead of treating each car

518
00:29:38,000 --> 00:29:43,030
as an individual agent
and simulate that, we just

519
00:29:43,030 --> 00:29:47,030
look at from a
macroscopic point of view

520
00:29:47,030 --> 00:29:51,030
and simulate this
like gas dynamics,

521
00:29:51,030 --> 00:29:54,850
where each car is
like a molecule.

522
00:29:54,850 --> 00:29:55,780
OK.

523
00:29:55,780 --> 00:30:01,020
And so a macroscopic description
of the cars on a highway

524
00:30:01,020 --> 00:30:06,570
is just P as a function
of x and t, right?

525
00:30:06,570 --> 00:30:08,290
So x is space.

526
00:30:08,290 --> 00:30:09,385
t is time.

527
00:30:09,385 --> 00:30:18,790
And P is the number of cars
per length of the highway.

528
00:30:18,790 --> 00:30:21,739
So P is a converse quantity,
because cars are converse,

529
00:30:21,739 --> 00:30:22,239
right?

530
00:30:22,239 --> 00:30:26,780
Cars are not disappearing
into [INAUDIBLE].

531
00:30:26,780 --> 00:30:29,255
Yeah.

532
00:30:29,255 --> 00:30:32,491
Cars are not disappearing
as airlines sometimes do.

533
00:30:32,491 --> 00:30:35,544


534
00:30:35,544 --> 00:30:36,419
AUDIENCE: [INAUDIBLE]

535
00:30:36,419 --> 00:30:41,478


536
00:30:41,478 --> 00:30:42,311
[INTERPOSING VOICES]

537
00:30:42,311 --> 00:30:46,147


538
00:30:46,147 --> 00:30:46,730
QIQI WANG: OK.

539
00:30:46,730 --> 00:30:51,450


540
00:30:51,450 --> 00:30:57,060
So we are also assuming
the velocity of cars

541
00:30:57,060 --> 00:30:59,420
depends on the density.

542
00:30:59,420 --> 00:31:05,940
Well, let's first say so P
equal to 0 is an empty highway.

543
00:31:05,940 --> 00:31:09,170
P equal to 1 is this.

544
00:31:09,170 --> 00:31:09,960
OK.

545
00:31:09,960 --> 00:31:13,112
P equal to 1 is
like a parking lot.

546
00:31:13,112 --> 00:31:16,660
P equal to 1 is like
the maximum density

547
00:31:16,660 --> 00:31:19,230
you can pack onto a highway.

548
00:31:19,230 --> 00:31:19,730
OK.

549
00:31:19,730 --> 00:31:23,330
So this is P. And we are
assuming the velocity

550
00:31:23,330 --> 00:31:26,990
of the cars is a functional P.

551
00:31:26,990 --> 00:31:31,840
So again, a velocity
equal to 0 is like this.

552
00:31:31,840 --> 00:31:38,760
So this car cannot move when
the density is at [INAUDIBLE].

553
00:31:38,760 --> 00:31:44,600
And let's pick the
unit of velocity,

554
00:31:44,600 --> 00:31:46,940
so that velocity equal
to 1 is something

555
00:31:46,940 --> 00:31:55,990
like 65 miles per hour, where
you would have-- or 80--

556
00:31:55,990 --> 00:32:01,360
where you would be driving
if there is no other cars.

557
00:32:01,360 --> 00:32:02,300
Or 85.

558
00:32:02,300 --> 00:32:04,570
OK.

559
00:32:04,570 --> 00:32:06,690
OK.

560
00:32:06,690 --> 00:32:09,550
So what would be an
easy way or intuitive

561
00:32:09,550 --> 00:32:14,565
way you would model
u as a function P?

562
00:32:14,565 --> 00:32:15,440
AUDIENCE: [INAUDIBLE]

563
00:32:15,440 --> 00:32:19,954


564
00:32:19,954 --> 00:32:20,620
QIQI WANG: Yeah.

565
00:32:20,620 --> 00:32:21,740
It can be more curved.

566
00:32:21,740 --> 00:32:23,800
But let's make it simple.

567
00:32:23,800 --> 00:32:25,400
Did you want to [INAUDIBLE]?

568
00:32:25,400 --> 00:32:27,720
So would you [INAUDIBLE]
code [INAUDIBLE]

569
00:32:27,720 --> 00:32:28,920
like something like this?

570
00:32:28,920 --> 00:32:32,220
Or would you [INAUDIBLE] coding
[INAUDIBLE] like U(P)= 1 minus

571
00:32:32,220 --> 00:32:32,720
P?

572
00:32:32,720 --> 00:32:38,600


573
00:32:38,600 --> 00:32:41,110
This captures the
right behavior.

574
00:32:41,110 --> 00:32:45,500
So when P is equal to 0, you
drive at 85 miles per hour.

575
00:32:45,500 --> 00:32:49,926
When P is equal to 1, even
if you want to drive faster,

576
00:32:49,926 --> 00:32:52,460
you're stuck, right?

577
00:32:52,460 --> 00:32:53,350
OK.

578
00:32:53,350 --> 00:32:55,280
Now, here's my first question.

579
00:32:55,280 --> 00:32:56,187
What is the flux?

580
00:32:56,187 --> 00:33:02,900


581
00:33:02,900 --> 00:33:03,695
What's F(P)?

582
00:33:03,695 --> 00:33:10,688


583
00:33:10,688 --> 00:33:13,511
AUDIENCE: [INAUDIBLE]

584
00:33:13,511 --> 00:33:14,386
QIQI WANG: Integral--

585
00:33:14,386 --> 00:33:15,261
AUDIENCE: [INAUDIBLE]

586
00:33:15,261 --> 00:33:17,934


587
00:33:17,934 --> 00:33:18,600
QIQI WANG: Yeah.

588
00:33:18,600 --> 00:33:19,766
I know what you're thinking.

589
00:33:19,766 --> 00:33:25,070
You are thinking about U
being the d f in P, right?

590
00:33:25,070 --> 00:33:26,340
So it has to be the integral.

591
00:33:26,340 --> 00:33:30,360
But that's not true.

592
00:33:30,360 --> 00:33:31,610
It's easier than that.

593
00:33:31,610 --> 00:33:35,085
You are thinking more--
you're too smart.

594
00:33:35,085 --> 00:33:36,379
AUDIENCE: [INAUDIBLE]

595
00:33:36,379 --> 00:33:38,420
QIQI WANG: P times U.
That's that correct answer.

596
00:33:38,420 --> 00:33:41,372
Can you say why
that is this case?

597
00:33:41,372 --> 00:33:43,268
AUDIENCE: [INAUDIBLE]

598
00:33:43,268 --> 00:33:44,841
QIQI WANG: That's [INAUDIBLE].

599
00:33:44,841 --> 00:33:45,340
Yeah.

600
00:33:45,340 --> 00:33:51,990
Because flux is number of cars,
the amount of converse quantity

601
00:33:51,990 --> 00:33:56,906
that flows through a point per
unit amount of time, right?

602
00:33:56,906 --> 00:34:01,855
So if you just [INAUDIBLE]
the control volume like this,

603
00:34:01,855 --> 00:34:06,010
this is my-- oh, I shouldn't use
green, because you can't see.

604
00:34:06,010 --> 00:34:10,882
If you true the control
volume like this--

605
00:34:10,882 --> 00:34:13,960
and the flux of this
[INAUDIBLE] is the number

606
00:34:13,960 --> 00:34:17,929
of cars that goes through this
line per unit amount of time.

607
00:34:17,929 --> 00:34:20,010
So this is cars that
have density of rho

608
00:34:20,010 --> 00:34:23,409
goes through the [INAUDIBLE]
U. Of course, the flux is

609
00:34:23,409 --> 00:34:26,530
going to be P times U.

610
00:34:26,530 --> 00:34:32,440
The flux is going to
be P times U, which is

611
00:34:32,440 --> 00:34:34,479
P minus P squared in this case.

612
00:34:34,479 --> 00:34:38,645
Because U is 1 minus P. P
times U is P minus P squared.

613
00:34:38,645 --> 00:34:41,909


614
00:34:41,909 --> 00:34:44,270
Is it clear?

615
00:34:44,270 --> 00:34:46,699
Any questions?

616
00:34:46,699 --> 00:34:51,564
So we derived our
differential equation, dP dt

617
00:34:51,564 --> 00:34:57,670
plus d of P-- I'm just going
to write F(P), it's easier.

618
00:34:57,670 --> 00:35:08,230
F(P) dx is equal to 0 where F(P)
is equal to P minus P squared.

619
00:35:08,230 --> 00:35:08,730
OK.

620
00:35:08,730 --> 00:35:11,290
Now, what is C?

621
00:35:11,290 --> 00:35:15,070
What is the [INAUDIBLE]
characteristic C?

622
00:35:15,070 --> 00:35:19,471


623
00:35:19,471 --> 00:35:23,840
Now, this becomes
highly [INAUDIBLE].

624
00:35:23,840 --> 00:35:25,901
What is C?

625
00:35:25,901 --> 00:35:27,650
AUDIENCE: 1 minus [INAUDIBLE].

626
00:35:27,650 --> 00:35:29,530
QIQI WANG: It's 1 minus 2P.

627
00:35:29,530 --> 00:35:32,980
It is equal to dF dP, right?

628
00:35:32,980 --> 00:35:37,710
So derivative of P
is 1 [INAUDIBLE].

629
00:35:37,710 --> 00:35:38,610
OK.

630
00:35:38,610 --> 00:35:43,360
So we have a relation like this.

631
00:35:43,360 --> 00:35:46,770
We have a relation
of F versus P.

632
00:35:46,770 --> 00:35:49,210
P is all the way from 0 to 1.

633
00:35:49,210 --> 00:35:57,060
F is something like this,
where the maximum takes 1/4.

634
00:35:57,060 --> 00:35:57,720
OK.

635
00:35:57,720 --> 00:36:02,460
And we have a C that
looks like this.

636
00:36:02,460 --> 00:36:06,440
We have C that is equal
to 1 when there is no cars

637
00:36:06,440 --> 00:36:09,080
and goes all the
way to minus 1 when

638
00:36:09,080 --> 00:36:13,100
there is a parking lot of cars.

639
00:36:13,100 --> 00:36:18,090
This is 1 and minus 1.

640
00:36:18,090 --> 00:36:19,300
OK.

641
00:36:19,300 --> 00:36:22,716
So what is happening?

642
00:36:22,716 --> 00:36:28,360
C, again, is the speed
of the local solution.

643
00:36:28,360 --> 00:36:31,330
It is intuitive that
when there are no cars,

644
00:36:31,330 --> 00:36:34,096
things move at 85
miles per hour.

645
00:36:34,096 --> 00:36:37,350
So the [INAUDIBLE] should be 1.

646
00:36:37,350 --> 00:36:38,560
What happens here?

647
00:36:38,560 --> 00:36:40,360
When there is a
parking lot of cars,

648
00:36:40,360 --> 00:36:43,236
things move backwards
at 85 miles an hour?

649
00:36:43,236 --> 00:36:47,637


650
00:36:47,637 --> 00:36:48,946
AUDIENCE: [INAUDIBLE]

651
00:36:48,946 --> 00:36:49,570
QIQI WANG: Yes.

652
00:36:49,570 --> 00:36:53,220
It's kind of the--
so this C here

653
00:36:53,220 --> 00:36:56,965
is not-- I mean, if
you look into physics

654
00:36:56,965 --> 00:36:59,006
you've got two kind of
speeds, like a group speed

655
00:36:59,006 --> 00:37:00,860
and a [INAUDIBLE] speed, right?

656
00:37:00,860 --> 00:37:05,520
So here, the cars move forward
always have a positive speed.

657
00:37:05,520 --> 00:37:09,940
But it doesn't mean if you
look at cars in the backwards

658
00:37:09,940 --> 00:37:11,910
[INAUDIBLE] point
of view, it doesn't

659
00:37:11,910 --> 00:37:16,950
mean the wave that appears
in the flow of traffic

660
00:37:16,950 --> 00:37:18,910
is going to move forward.

661
00:37:18,910 --> 00:37:21,020
Sometimes you are going
to see waves actually

662
00:37:21,020 --> 00:37:22,752
moving backwards.

663
00:37:22,752 --> 00:37:25,210
I mean, we're waiting to see
some examples when it actually

664
00:37:25,210 --> 00:37:26,209
solves that numerically.

665
00:37:26,209 --> 00:37:29,110


666
00:37:29,110 --> 00:37:30,550
OK.

667
00:37:30,550 --> 00:37:35,200
So the task for
you is to hold up

668
00:37:35,200 --> 00:37:39,540
a simulation of this thing in a
periodic [INAUDIBLE] solution.

669
00:37:39,540 --> 00:37:43,290
So let's say this
is our highway.

670
00:37:43,290 --> 00:37:46,280
This is our highway.

671
00:37:46,280 --> 00:37:48,380
It just goes in circles, right?

672
00:37:48,380 --> 00:37:51,380
Assuming you have a bunch
of cars driving in circles.

673
00:37:51,380 --> 00:37:53,380
AUDIENCE: I could never
get off the [INAUDIBLE].

674
00:37:53,380 --> 00:37:54,255
AUDIENCE: [INAUDIBLE]

675
00:37:54,255 --> 00:37:54,880
QIQI WANG: Huh?

676
00:37:54,880 --> 00:37:55,653
AUDIENCE: NASCAR.

677
00:37:55,653 --> 00:37:57,545
QIQI WANG: NASCAR,
yeah, yeah, right.

678
00:37:57,545 --> 00:37:59,910
Yeah.

679
00:37:59,910 --> 00:38:02,320
We have a lot of cars
driving in home circles.

680
00:38:02,320 --> 00:38:05,850
And let's just [INAUDIBLE]
how these work.

681
00:38:05,850 --> 00:38:07,690
And then at sometime,
we are going

682
00:38:07,690 --> 00:38:10,530
to put a red light over here.

683
00:38:10,530 --> 00:38:14,774
And say, OK, the cars stop here.

684
00:38:14,774 --> 00:38:16,774
What is the right way to
model that numerically?

685
00:38:16,774 --> 00:38:21,040


686
00:38:21,040 --> 00:38:23,610
You can set the flux
[INAUDIBLE] to 0, right?

687
00:38:23,610 --> 00:38:26,243
I mean, at some point, that's
a [INAUDIBLE] simulation.

688
00:38:26,243 --> 00:38:28,168
And for example,
time equal to 1,

689
00:38:28,168 --> 00:38:34,130
that's [INAUDIBLE] 0 [INAUDIBLE]
before the red light.

690
00:38:34,130 --> 00:38:39,370
And at some point, let's
set the light [INAUDIBLE].

691
00:38:39,370 --> 00:38:42,306
But let's do something
simple in the beginning.

692
00:38:42,306 --> 00:38:45,150
Let's just [INAUDIBLE]
the solution

693
00:38:45,150 --> 00:38:48,110
of [INAUDIBLE] with no red
lights or anything like that.

694
00:38:48,110 --> 00:38:50,675
I'm going to give you
the initial position.

695
00:38:50,675 --> 00:38:53,650
And we'll see how it evolves.

696
00:38:53,650 --> 00:38:57,590
And then we are
using final volume.

697
00:38:57,590 --> 00:39:01,770
And you are free to take--
the [INAUDIBLE] case

698
00:39:01,770 --> 00:39:15,050
we had in the last lecture
is in the shared folder.

699
00:39:15,050 --> 00:39:17,900
And I'm just going to make
another shared folder here.

700
00:39:17,900 --> 00:39:23,780


701
00:39:23,780 --> 00:39:24,700
OK.

702
00:39:24,700 --> 00:39:32,210
I'm just going to take a
[INAUDIBLE] and the du dt.

703
00:39:32,210 --> 00:39:40,356
I'm just going to take the
[INAUDIBLE] stamp [INAUDIBLE]

704
00:39:40,356 --> 00:39:41,160
to this.

705
00:39:41,160 --> 00:39:43,182
Because I know it's working.

706
00:39:43,182 --> 00:39:52,004
And [INAUDIBLE] start by
modifying [INAUDIBLE].

707
00:39:52,004 --> 00:39:58,470


708
00:39:58,470 --> 00:40:01,210
So instead of u_bar, you're
going to solving for P_bar.

709
00:40:01,210 --> 00:40:05,270


710
00:40:05,270 --> 00:40:14,843
And what is the first thing
you would want changed in this?

711
00:40:14,843 --> 00:40:15,718
AUDIENCE: [INAUDIBLE]

712
00:40:15,718 --> 00:40:21,027


713
00:40:21,027 --> 00:40:21,860
QIQI WANG: Oh, yeah.

714
00:40:21,860 --> 00:40:22,790
Yeah, make it upwind.

715
00:40:22,790 --> 00:40:23,580
Yeah, right.

716
00:40:23,580 --> 00:40:27,478
To make it upwind
there are [INAUDIBLE].

717
00:40:27,478 --> 00:40:28,464
Huh?

718
00:40:28,464 --> 00:40:30,484
AUDIENCE: [INAUDIBLE]

719
00:40:30,484 --> 00:40:32,150
QIQI WANG: It's no
longer that equation.

720
00:40:32,150 --> 00:40:33,370
The equation is [INAUDIBLE].

721
00:40:33,370 --> 00:40:35,600
But there is only
one line we can

722
00:40:35,600 --> 00:40:39,030
change to solve for a different
equation and find the volume.

723
00:40:39,030 --> 00:40:40,041
What line is that?

724
00:40:40,041 --> 00:40:40,916
AUDIENCE: [INAUDIBLE]

725
00:40:40,916 --> 00:40:44,820


726
00:40:44,820 --> 00:40:47,290
QIQI WANG: Second line.

727
00:40:47,290 --> 00:40:53,640
The only thing that satisfies a
different equation is the flux.

728
00:40:53,640 --> 00:40:58,150
You change the flux, you are now
solving a different equation.

729
00:40:58,150 --> 00:40:59,790
OK.

730
00:40:59,790 --> 00:41:01,710
Of course, another
thing is what is

731
00:41:01,710 --> 00:41:05,310
upwind direction [INAUDIBLE].

732
00:41:05,310 --> 00:41:09,155
In [INAUDIBLE] equation, we know
U is positive plus [INAUDIBLE]

733
00:41:09,155 --> 00:41:10,010
the right.

734
00:41:10,010 --> 00:41:11,940
U is negative [INAUDIBLE].

735
00:41:11,940 --> 00:41:15,260
Now, this is different.

736
00:41:15,260 --> 00:41:19,150
Now, this is different,
because we have [INAUDIBLE].

737
00:41:19,150 --> 00:41:21,265
What is left and
what is right all

738
00:41:21,265 --> 00:41:25,456
depends on the [INAUDIBLE]
instead of the [INAUDIBLE].

739
00:41:25,456 --> 00:41:31,865


740
00:41:31,865 --> 00:41:33,946
So any questions?

741
00:41:33,946 --> 00:41:34,446
OK.

742
00:41:34,446 --> 00:41:39,220
If you are done with the-- If
you get the solution going,

743
00:41:39,220 --> 00:41:43,810
try to put a red
light somewhere.

744
00:41:43,810 --> 00:41:48,855
Let me kind of illustrate how
I should modify the driver.

745
00:41:48,855 --> 00:41:53,460


746
00:41:53,460 --> 00:41:55,746
So I have this here.

747
00:41:55,746 --> 00:41:58,590


748
00:41:58,590 --> 00:42:00,110
That, I didn't call
it [INAUDIBLE].

749
00:42:00,110 --> 00:42:02,890
Because it's no
longer [INAUDIBLE].

750
00:42:02,890 --> 00:42:03,830
Let me [INAUDIBLE].

751
00:42:03,830 --> 00:42:07,120


752
00:42:07,120 --> 00:42:08,140
trafficDriver.

753
00:42:08,140 --> 00:42:09,122
OK, trafficDriver.

754
00:42:09,122 --> 00:42:12,070


755
00:42:12,070 --> 00:42:14,230
And let me see.

756
00:42:14,230 --> 00:42:25,777
So in this ODE 45, I'm going to
set t0=(i-1) times dt and t1=i

757
00:42:25,777 --> 00:42:26,910
times dt.

758
00:42:26,910 --> 00:42:30,460
So this is like the beginning
and end of this interval.

759
00:42:30,460 --> 00:42:32,560
And when I'm solving
it, I'm going

760
00:42:32,560 --> 00:42:35,570
to be solving it from t0 to t1.

761
00:42:35,570 --> 00:42:49,800
So now the difference is if I
look at-- so let me do this.

762
00:42:49,800 --> 00:42:57,336


763
00:42:57,336 --> 00:42:59,220
OK.

764
00:42:59,220 --> 00:43:05,170
So I just want to be able
to passing the actual time.

765
00:43:05,170 --> 00:43:09,870
So I'm just going to be
passing also time into this.

766
00:43:09,870 --> 00:43:12,820
So I also have a time into this.

767
00:43:12,820 --> 00:43:17,670
And then now if--
OK, so let me just

768
00:43:17,670 --> 00:43:21,090
go through what
you need to modify.

769
00:43:21,090 --> 00:43:26,340
So first of all, I'm
going to modify this,

770
00:43:26,340 --> 00:43:28,980
which is the different flux.

771
00:43:28,980 --> 00:43:30,360
Right?

772
00:43:30,360 --> 00:43:34,670
Instead of half of u squared,
I have u minus u squared.

773
00:43:34,670 --> 00:43:35,210
OK.

774
00:43:35,210 --> 00:43:37,360
k_plus_half is still the same.

775
00:43:37,360 --> 00:43:42,840
I'm going to-- so if I
don't modify anything,

776
00:43:42,840 --> 00:43:44,110
I think it all wrong.

777
00:43:44,110 --> 00:43:47,430
But like once a shock wave
forms, it'll go unstable.

778
00:43:47,430 --> 00:43:50,022
So let me try it.

779
00:43:50,022 --> 00:43:51,190
I doesn't even wrong.

780
00:43:51,190 --> 00:43:55,380


781
00:43:55,380 --> 00:43:58,930
[INAUDIBLE] dimension
matrices being concatenated

782
00:43:58,930 --> 00:43:59,859
are not consistent.

783
00:43:59,859 --> 00:44:08,002


784
00:44:08,002 --> 00:44:10,260
AUDIENCE: [INAUDIBLE]

785
00:44:10,260 --> 00:44:11,010
QIQI WANG: Oh, OK.

786
00:44:11,010 --> 00:44:12,070
I'm [INAUDIBLE]
different [INAUDIBLE].

787
00:44:12,070 --> 00:44:13,070
Yes, thank you.

788
00:44:13,070 --> 00:44:23,846


789
00:44:23,846 --> 00:44:24,794
I save this.

790
00:44:24,794 --> 00:44:27,578


791
00:44:27,578 --> 00:44:30,370
AUDIENCE: [INAUDIBLE]

792
00:44:30,370 --> 00:44:33,150
QIQI WANG: Oh, i'm solving--
yeah-- from t0 to t1.

793
00:44:33,150 --> 00:44:34,470
Yeah, thank you.

794
00:44:34,470 --> 00:44:35,000
OK.

795
00:44:35,000 --> 00:44:37,050
So this thing goes.

796
00:44:37,050 --> 00:44:42,380
And we're going to see
[INAUDIBLE]-- wait.

797
00:44:42,380 --> 00:44:44,710
I shouldn't initialize
it to 2, right?

798
00:44:44,710 --> 00:44:46,300
Because 1 is parking lot.

799
00:44:46,300 --> 00:44:47,440
What is 2?

800
00:44:47,440 --> 00:44:47,940
OK.

801
00:44:47,940 --> 00:44:50,690
So I need to also modify
the initial condition.

802
00:44:50,690 --> 00:44:55,120
So what I can do is I can set
this, let's say, divide by 4.

803
00:44:55,120 --> 00:44:55,620
OK.

804
00:44:55,620 --> 00:44:59,820
So this is going to be my
initial condition [INAUDIBLE].

805
00:44:59,820 --> 00:45:00,320
OK.

806
00:45:00,320 --> 00:45:06,180
I also want to-- my ylim,
when I show my solution,

807
00:45:06,180 --> 00:45:09,780
I just want to go
from 0 to 1, right?

808
00:45:09,780 --> 00:45:10,280
OK.

809
00:45:10,280 --> 00:45:12,450
So let me run again.

810
00:45:12,450 --> 00:45:12,950
OK.

811
00:45:12,950 --> 00:45:15,374
So this is how my
solution looks like.

812
00:45:15,374 --> 00:45:17,968
This part goes really fast.

813
00:45:17,968 --> 00:45:21,700
[INAUDIBLE] unstable.

814
00:45:21,700 --> 00:45:23,890
And I need to change
this to upwinding.

815
00:45:23,890 --> 00:45:27,230
But we have the [INAUDIBLE]
part of the solution correct.

816
00:45:27,230 --> 00:45:30,355
Every [INAUDIBLE] towards
a positive direction when

817
00:45:30,355 --> 00:45:32,746
the [INAUDIBLE] move faster.

818
00:45:32,746 --> 00:45:34,910
That's [INAUDIBLE] car
density, cars move slower.

819
00:45:34,910 --> 00:45:39,032
So sometimes, a shock
wave is going to form.

820
00:45:39,032 --> 00:45:41,362
Because the faster
parts is going

821
00:45:41,362 --> 00:45:46,752
to [INAUDIBLE] to the
slower parts and decelerate.

822
00:45:46,752 --> 00:45:48,690
So it becomes a shock wave.

823
00:45:48,690 --> 00:45:51,270


824
00:45:51,270 --> 00:45:54,790
So let's start to modify
this to an upwinding scheme.

825
00:45:54,790 --> 00:45:57,640


826
00:45:57,640 --> 00:45:59,620
We computed F_bar.

827
00:45:59,620 --> 00:46:01,130
And we have a c.

828
00:46:01,130 --> 00:46:02,380
c is equal to 1-u_bar times 2.

829
00:46:02,380 --> 00:46:07,730


830
00:46:07,730 --> 00:46:08,760
OK.

831
00:46:08,760 --> 00:46:21,720
And the c at k_plus_half
is equal to c plus-- OK.

832
00:46:21,720 --> 00:46:24,150
Let me do like this.

833
00:46:24,150 --> 00:46:32,280
So in the condition, instead
of u_bar, I'm going to put c.

834
00:46:32,280 --> 00:46:38,640


835
00:46:38,640 --> 00:46:39,140
OK.

836
00:46:39,140 --> 00:46:40,490
Here, too.

837
00:46:40,490 --> 00:46:43,783
Instead of the u_bar,
I'm going to put c.

838
00:46:43,783 --> 00:46:47,170


839
00:46:47,170 --> 00:46:48,930
OK.

840
00:46:48,930 --> 00:46:51,220
So this completes my upwinding.

841
00:46:51,220 --> 00:46:53,180
And let's try it again.

842
00:46:53,180 --> 00:46:56,400


843
00:46:56,400 --> 00:46:58,275
We still have the solution.

844
00:46:58,275 --> 00:47:01,110
But the lower
density goes faster.

845
00:47:01,110 --> 00:47:06,370
High density goes slower
[INAUDIBLE] high density,

846
00:47:06,370 --> 00:47:09,844
but now it's
[INAUDIBLE] stability.

847
00:47:09,844 --> 00:47:10,792
So that's good.

848
00:47:10,792 --> 00:47:13,640


849
00:47:13,640 --> 00:47:15,830
And if you want to
get better solution,

850
00:47:15,830 --> 00:47:18,124
I can change this to 256.

851
00:47:18,124 --> 00:47:20,630
And I round this.

852
00:47:20,630 --> 00:47:22,884
And I [INAUDIBLE]
the top [INAUDIBLE].

853
00:47:22,884 --> 00:47:31,841


854
00:47:31,841 --> 00:47:32,340
OK.

855
00:47:32,340 --> 00:47:38,330
So now, let's say at this point,
so at either the beginning

856
00:47:38,330 --> 00:47:44,190
[INAUDIBLE], I have time equal
to 1, we get a red light.

857
00:47:44,190 --> 00:47:47,900
And at time equal to 2, we
turn the light to green.

858
00:47:47,900 --> 00:47:49,860
[INAUDIBLE] there.

859
00:47:49,860 --> 00:47:51,820
OK.

860
00:47:51,820 --> 00:47:54,130
So we do that.

861
00:47:54,130 --> 00:47:56,670
[INAUDIBLE] then
that [INAUDIBLE]

862
00:47:56,670 --> 00:48:01,530
what we need to do
to make that happen.

863
00:48:01,530 --> 00:48:04,750
The key thing is
now you have time.

864
00:48:04,750 --> 00:48:07,130
You have t available to you.

865
00:48:07,130 --> 00:48:12,950
So you can basically do a
conditional statement on t.

866
00:48:12,950 --> 00:48:18,932
If t is greater than 1 and t is
less than 2, set the red light.

867
00:48:18,932 --> 00:48:22,916


868
00:48:22,916 --> 00:48:23,533
[INAUDIBLE]

869
00:48:23,533 --> 00:48:24,908
AUDIENCE: What
about [INAUDIBLE]?

870
00:48:24,908 --> 00:48:29,400


871
00:48:29,400 --> 00:48:31,600
QIQI WANG: Oh.

872
00:48:31,600 --> 00:48:32,290
Thank you.

873
00:48:32,290 --> 00:48:33,050
That is wrong.

874
00:48:33,050 --> 00:48:34,070
So it should be F_bar_k.

875
00:48:34,070 --> 00:48:38,690


876
00:48:38,690 --> 00:48:39,876
Yeah, it should be F_bar_k.

877
00:48:39,876 --> 00:48:43,620


878
00:48:43,620 --> 00:48:46,350
Thanks.

879
00:48:46,350 --> 00:48:47,420
Yeah.

880
00:48:47,420 --> 00:48:49,720
I should just rename
it to something else,

881
00:48:49,720 --> 00:48:58,335
so that-- I'm just going
to rename it as 2t.

882
00:48:58,335 --> 00:49:02,290


883
00:49:02,290 --> 00:49:02,790
All right.

884
00:49:02,790 --> 00:49:05,380


885
00:49:05,380 --> 00:49:06,255
AUDIENCE: [INAUDIBLE]

886
00:49:06,255 --> 00:49:11,700


887
00:49:11,700 --> 00:49:14,290
QIQI WANG: Which
one [INAUDIBLE]?

888
00:49:14,290 --> 00:49:15,165
AUDIENCE: [INAUDIBLE]

889
00:49:15,165 --> 00:49:19,150


890
00:49:19,150 --> 00:49:19,900
QIQI WANG: Sorry?

891
00:49:19,900 --> 00:49:22,067
AUDIENCE: [INAUDIBLE]

892
00:49:22,067 --> 00:49:22,900
QIQI WANG: This one?

893
00:49:22,900 --> 00:49:26,160
AUDIENCE: Yeah. [INAUDIBLE]

894
00:49:26,160 --> 00:49:28,430
QIQI WANG: Yeah.
[INAUDIBLE] plus 1, right?

895
00:49:28,430 --> 00:49:31,530
So because I'm looking, I
[INAUDIBLE] the interface.

896
00:49:31,530 --> 00:49:33,800
I want to look at the
interface [INAUDIBLE]

897
00:49:33,800 --> 00:49:38,878
decide which side I should
upwind the flux at k_plus_half.

898
00:49:38,878 --> 00:49:42,200


899
00:49:42,200 --> 00:49:48,335
In order to make a decision,
I want my [INAUDIBLE] speed

900
00:49:48,335 --> 00:49:50,440
at k_plus_half.

901
00:49:50,440 --> 00:49:53,120
And the right way to [INAUDIBLE]
the other [INAUDIBLE]

902
00:49:53,120 --> 00:49:57,308
here k_plus_half is take the
average of the [INAUDIBLE]

903
00:49:57,308 --> 00:50:00,100
speed [INAUDIBLE]
at k(and) k plus 1.

904
00:50:00,100 --> 00:50:05,000


905
00:50:05,000 --> 00:50:07,120
Just to make the code
easier to understand,

906
00:50:07,120 --> 00:50:19,680
I just want to change-- I
think I want to replace-- oh.

907
00:50:19,680 --> 00:50:24,300
I think I want to
replace my i with k.

908
00:50:24,300 --> 00:50:27,891


909
00:50:27,891 --> 00:50:28,390
[INAUDIBLE]

910
00:50:28,390 --> 00:50:32,910


911
00:50:32,910 --> 00:50:35,660
All right.

912
00:50:35,660 --> 00:50:36,160
Yeah.

913
00:50:36,160 --> 00:50:40,360
So instead of-- it's easier
to replace [INAUDIBLE]

914
00:50:40,360 --> 00:50:41,884
k_plus_half than [INAUDIBLE].

915
00:50:41,884 --> 00:50:46,132


916
00:50:46,132 --> 00:50:48,970
AUDIENCE: [INAUDIBLE]

917
00:50:48,970 --> 00:50:51,440
QIQI WANG: Why did I half t?

918
00:50:51,440 --> 00:50:55,800
I wanted to half t, because
what I like you to do now

919
00:50:55,800 --> 00:51:00,390
is to set a [INAUDIBLE]
when t is [INAUDIBLE].

920
00:51:00,390 --> 00:51:18,210


921
00:51:18,210 --> 00:51:18,710
OK.

922
00:51:18,710 --> 00:51:22,180
So what happens
on the red light?

923
00:51:22,180 --> 00:51:23,680
AUDIENCE: [INAUDIBLE]

924
00:51:23,680 --> 00:51:24,601
QIQI WANG: Huh?

925
00:51:24,601 --> 00:51:26,710
AUDIENCE: [INAUDIBLE]

926
00:51:26,710 --> 00:51:28,570
QIQI WANG: Yeah, the
[INAUDIBLE] goes to 0.

927
00:51:28,570 --> 00:51:30,040
That's one way to say it.

928
00:51:30,040 --> 00:51:34,549
Another way to say it is
no cars can pass, right?

929
00:51:34,549 --> 00:51:37,208
If no cars can pass,
then the [INAUDIBLE]

930
00:51:37,208 --> 00:51:41,030
pass through the
red light is what?

931
00:51:41,030 --> 00:51:41,966
0, right?

932
00:51:41,966 --> 00:51:44,880


933
00:51:44,880 --> 00:51:51,150
So what I can do here is that
before I compute k minus half,

934
00:51:51,150 --> 00:51:54,870
I just want to say
if k is greater

935
00:51:54,870 --> 00:52:04,430
than 1 t is less than 2,
F_k_plus_half at the end--

936
00:52:04,430 --> 00:52:07,820
so k_plus_half-- the last one
is basically the right hand

937
00:52:07,820 --> 00:52:12,818
and also the left
hand-- is equal to 0.

938
00:52:12,818 --> 00:52:16,500
I think I [INAUDIBLE].

939
00:52:16,500 --> 00:52:17,000
OK.

940
00:52:17,000 --> 00:52:19,868
That's it.

941
00:52:19,868 --> 00:52:22,360
That's what I need to do.

942
00:52:22,360 --> 00:52:31,620
Now, in the trafficDriver,
let me set it to-- ah, ah, ah.

943
00:52:31,620 --> 00:52:34,056
Let me copy this trafficDriver
for the [INAUDIBLE].

944
00:52:34,056 --> 00:52:39,020


945
00:52:39,020 --> 00:52:39,520
Yeah.

946
00:52:39,520 --> 00:52:41,690
I think we're
overwriting [INAUDIBLE].

947
00:52:41,690 --> 00:52:46,460
And it's going to save
trafficDriver [INAUDIBLE].

948
00:52:46,460 --> 00:52:54,270


949
00:52:54,270 --> 00:52:54,770
OK.

950
00:52:54,770 --> 00:52:58,290


951
00:52:58,290 --> 00:52:59,965
Now, instead of 1
over dt, I'm going

952
00:52:59,965 --> 00:53:03,840
to simulate it for
five time units,

953
00:53:03,840 --> 00:53:07,542
so that I actually
see the light.

954
00:53:07,542 --> 00:53:10,601


955
00:53:10,601 --> 00:53:11,100
OK.

956
00:53:11,100 --> 00:53:16,362
We see the same thing
here [INAUDIBLE].

957
00:53:16,362 --> 00:53:18,400
At sometimes, the
light [INAUDIBLE].

958
00:53:18,400 --> 00:53:22,450


959
00:53:22,450 --> 00:53:23,390
Yeah, OK.

960
00:53:23,390 --> 00:53:26,660
See, the lights
is in effect here.

961
00:53:26,660 --> 00:53:28,590
All the [INAUDIBLE]
simulates here.

962
00:53:28,590 --> 00:53:31,280
And then we have
even [INAUDIBLE]

963
00:53:31,280 --> 00:53:34,490
that actually goes backward.

964
00:53:34,490 --> 00:53:34,990
OK.

965
00:53:34,990 --> 00:53:38,280
Now, there is no cars moving.

966
00:53:38,280 --> 00:53:40,116
All the cars are
stopped over there.

967
00:53:40,116 --> 00:53:42,870


968
00:53:42,870 --> 00:53:46,680
So this is a parking lot
while there is no cars here.

969
00:53:46,680 --> 00:53:48,690
And at [INAUDIBLE] equal
to 2 or [INAUDIBLE].

970
00:53:48,690 --> 00:53:51,308


971
00:53:51,308 --> 00:53:52,183
AUDIENCE: [INAUDIBLE]

972
00:53:52,183 --> 00:54:03,660


973
00:54:03,660 --> 00:54:15,960
QIQI WANG: Did I-- did
I-- what's happening?

974
00:54:15,960 --> 00:54:16,460
OK.

975
00:54:16,460 --> 00:54:20,962
If t is greater than 1
and t is less than 2--

976
00:54:20,962 --> 00:54:23,332
AUDIENCE: [INAUDIBLE]

977
00:54:23,332 --> 00:54:25,580
QIQI WANG: And never goes
passed 2, are you sure?

978
00:54:25,580 --> 00:54:29,624


979
00:54:29,624 --> 00:54:30,499
AUDIENCE: [INAUDIBLE]

980
00:54:30,499 --> 00:54:53,797


981
00:54:53,797 --> 00:54:54,380
QIQI WANG: OK.

982
00:54:54,380 --> 00:54:58,028
I think I know what's happening.

983
00:54:58,028 --> 00:54:59,920
So I actually have
come to a situation

984
00:54:59,920 --> 00:55:03,784
where-- hm, let me think.

985
00:55:03,784 --> 00:55:21,988


986
00:55:21,988 --> 00:55:24,450
AUDIENCE: [INAUDIBLE].

987
00:55:24,450 --> 00:55:25,330
What

988
00:55:25,330 --> 00:55:26,896
QIQI WANG: Yeah.

989
00:55:26,896 --> 00:55:28,590
Let me do this.

990
00:55:28,590 --> 00:55:32,730
I'm actually encountering
a situation that cannot be

991
00:55:32,730 --> 00:55:34,430
handled by this [INAUDIBLE].

992
00:55:34,430 --> 00:55:39,610


993
00:55:39,610 --> 00:55:46,780
What is happening is that
[INAUDIBLE]-- let me do this.

994
00:55:46,780 --> 00:55:51,252
I think I can change this
around by doing this.

995
00:55:51,252 --> 00:55:54,450


996
00:55:54,450 --> 00:55:59,860
I can change this around by
not letting the red light

997
00:55:59,860 --> 00:56:01,280
to stop for that long.

998
00:56:01,280 --> 00:56:11,040
So I only let the red
light to-- if I only

999
00:56:11,040 --> 00:56:17,810
let the red light
go from 0.2, I think

1000
00:56:17,810 --> 00:56:19,890
I'm going to get
rid of the problem.

1001
00:56:19,890 --> 00:56:24,950
The problem is that when
the red light happens here,

1002
00:56:24,950 --> 00:56:31,200
the flux-- the P at one
side shock wave is 1.

1003
00:56:31,200 --> 00:56:34,470
At the other side of the shock
wave, the P is equal to 0.

1004
00:56:34,470 --> 00:56:37,820


1005
00:56:37,820 --> 00:56:44,718
So both values of P
[INAUDIBLE] equal to 0.

1006
00:56:44,718 --> 00:56:49,856
Because the flux is P times 1
minus P or P minus P squared.

1007
00:56:49,856 --> 00:56:52,640
So the other P is 1 or zero.

1008
00:56:52,640 --> 00:56:54,810
The flux is equal to 0.

1009
00:56:54,810 --> 00:56:58,138
That's why no matter if you
take upwinding or downwinding,

1010
00:56:58,138 --> 00:56:59,840
you'll get 0 flux.

1011
00:56:59,840 --> 00:57:02,692
So this is a case where
[INAUDIBLE] upwinding

1012
00:57:02,692 --> 00:57:04,960
wouldn't work.

1013
00:57:04,960 --> 00:57:07,894
It's like a singular case
where [INAUDIBLE] upwinding

1014
00:57:07,894 --> 00:57:10,800
wouldn't work over here.

1015
00:57:10,800 --> 00:57:13,550
So what I'm going to
do here is am going

1016
00:57:13,550 --> 00:57:16,670
to set something artificial.

1017
00:57:16,670 --> 00:57:23,445
And let's say when
there is a red light,

1018
00:57:23,445 --> 00:57:26,880
I think I'm just going to
set the flux equal to 0.01.

1019
00:57:26,880 --> 00:57:33,790
So that there is a small leakage
of cars over the red light,

1020
00:57:33,790 --> 00:57:36,060
so that we actually
want to see what

1021
00:57:36,060 --> 00:57:38,560
is going to happen [INAUDIBLE].

1022
00:57:38,560 --> 00:57:41,589


1023
00:57:41,589 --> 00:57:42,464
AUDIENCE: [INAUDIBLE]

1024
00:57:42,464 --> 00:57:48,012


1025
00:57:48,012 --> 00:57:48,678
QIQI WANG: Yeah.

1026
00:57:48,678 --> 00:57:51,082
So if you take
[INAUDIBLE], you're

1027
00:57:51,082 --> 00:57:58,040
going to learn about the entropy
conditions of [INAUDIBLE].

1028
00:57:58,040 --> 00:58:00,122
And the entropy
condition tells you

1029
00:58:00,122 --> 00:58:04,800
that there is a
physical solution.

1030
00:58:04,800 --> 00:58:07,072
There is a
non-physical solution.

1031
00:58:07,072 --> 00:58:09,840
And over here [INAUDIBLE].

1032
00:58:09,840 --> 00:58:12,264
But then over here it
is a physical solution.

1033
00:58:12,264 --> 00:58:15,715
While this kind
of an [INAUDIBLE]

1034
00:58:15,715 --> 00:58:17,623
is going to be a
non-physical solution.

1035
00:58:17,623 --> 00:58:20,970


1036
00:58:20,970 --> 00:58:27,160
And that cannot be dealt with
by the [INAUDIBLE] scheme.

1037
00:58:27,160 --> 00:58:31,480
And you nee a more
complex flux scheme

1038
00:58:31,480 --> 00:58:34,990
to deal with
situations like that.

1039
00:58:34,990 --> 00:58:35,865
AUDIENCE: [INAUDIBLE]

1040
00:58:35,865 --> 00:58:39,650


1041
00:58:39,650 --> 00:58:41,170
QIQI WANG: There
is no downwinding.

1042
00:58:41,170 --> 00:58:45,050
And what you need to do is this.

1043
00:58:45,050 --> 00:58:47,860


1044
00:58:47,860 --> 00:58:51,490
What you need do is
you need add something

1045
00:58:51,490 --> 00:58:55,340
called numerical
dispositing into this.

1046
00:58:55,340 --> 00:59:00,413
So you need to do
F_k_plus_half is

1047
00:59:00,413 --> 00:59:10,074
equal to F_k_plus_half plus
0.5 times absolute value of c.

1048
00:59:10,074 --> 00:59:13,790


1049
00:59:13,790 --> 00:59:21,480
So instead of
taking the c's, you

1050
00:59:21,480 --> 00:59:26,170
want to take the absolute
value of c's and divide by 2.

1051
00:59:26,170 --> 00:59:30,502
So this is like a very
similar concept to upwinding.

1052
00:59:30,502 --> 00:59:33,080
But it operates differently.

1053
00:59:33,080 --> 00:59:37,265
You need to take the
absolute value of c,

1054
00:59:37,265 --> 00:59:48,722
but multiply that with
u_bar and plus_half.

1055
00:59:48,722 --> 00:59:58,045
So it's minus [INAUDIBLE]
u_bar minus 1 0, minus 1.

1056
00:59:58,045 --> 01:00:00,910


1057
01:00:00,910 --> 01:00:05,910
So you need to kind of-- so
instead of doing upwinding,

1058
01:00:05,910 --> 01:00:10,660
you need to use a
numerical dispositive

1059
01:00:10,660 --> 01:00:18,930
or a numerical diffusion
to basically achieve

1060
01:00:18,930 --> 01:00:22,680
the impact of upwinding, but do
this a little bit differently.

1061
01:00:22,680 --> 01:00:26,680
But that's kind of beyond
the scope of this class.

1062
01:00:26,680 --> 01:00:29,540
So, yeah, If you
want to learn that,

1063
01:00:29,540 --> 01:00:32,110
you're welcome to
take the [INAUDIBLE].

1064
01:00:32,110 --> 01:00:36,675
We are going to be learning
all about [INAUDIBLE] speed.

1065
01:00:36,675 --> 01:00:39,342
So let's see if this
goes [INAUDIBLE].

1066
01:00:39,342 --> 01:00:44,794


1067
01:00:44,794 --> 01:00:45,294
OK.

1068
01:00:45,294 --> 01:00:46,782
So we have the [INAUDIBLE].

1069
01:00:46,782 --> 01:00:51,738


1070
01:00:51,738 --> 01:00:52,238
OK.

1071
01:00:52,238 --> 01:00:53,950
Now, the cars are
starting to move.

1072
01:00:53,950 --> 01:00:57,282


1073
01:00:57,282 --> 01:01:05,104
[INAUDIBLE] the light goes
green, the cars start to move.

1074
01:01:05,104 --> 01:01:12,000


1075
01:01:12,000 --> 01:01:15,110
To do it properly, you
have to change this back

1076
01:01:15,110 --> 01:01:17,500
to the central average.

1077
01:01:17,500 --> 01:01:20,954
So you do upwinding or
add numerical dispositive,

1078
01:01:20,954 --> 01:01:21,850
but don't do both.

1079
01:01:21,850 --> 01:01:26,050
And doing both, so
the scheme works,

1080
01:01:26,050 --> 01:01:28,446
but it's not as accurate.

1081
01:01:28,446 --> 01:01:34,490
It's suffers from additional
numerical dissipation.

1082
01:01:34,490 --> 01:01:35,130
OK.

1083
01:01:35,130 --> 01:01:43,230
So I hope you have fun playing
with this small example here.

1084
01:01:43,230 --> 01:01:47,250
So starting from our next
lecture, the next lecture

1085
01:01:47,250 --> 01:01:50,822
we're going to be doing
a review for the midterm.

1086
01:01:50,822 --> 01:01:52,530
But starting from
after the midterm,

1087
01:01:52,530 --> 01:01:56,150
we're going to be talking
about first of all, stability

1088
01:01:56,150 --> 01:01:58,630
of solution of positive
differential equations,

1089
01:01:58,630 --> 01:02:02,032
and then talk about finite
[INAUDIBLE] method, which

1090
01:02:02,032 --> 01:02:03,740
is the third method
we are going to start

1091
01:02:03,740 --> 01:02:04,965
in fall for [INAUDIBLE].

1092
01:02:04,965 --> 01:02:08,430


1093
01:02:08,430 --> 01:02:09,330
All right.

1094
01:02:09,330 --> 01:02:11,480
We'll see you next week.

1095
01:02:11,480 --> 01:02:21,012