1 00:00:00,000 --> 00:00:01,655 NARRATOR: The following content is 2 00:00:01,655 --> 00:00:04,966 provided by MIT OpenCourseWare under a Creative Commons 3 00:00:04,966 --> 00:00:06,060 license. 4 00:00:06,060 --> 00:00:08,230 Additional information about our license 5 00:00:08,230 --> 00:00:10,550 and MIT OpenCourseWare in general 6 00:00:10,550 --> 00:00:11,770 is available at ocw.mit.edu. 7 00:00:15,460 --> 00:00:21,760 PROFESSOR: So, you see that -- I thought maybe we've got enough 8 00:00:21,760 --> 00:00:27,740 good examples in theory where we need some practice. 9 00:00:27,740 --> 00:00:34,680 We need some experiments with some of the finite difference 10 00:00:34,680 --> 00:00:37,320 methods that we've spoken about. 11 00:00:37,320 --> 00:00:42,160 So I created this homework in which you would like choose 12 00:00:42,160 --> 00:00:45,610 one of these that we've done. 13 00:00:45,610 --> 00:00:50,690 Well, actually, number two, the Schrodinger equation, 14 00:00:50,690 --> 00:00:54,860 which has that imaginary i in there, 15 00:00:54,860 --> 00:00:57,800 we haven't spoken about yet, but you 16 00:00:57,800 --> 00:01:02,160 could do the kind of Fourier analysis 17 00:01:02,160 --> 00:01:04,240 that we've done for the heat equation. 18 00:01:04,240 --> 00:01:09,790 But, of course, it's going to be different because of that i. 19 00:01:09,790 --> 00:01:14,390 You'll get an e to the minus i k squared t, I guess. 20 00:01:14,390 --> 00:01:18,610 And that means that's a number of absolute value 21 00:01:18,610 --> 00:01:23,550 1 instead of a rapidly decaying coefficient. 22 00:01:23,550 --> 00:01:26,990 OK, so my idea is to choose one of those groups, 23 00:01:26,990 --> 00:01:30,300 or a number five, if you want. 24 00:01:30,300 --> 00:01:36,320 And say, by next Friday, just give me 25 00:01:36,320 --> 00:01:41,300 some experiment with a code. 26 00:01:43,850 --> 00:01:47,150 In this case, you could use -- well, 27 00:01:47,150 --> 00:01:52,150 we have four methods and three of them are already in code, 28 00:01:52,150 --> 00:01:54,590 and you're welcome to use those or your own. 29 00:01:54,590 --> 00:01:57,050 And I raised questions last time -- 30 00:01:57,050 --> 00:02:01,740 and I'll put this on the website with a little more detail. 31 00:02:01,740 --> 00:02:04,880 So just to give you an idea of what's coming. 32 00:02:04,880 --> 00:02:07,550 So this is the problem that we're 33 00:02:07,550 --> 00:02:10,890 in the middle of speaking about, near the end, 34 00:02:10,890 --> 00:02:15,560 the mixture of convection and diffusion. 35 00:02:15,560 --> 00:02:19,140 And this is Schrodinger, which is new. 36 00:02:19,140 --> 00:02:23,430 This is the advection equation, or the one-way wave equation, 37 00:02:23,430 --> 00:02:27,750 which we've started with, and this is what's -- 38 00:02:27,750 --> 00:02:32,320 this'll be the subject of today's lecture and next time. 39 00:02:32,320 --> 00:02:39,100 And the best reference I can give, as well as the textbook, 40 00:02:39,100 --> 00:02:45,440 the applied math book has significant discussion 41 00:02:45,440 --> 00:02:48,660 of this conservation law. 42 00:02:48,660 --> 00:02:54,320 But also it's very well presented in the 16.920 lecture 43 00:02:54,320 --> 00:02:57,630 notes that are on OpenCourseWare, 44 00:02:57,630 --> 00:02:59,070 lectures 11 and 12. 45 00:02:59,070 --> 00:03:04,770 So I'm going to use those as a guide in this lecture. 46 00:03:04,770 --> 00:03:14,730 Anyway, I thought I'd just put that up as, you 47 00:03:14,730 --> 00:03:21,290 know, time to see what various difference methods will do, 48 00:03:21,290 --> 00:03:24,410 And again, it'll be on the website. 49 00:03:24,410 --> 00:03:25,760 OK. 50 00:03:25,760 --> 00:03:30,640 Now to say another word about this one, 51 00:03:30,640 --> 00:03:36,120 this convection-diffusion equation. 52 00:03:36,120 --> 00:03:42,970 So, last time I wrote down a difference equation very much 53 00:03:42,970 --> 00:03:47,260 like that except what I wrote down last time 54 00:03:47,260 --> 00:03:51,170 was upwinded so that was just j. 55 00:03:51,170 --> 00:03:53,970 Now it's j minus 1. 56 00:03:53,970 --> 00:03:59,920 And the choice there is not clear. 57 00:03:59,920 --> 00:04:04,140 This gives us higher accuracy, of course, 58 00:04:04,140 --> 00:04:09,270 for that x difference, because it's centered at j. 59 00:04:11,880 --> 00:04:15,090 And we still have this centered, of course. 60 00:04:15,090 --> 00:04:17,740 The second difference is centered. 61 00:04:17,740 --> 00:04:20,890 So I wrote down here what the coefficients 62 00:04:20,890 --> 00:04:28,170 are of the three old values at j plus 1, at j, and at j minus 1. 63 00:04:28,170 --> 00:04:32,230 But there is one important point here. 64 00:04:32,230 --> 00:04:36,530 The price for that extra accuracy 65 00:04:36,530 --> 00:04:41,650 may possibly be a negative coefficient. 66 00:04:41,650 --> 00:04:45,310 This could be negative, or it could be positive. 67 00:04:45,310 --> 00:04:47,840 It depends which is more important: 68 00:04:47,840 --> 00:04:54,280 the small r, the convection part, or the big R. 69 00:04:54,280 --> 00:04:58,100 And let me follow up. 70 00:04:58,100 --> 00:05:00,730 So you remember the price. 71 00:05:00,730 --> 00:05:04,410 If we see a negative coefficient, 72 00:05:04,410 --> 00:05:10,610 then we expect some oscillations to appear. 73 00:05:10,610 --> 00:05:13,870 And actually, they were supposed to appear in Figure 5.12, 74 00:05:13,870 --> 00:05:15,710 but it hasn't been done yet. 75 00:05:15,710 --> 00:05:19,470 And if you would choose this one and do Figure 5.12, 76 00:05:19,470 --> 00:05:22,320 I'll be incredibly grateful. 77 00:05:22,320 --> 00:05:29,060 So Figure 5.12 was to show the evolution from a u naught 78 00:05:29,060 --> 00:05:32,220 for the convection-diffusion. 79 00:05:32,220 --> 00:05:38,670 OK, so can I just -- do you see that in comparing R with r over 80 00:05:38,670 --> 00:05:43,970 2, those are dimensionless quantities, right? 81 00:05:43,970 --> 00:05:46,760 Can I take another minute to try? 82 00:05:46,760 --> 00:05:53,140 I'm a total amateur at dimensional analysis. 83 00:05:53,140 --> 00:05:59,043 I did always think, however, that Einstein could have proved 84 00:05:59,043 --> 00:06:04,320 -- well, not maybe proved, but could have conjectured that E 85 00:06:04,320 --> 00:06:08,210 was m c squared, the most important equation 86 00:06:08,210 --> 00:06:13,030 in relativity, just by checking the dimensions. 87 00:06:13,030 --> 00:06:15,360 I don't know if that would -- I mean, right? 88 00:06:15,360 --> 00:06:19,410 That would like -- might have occurred to him to realize that 89 00:06:19,410 --> 00:06:22,940 m c squared has the dimensions, if we wrote those out, 90 00:06:22,940 --> 00:06:24,810 of energy. 91 00:06:24,810 --> 00:06:34,410 I don't know if that put an idea into the system which produced, 92 00:06:34,410 --> 00:06:37,910 of course, the fact that -- and, of course, 93 00:06:37,910 --> 00:06:39,690 there would be a dimensionless constant, 94 00:06:39,690 --> 00:06:43,000 which amazingly happens to be 1. 95 00:06:43,000 --> 00:06:47,080 OK, so dimensional analysis could, like, change the world. 96 00:06:47,080 --> 00:06:48,820 This won't quite change the world, 97 00:06:48,820 --> 00:06:54,040 but a little touch of something. 98 00:06:54,040 --> 00:06:59,750 OK, so can I just -- you remember that this Peclet 99 00:06:59,750 --> 00:07:03,370 number, and I'll write his name down again. 100 00:07:03,370 --> 00:07:07,240 P-E-C-L-E-T. The Peclet number. 101 00:07:07,240 --> 00:07:11,890 In the differential equation, the Peclet number 102 00:07:11,890 --> 00:07:20,000 was a ratio, essentially a ratio, of c to d. 103 00:07:20,000 --> 00:07:24,650 But to get the -- I think we needed -- 104 00:07:24,650 --> 00:07:27,440 was that what we needed? 105 00:07:27,440 --> 00:07:32,810 We needed a length to make it dimensionless. 106 00:07:32,810 --> 00:07:37,670 c is the coefficient for -- c over d sort of tells us roughly 107 00:07:37,670 --> 00:07:40,070 the importance of convection and diffusion, 108 00:07:40,070 --> 00:07:43,470 but there's a length scale to be included. 109 00:07:43,470 --> 00:07:52,080 Now the cell Peclet number, which I'll just call P, 110 00:07:52,080 --> 00:07:55,427 I'm just going to take to be r over R 111 00:07:55,427 --> 00:07:56,760 because those are dimensionless. 112 00:07:59,370 --> 00:08:04,410 And you see that it's going to play a part here. 113 00:08:04,410 --> 00:08:06,460 Let me see, I forgot. 114 00:08:06,460 --> 00:08:13,420 I think I want r over R. Maybe I want a 2 just because I would 115 00:08:13,420 --> 00:08:19,270 like that to -- I'd like it to come out simply there. 116 00:08:19,270 --> 00:08:23,380 OK, so I may put in a 2. 117 00:08:23,380 --> 00:08:26,100 So that r would be 2R*p. 118 00:08:26,100 --> 00:08:28,760 Where did that 2 come from, by the way? 119 00:08:28,760 --> 00:08:32,350 Oh, I guess it came from there, right? 120 00:08:32,350 --> 00:08:35,940 It came from that minus 1/2. 121 00:08:35,940 --> 00:08:38,440 So probably I want a 2 there just so 122 00:08:38,440 --> 00:08:41,760 that I have a very simple decision 123 00:08:41,760 --> 00:08:49,170 of whether this coefficient is positive or negative, yeah. 124 00:08:49,170 --> 00:08:52,580 So let me put in a 2. 125 00:08:52,580 --> 00:08:55,920 OK, so r over 2 is R*P then. 126 00:08:55,920 --> 00:09:02,100 This quantity is R times P, so this is R minus R times P, 127 00:09:02,100 --> 00:09:09,220 or it's R, which we know is OK, times 1 minus P. 128 00:09:09,220 --> 00:09:15,330 In other words, P smaller than 1, that coefficient -- 129 00:09:15,330 --> 00:09:17,370 all the three coefficients are positive. 130 00:09:17,370 --> 00:09:18,770 It's certainly stable. 131 00:09:18,770 --> 00:09:22,010 Not only stable, it can't oscillate. 132 00:09:22,010 --> 00:09:26,950 We can't have -- we're taking a combination of the three old 133 00:09:26,950 --> 00:09:31,550 values and oscillations can't show up. 134 00:09:31,550 --> 00:09:34,420 OK, so just to see, what is this? 135 00:09:34,420 --> 00:09:40,720 And you remember that r is c delta t over delta x, 136 00:09:40,720 --> 00:09:50,730 and capital R is d delta t over delta x squared. 137 00:09:50,730 --> 00:09:56,650 So just to complete this, the delta t's cancel. 138 00:09:56,650 --> 00:09:58,310 One of the delta x cancels. 139 00:09:58,310 --> 00:09:59,360 The other one comes up. 140 00:09:59,360 --> 00:10:04,850 So it's c over d, and then we have a delta x and a 2. 141 00:10:08,780 --> 00:10:11,300 So that's why it's called the cell Peclet number, 142 00:10:11,300 --> 00:10:16,680 because it's the cell size, or in my convention, 143 00:10:16,680 --> 00:10:24,040 half the cell size that gives the length scale and decides 144 00:10:24,040 --> 00:10:27,860 whether P is -- so I'm just going to write down 145 00:10:27,860 --> 00:10:31,680 the conclusion that the figure should show. 146 00:10:31,680 --> 00:10:35,400 P smaller than 1. 147 00:10:35,400 --> 00:10:39,780 That means that this coefficient is also positive. 148 00:10:39,780 --> 00:10:45,380 So that's positive coefficients, no oscillation. 149 00:10:50,820 --> 00:10:58,740 And I think you'll see from the output that P bigger than 1 150 00:10:58,740 --> 00:11:04,520 produces oscillation we don't like. 151 00:11:04,520 --> 00:11:07,990 In other words, we really don't want P larger than 1. 152 00:11:11,630 --> 00:11:15,520 And of course you could say, well, 153 00:11:15,520 --> 00:11:18,660 look at P. It's got a delta x in it. 154 00:11:18,660 --> 00:11:20,740 That's not going to be a big number. 155 00:11:23,580 --> 00:11:31,600 And the method will work fine as delta x and delta t go to zero, 156 00:11:31,600 --> 00:11:36,460 but we're going to do a finite calculation. 157 00:11:36,460 --> 00:11:39,290 And so we're seeing for the first time, 158 00:11:39,290 --> 00:11:42,730 like, because we have two terms that are not dimensionally 159 00:11:42,730 --> 00:11:52,780 identical, c and d, that in that competition between convection 160 00:11:52,780 --> 00:11:55,890 and diffusion, the length scale is crucial, 161 00:11:55,890 --> 00:12:00,590 and the mesh size is crucial. 162 00:12:00,590 --> 00:12:04,510 So I guess the answer is we can resolve it 163 00:12:04,510 --> 00:12:09,060 by taking delta x small enough, and hopefully, 164 00:12:09,060 --> 00:12:10,090 that's not a problem. 165 00:12:10,090 --> 00:12:18,380 But, of course, you know that there are many physical cases 166 00:12:18,380 --> 00:12:24,050 in which the diffusion coefficient or the viscosity 167 00:12:24,050 --> 00:12:28,230 coefficient is small, is quite small. 168 00:12:28,230 --> 00:12:31,700 And so getting that below 1 might be hard, 169 00:12:31,700 --> 00:12:34,200 because this d could be a very small number. 170 00:12:37,090 --> 00:12:43,740 OK, so maybe -- can I also mention on the -- 171 00:12:43,740 --> 00:12:46,580 as something that we haven't done here, 172 00:12:46,580 --> 00:12:57,210 was to move the second derivative, this guy, 173 00:12:57,210 --> 00:13:03,390 to time n plus 1, make it implicit. 174 00:13:03,390 --> 00:13:07,540 In the diffusion part, one often does that. 175 00:13:07,540 --> 00:13:11,320 The diffusion is the term, is the one 176 00:13:11,320 --> 00:13:19,210 that's forcing us to take this coefficient R 177 00:13:19,210 --> 00:13:23,160 below a half, delta t of order delta 178 00:13:23,160 --> 00:13:26,630 x squared, and the way to avoid that is 179 00:13:26,630 --> 00:13:31,670 to move the viscosity term, at least half of it, 180 00:13:31,670 --> 00:13:35,320 up to the new time. 181 00:13:35,320 --> 00:13:37,690 OK, so that's a third method. 182 00:13:37,690 --> 00:13:44,820 So we've done upwind last time, centered, explicit 183 00:13:44,820 --> 00:13:48,360 this time, but then that's a third one that 184 00:13:48,360 --> 00:13:50,690 has to be looked at. 185 00:13:50,690 --> 00:13:58,920 OK, so that's where we stand on linear problems. 186 00:14:03,410 --> 00:14:10,310 I'm sort of ready to jump into nonlinear ones, 187 00:14:10,310 --> 00:14:13,830 recognizing that -- well, there's always more to say 188 00:14:13,830 --> 00:14:14,920 about linear ones. 189 00:14:14,920 --> 00:14:20,900 What are some of the things that we have not properly discussed? 190 00:14:20,900 --> 00:14:24,170 Boundary conditions, above all. 191 00:14:24,170 --> 00:14:31,920 Boundary conditions, and we could do something with those, 192 00:14:31,920 --> 00:14:37,330 and the notes will say more about boundary conditions. 193 00:14:37,330 --> 00:14:39,580 So there are several types of boundary conditions. 194 00:14:39,580 --> 00:14:42,400 I guess I'm going to say something about them now, 195 00:14:42,400 --> 00:14:45,650 but this'll be a board that kind of gets covered up. 196 00:14:45,650 --> 00:14:51,050 So boundary conditions: So what are the different types? 197 00:14:51,050 --> 00:14:55,220 Well, at a real physical boundary, 198 00:14:55,220 --> 00:14:58,600 we might have u equals 0. 199 00:14:58,600 --> 00:15:05,520 Say, in heat flow, we might hold the boundary -- 200 00:15:05,520 --> 00:15:09,340 it might be the edge of a freezer or something -- 201 00:15:09,340 --> 00:15:14,030 we might hold the temperature at zero, or at some given value. 202 00:15:14,030 --> 00:15:17,300 We might prescribe u at a boundary. 203 00:15:17,300 --> 00:15:21,450 Or we might prescribe the derivative. 204 00:15:21,450 --> 00:15:24,460 And just in case we're in several dimensions, 205 00:15:24,460 --> 00:15:27,380 I'd better write it as the outgoing derivative. 206 00:15:27,380 --> 00:15:29,090 It could be prescribed, say, 0. 207 00:15:32,680 --> 00:15:35,440 So what are those two types of boundary conditions called? 208 00:15:35,440 --> 00:15:38,920 One's called absorbing, an absorbing boundary, 209 00:15:38,920 --> 00:15:41,810 and the other is an insulated boundary 210 00:15:41,810 --> 00:15:46,450 where no heat goes through. 211 00:15:46,450 --> 00:15:50,390 To maintain this, heat probably does go through. 212 00:15:50,390 --> 00:15:54,450 If our body is warm and we're holding it 213 00:15:54,450 --> 00:15:59,590 at zero on its boundary, then heat is going to travel out. 214 00:15:59,590 --> 00:16:01,660 And in fact, as time goes to infinity, 215 00:16:01,660 --> 00:16:07,910 we're going to approach cold -- a uniformly cold. 216 00:16:07,910 --> 00:16:10,420 Here, heat's not allowed to travel out, 217 00:16:10,420 --> 00:16:12,880 so what will happen as time goes on 218 00:16:12,880 --> 00:16:19,590 is it will approach a constant. 219 00:16:19,590 --> 00:16:23,800 The temperature will approach a constant because it can't leave 220 00:16:23,800 --> 00:16:25,570 and it diffuses. 221 00:16:25,570 --> 00:16:33,160 And now I wanted to mention a third type 222 00:16:33,160 --> 00:16:38,420 of boundary, which is a purely computational boundary. 223 00:16:38,420 --> 00:16:42,730 We talked last time about Maxwell's equations 224 00:16:42,730 --> 00:16:45,570 in free space, and the big application 225 00:16:45,570 --> 00:16:49,120 of Maxwell's equations, or one of the big ones historically, 226 00:16:49,120 --> 00:16:53,860 has been to, like, the exterior of an airplane, 227 00:16:53,860 --> 00:16:58,290 to radar, or other electromagnetic signals. 228 00:16:58,290 --> 00:17:01,690 So what do we do if we have an exterior problem? 229 00:17:01,690 --> 00:17:04,940 See, here, I was always thinking about an interior problem. 230 00:17:04,940 --> 00:17:09,300 You know, we're inside 0 to L or something. 231 00:17:09,300 --> 00:17:15,320 But now, what do we do if a region is the outside, 232 00:17:15,320 --> 00:17:17,330 it's free space all the way? 233 00:17:17,330 --> 00:17:22,980 Well, we have to create a computational box of some sort. 234 00:17:22,980 --> 00:17:28,510 So we create a box in which we compute, 235 00:17:28,510 --> 00:17:32,960 but the boundaries of this box, the x-boundaries of it, 236 00:17:32,960 --> 00:17:35,910 are purely artificial. 237 00:17:35,910 --> 00:17:39,960 They're just because we can't compute to infinity. 238 00:17:39,960 --> 00:17:46,630 So waves travel in this box, and they travel to the -- 239 00:17:46,630 --> 00:17:49,250 so you see what I mean? 240 00:17:49,250 --> 00:17:52,450 We're solving it in all this space, except that's too much, 241 00:17:52,450 --> 00:17:57,800 so that's meant to be a very large box, not a small one, 242 00:17:57,800 --> 00:18:00,690 as large as we can afford. 243 00:18:00,690 --> 00:18:06,780 But waves are still going to hit the boundary of that box. 244 00:18:06,780 --> 00:18:09,480 OK, what do we choose as boundary condition there? 245 00:18:12,260 --> 00:18:16,650 We don't want it to reflect back because it's not 246 00:18:16,650 --> 00:18:18,160 a real boundary. 247 00:18:18,160 --> 00:18:24,650 So we have to create -- I'll put ABC for absorbing boundary 248 00:18:24,650 --> 00:18:25,710 condition. 249 00:18:25,710 --> 00:18:28,090 Absorbing. 250 00:18:28,090 --> 00:18:31,240 That A is for an absorbing boundary condition. 251 00:18:36,700 --> 00:18:39,670 And the creation of these boundary conditions 252 00:18:39,670 --> 00:18:44,310 is an important topic within finite differences. 253 00:18:44,310 --> 00:18:50,350 For Maxwell's equations, a good idea 254 00:18:50,350 --> 00:18:54,370 was produced and keeps being developed, 255 00:18:54,370 --> 00:18:57,050 called perfectly matched layer. 256 00:18:57,050 --> 00:18:59,720 The idea in that perfectly matched layer -- 257 00:18:59,720 --> 00:19:06,100 let me just write PML -- and I haven't seen this so much 258 00:19:06,100 --> 00:19:10,150 in other application areas. 259 00:19:10,150 --> 00:19:12,630 The idea of this perfectly matched layer 260 00:19:12,630 --> 00:19:19,020 is to create an artificial dielectric, I guess, 261 00:19:19,020 --> 00:19:23,520 a dielectric medium it would be, in the electromagnetic case. 262 00:19:23,520 --> 00:19:31,840 An artificial thin layer, which has just the right properties 263 00:19:31,840 --> 00:19:36,240 to stop the wave and swallow it. 264 00:19:36,240 --> 00:19:44,140 OK, I'll put on the website references to that. 265 00:19:44,140 --> 00:19:52,530 It's a bright idea, but not totally simple, 266 00:19:52,530 --> 00:19:56,710 and when we change difference methods, then that perfectly 267 00:19:56,710 --> 00:19:58,250 matched layer has to go with it. 268 00:19:58,250 --> 00:20:02,640 OK, that's the end of my thoughts 269 00:20:02,640 --> 00:20:07,060 about linear problems for now. 270 00:20:07,060 --> 00:20:14,020 We're coming back, of course, in a few weeks to solving 271 00:20:14,020 --> 00:20:19,270 large systems, large linear systems, the kind that we would 272 00:20:19,270 --> 00:20:22,820 meet if we use an implicit method, 273 00:20:22,820 --> 00:20:24,310 or the kind that we meet if we're 274 00:20:24,310 --> 00:20:26,050 solving a steady-state problem. 275 00:20:26,050 --> 00:20:33,660 But for now, let me go to that model equation. 276 00:20:36,640 --> 00:20:38,780 And you see, right away, the difference 277 00:20:38,780 --> 00:20:45,060 between that equation and the advection equation that 278 00:20:45,060 --> 00:20:47,700 had a constant velocity. 279 00:20:47,700 --> 00:20:56,360 Here we have of a velocity that -- so c is minus u, I guess. 280 00:20:56,360 --> 00:21:01,330 c is minus -- think of c -- this u as being comparable to minus 281 00:21:01,330 --> 00:21:03,340 c. 282 00:21:03,340 --> 00:21:06,340 So, in other words, if u is positive, 283 00:21:06,340 --> 00:21:10,060 I'm now going to have waves going to the right. 284 00:21:12,680 --> 00:21:16,600 So let me just put this down. u greater than 0 285 00:21:16,600 --> 00:21:24,830 will mean waves going to increasing x, to the right. 286 00:21:24,830 --> 00:21:32,280 And the idea will be -- so we have to discuss that problem. 287 00:21:32,280 --> 00:21:37,320 And as always, we would like to solve it analytically. 288 00:21:37,320 --> 00:21:42,500 Find whatever formulas we can, understand what's happening 289 00:21:42,500 --> 00:21:47,080 in terms of these characteristic lines, because -- 290 00:21:47,080 --> 00:21:52,520 when we have one equation and one space variable, 291 00:21:52,520 --> 00:21:59,000 we really can catch the essence of the solution 292 00:21:59,000 --> 00:22:03,400 by understanding the characteristic lines. 293 00:22:03,400 --> 00:22:07,760 And then how do we do it numerically? 294 00:22:07,760 --> 00:22:14,850 OK, before class, I wrote just a few equivalent forms of it. 295 00:22:14,850 --> 00:22:17,470 This is an equivalent form, and this is, so to speak, 296 00:22:17,470 --> 00:22:18,280 the right form. 297 00:22:22,420 --> 00:22:27,780 This is the form where we see a time derivative of u, 298 00:22:27,780 --> 00:22:33,590 as always, and we see a space derivative of u squared over 2, 299 00:22:33,590 --> 00:22:35,800 which, of course, produces that. 300 00:22:35,800 --> 00:22:41,780 But somehow this is the quantity -- it's called the flux -- 301 00:22:41,780 --> 00:22:47,180 that's physically meaningful, f of u. 302 00:22:47,180 --> 00:22:51,400 So I've chosen a simple flux function, u squared over 2. 303 00:22:57,490 --> 00:23:00,270 So that's the differential equation. 304 00:23:00,270 --> 00:23:08,580 But you'll see that the differential equation can 305 00:23:08,580 --> 00:23:13,170 produce two solutions at the same point, 306 00:23:13,170 --> 00:23:16,670 and we have to choose. 307 00:23:16,670 --> 00:23:18,810 In other words, the differential -- 308 00:23:18,810 --> 00:23:21,220 the solution can become discontinuous. 309 00:23:21,220 --> 00:23:29,350 A perfectly smooth starting function, after a while -- 310 00:23:29,350 --> 00:23:33,760 this is the essence of the problem now. 311 00:23:33,760 --> 00:23:37,670 I might have a smooth starting function, at least continuous. 312 00:23:40,460 --> 00:23:46,050 The solution is constant along characteristic lines. 313 00:23:46,050 --> 00:23:49,310 But what happens if two characteristic lines 314 00:23:49,310 --> 00:23:50,830 run into each other? 315 00:23:50,830 --> 00:23:52,750 I've drawn that possibility here. 316 00:23:55,510 --> 00:24:02,020 And it'll take a little time to see exactly what's happening, 317 00:24:02,020 --> 00:24:04,570 but the characteristics we can see here. 318 00:24:04,570 --> 00:24:07,660 So the starting values were 1 up to that point. 319 00:24:10,770 --> 00:24:16,810 So when the starting value is 1 and u 320 00:24:16,810 --> 00:24:22,380 is constant along characteristic lines, then u is 1, 321 00:24:22,380 --> 00:24:27,540 and those lines, it's just like having c equal to minus 1, 322 00:24:27,540 --> 00:24:30,720 I guess, in the one-way wave equation. 323 00:24:30,720 --> 00:24:32,920 We have a one-way wave going to the right 324 00:24:32,920 --> 00:24:36,160 now along characteristic lines. 325 00:24:36,160 --> 00:24:47,990 So u is 1 in this part of the xt-plane, right? 326 00:24:47,990 --> 00:24:49,960 We have to come back to the formulas 327 00:24:49,960 --> 00:24:53,990 to confirm what I just said. 328 00:24:53,990 --> 00:25:02,670 And then, over here on the far right, the starting value is 0, 329 00:25:02,670 --> 00:25:09,680 and the characteristics in that case, 330 00:25:09,680 --> 00:25:15,240 the c associated with that is 0, so it's a standing -- 331 00:25:15,240 --> 00:25:16,300 it's not moving. 332 00:25:16,300 --> 00:25:20,084 The characteristics are just straight lines 333 00:25:20,084 --> 00:25:21,000 in the time direction. 334 00:25:21,000 --> 00:25:22,180 Nothing's happening, right? 335 00:25:22,180 --> 00:25:25,650 When you think of a characteristic that's 336 00:25:25,650 --> 00:25:30,410 going straight up, it's just carrying the value of u 337 00:25:30,410 --> 00:25:34,490 without moving left or right, so u stays 0. 338 00:25:34,490 --> 00:25:38,280 So u will be 1 here, u will be 0 here, 339 00:25:38,280 --> 00:25:41,240 and now something's happening in here, 340 00:25:41,240 --> 00:25:45,740 and we have to figure out what it is. 341 00:25:45,740 --> 00:25:50,740 But maybe I can anticipate the main point before any algebra. 342 00:25:50,740 --> 00:25:55,350 The characteristic lines in this region 343 00:25:55,350 --> 00:26:01,450 have a slope somewhere between the slope 1 and the slope 0. 344 00:26:01,450 --> 00:26:05,740 You see, we're starting -- let me graph u_0 of x. 345 00:26:05,740 --> 00:26:19,500 So let me graph. u_0 of x is -- the key points are 0 and 1, 346 00:26:19,500 --> 00:26:20,930 0 and 1. 347 00:26:20,930 --> 00:26:33,500 So u_0 of x, I start with a constant initial function, 348 00:26:33,500 --> 00:26:37,750 a constant profile to the left of 0, and a constant profile 349 00:26:37,750 --> 00:26:43,100 to the right of 1, and linear between. 350 00:26:43,100 --> 00:26:46,990 OK, so it's that starting values that I wrote here: 351 00:26:46,990 --> 00:26:52,040 1, 1 minus x, and 0 are these three pieces. 352 00:26:52,040 --> 00:26:54,790 And then here I'm in the xt-plane 353 00:26:54,790 --> 00:26:59,390 where here I'm in the xu-plane. 354 00:26:59,390 --> 00:27:04,910 So I drew characteristics, and we still have to track -- 355 00:27:04,910 --> 00:27:07,560 we have to say more what characteristics are 356 00:27:07,560 --> 00:27:11,210 and understand that these have slopes in between, 357 00:27:11,210 --> 00:27:15,270 but you're not surprised because the values of u are in between 358 00:27:15,270 --> 00:27:16,630 1 and 0. 359 00:27:16,630 --> 00:27:22,160 So they go that way and they meet here. 360 00:27:22,160 --> 00:27:24,670 So up to time t equal 1, characteristics 361 00:27:24,670 --> 00:27:26,250 tell everything. 362 00:27:26,250 --> 00:27:29,370 Up to t equal 1, I know that u will 363 00:27:29,370 --> 00:27:30,946 be 1 along these characteristics, 364 00:27:30,946 --> 00:27:33,730 and when I find these characteristics, 365 00:27:33,730 --> 00:27:37,030 u is whatever it starts at. 366 00:27:37,030 --> 00:27:39,870 And u is 0 up all of these characteristics, 367 00:27:39,870 --> 00:27:44,850 so up to that time, the algebra -- 368 00:27:44,850 --> 00:27:48,880 the differential equation is going to be totally OK. 369 00:27:51,580 --> 00:27:54,830 But after that time, the differential equation 370 00:27:54,830 --> 00:27:58,380 is in difficulty. 371 00:27:58,380 --> 00:28:05,130 Because at that moment t -- so what's happening as t increases 372 00:28:05,130 --> 00:28:07,280 from 0 to 1? 373 00:28:07,280 --> 00:28:10,430 This is the picture at t equals 0. 374 00:28:10,430 --> 00:28:14,900 The picture at t equal to 1 -- t equal to half, let's say. 375 00:28:14,900 --> 00:28:17,030 What would be the picture at t equal to half? 376 00:28:17,030 --> 00:28:21,380 Well, at t equal to half, this state 1, 377 00:28:21,380 --> 00:28:28,940 has penetrated this far, halfway. 378 00:28:28,940 --> 00:28:34,190 Then, in here is this converging characteristic that 379 00:28:34,190 --> 00:28:38,150 come from a linear profile. 380 00:28:38,150 --> 00:28:40,870 And over here, it's all zeros. 381 00:28:40,870 --> 00:28:44,180 So that zero is still there, but -- you see what's happening? 382 00:28:44,180 --> 00:28:50,210 This guy is -- the profile is steepening. 383 00:28:50,210 --> 00:28:51,620 This is at t equal to half. 384 00:28:55,341 --> 00:28:55,840 Steeper. 385 00:29:01,050 --> 00:29:02,950 We still have the algebra to do, but I 386 00:29:02,950 --> 00:29:08,360 think the first thing is to see roughly what's happening. 387 00:29:08,360 --> 00:29:14,030 Then at t equal to 1, this is totally steep. 388 00:29:14,030 --> 00:29:19,470 So this is the situation at t equal to 1. 389 00:29:19,470 --> 00:29:24,650 And I could've given you that as the initial condition, 390 00:29:24,650 --> 00:29:30,360 but I and the 16.920 notes, thought, OK, 391 00:29:30,360 --> 00:29:33,090 let's have a little peace for awhile, 392 00:29:33,090 --> 00:29:37,360 see what those characteristics are doing when 393 00:29:37,360 --> 00:29:43,010 they're converging, but they're not crossing. 394 00:29:43,010 --> 00:29:48,460 The trouble is, after t equal to 1, they are crossing. 395 00:29:48,460 --> 00:29:57,950 And now, what's the solution in this crosshatched region? 396 00:29:57,950 --> 00:30:01,820 Do I take the value of 0 that's coming up 397 00:30:01,820 --> 00:30:03,700 along these characteristics? 398 00:30:03,700 --> 00:30:05,730 Do I take the value 1 that's coming 399 00:30:05,730 --> 00:30:07,520 along these characteristics? 400 00:30:07,520 --> 00:30:10,270 What's happening in here? 401 00:30:10,270 --> 00:30:13,360 So there's a crosshatched region there, 402 00:30:13,360 --> 00:30:21,850 which is a big question mark because my rule 403 00:30:21,850 --> 00:30:24,640 of following characteristics has led me to two answers. 404 00:30:28,720 --> 00:30:39,030 And we want a physically relevant answer. 405 00:30:39,030 --> 00:30:41,660 We're sort of expecting, and we'll 406 00:30:41,660 --> 00:30:51,840 see, that this step function is a step function. 407 00:30:51,840 --> 00:30:55,190 I mean, the answer will be a step function. 408 00:30:55,190 --> 00:31:03,220 It will be 1 and then drop to 0 along some path 409 00:31:03,220 --> 00:31:07,150 in the xt-plane, which is not a characteristic. 410 00:31:07,150 --> 00:31:13,600 The characteristics are going at 45 degrees and at 90 degrees. 411 00:31:13,600 --> 00:31:18,520 And actually, so this is the region I'm interested in. 412 00:31:18,520 --> 00:31:20,750 So can I blow up that region? 413 00:31:20,750 --> 00:31:26,010 So if I blow up that region, that's where the shocks -- 414 00:31:26,010 --> 00:31:30,620 the shocks are meeting everywhere in this region. 415 00:31:30,620 --> 00:31:33,010 This was the point -- what was that point? 416 00:31:33,010 --> 00:31:38,650 I guess that was the point x equals 0 and t equal to 1, 417 00:31:38,650 --> 00:31:39,690 there. 418 00:31:39,690 --> 00:31:44,450 OK, and the question is -- and I know that I have u equals 0 419 00:31:44,450 --> 00:31:48,210 over here, and I know -- oops, u equal 1 over here, 420 00:31:48,210 --> 00:31:50,070 and I know that I have u equal 0 here, 421 00:31:50,070 --> 00:31:54,600 but the question is where, and -- well, 422 00:31:54,600 --> 00:31:55,650 what happens in between? 423 00:31:59,630 --> 00:32:07,920 You can think of at least two possible scenarios, 424 00:32:07,920 --> 00:32:15,450 and one of those is that there's a shock line that goes, 425 00:32:15,450 --> 00:32:18,610 let's say, somewhere in the middle. 426 00:32:18,610 --> 00:32:23,390 And so u stays 1 all the way to that line 427 00:32:23,390 --> 00:32:25,290 and drops to 0 after it. 428 00:32:25,290 --> 00:32:35,240 In other words, this wall of water travels to the right. 429 00:32:35,240 --> 00:32:41,120 The shock, the discontinuity, moves along, 430 00:32:41,120 --> 00:32:44,920 not necessarily at the speed that this one -- 431 00:32:44,920 --> 00:32:50,180 this would be moving along the speed of 1, with this pushing. 432 00:32:50,180 --> 00:32:53,410 Not stationary either, which is what this right side would 433 00:32:53,410 --> 00:32:55,440 like, but somewhere between. 434 00:32:55,440 --> 00:32:57,650 So that's a shock line. 435 00:32:57,650 --> 00:32:59,880 I'll just, for short, call it a shock. 436 00:33:05,140 --> 00:33:06,870 Then there's another possibility. 437 00:33:06,870 --> 00:33:09,680 And that will happen in this case, actually. 438 00:33:09,680 --> 00:33:12,540 That will happen in this case. 439 00:33:12,540 --> 00:33:14,980 But there is another possibility that 440 00:33:14,980 --> 00:33:18,860 will happen in other cases, and that would 441 00:33:18,860 --> 00:33:26,170 be a fan of characteristics. 442 00:33:26,170 --> 00:33:30,310 Let me see that when it comes, but I'm just mentioning it 443 00:33:30,310 --> 00:33:33,540 to say that this business of the shock line, 444 00:33:33,540 --> 00:33:35,990 which is what we'll pursue now, which -- 445 00:33:35,990 --> 00:33:39,540 because it's what happens with this problem -- 446 00:33:39,540 --> 00:33:47,240 is not the only -- is not always the way to match, 447 00:33:47,240 --> 00:33:55,350 the way to deal with a problem of lack of information coming 448 00:33:55,350 --> 00:33:56,610 from the characteristics. 449 00:33:56,610 --> 00:34:05,870 OK, so I guess our question is, what is that shock line? 450 00:34:05,870 --> 00:34:06,730 What's the path? 451 00:34:06,730 --> 00:34:13,500 What's the slope, and what controls it? 452 00:34:13,500 --> 00:34:16,720 And the answer will be that what controls -- 453 00:34:16,720 --> 00:34:21,140 we run into problems with the differential equations. 454 00:34:21,140 --> 00:34:28,430 And the way to see what continues to happen 455 00:34:28,430 --> 00:34:31,900 is to go to the integrated form, the integral form 456 00:34:31,900 --> 00:34:37,470 of the equation, where, of course, by taking an integral, 457 00:34:37,470 --> 00:34:41,220 I make everything smoother, and also I 458 00:34:41,220 --> 00:34:45,770 preserve the physical law, the conservation law. 459 00:34:45,770 --> 00:34:48,360 So this is really the conservation law. 460 00:34:52,250 --> 00:34:53,330 What is this? 461 00:34:53,330 --> 00:34:55,620 Well, let's first of all see where it comes from. 462 00:34:55,620 --> 00:35:02,970 It just comes from integrating this equation with respect 463 00:35:02,970 --> 00:35:04,210 to x. 464 00:35:04,210 --> 00:35:07,980 So I have a time derivative that I keep out here, 465 00:35:07,980 --> 00:35:10,780 but my space derivative is going to disappear because I'm 466 00:35:10,780 --> 00:35:12,240 integrating with respect to x. 467 00:35:12,240 --> 00:35:18,075 So I can call d by dt or partial d by dt the integral of u with 468 00:35:18,075 --> 00:35:21,310 respect to x, and I'm integrating from somewhere -- 469 00:35:21,310 --> 00:35:24,040 from some point to some other point. 470 00:35:24,040 --> 00:35:28,440 Instead of a and b, I'll call them -- I really should -- 471 00:35:28,440 --> 00:35:33,740 I'm sorry, I should say x on the left and x on the right. 472 00:35:33,740 --> 00:35:40,800 Some point x at the left and some point x at the right. 473 00:35:40,800 --> 00:35:43,370 And now I've integrated this one. 474 00:35:43,370 --> 00:35:50,870 I've called -- this is f of u so I'm now thinking of any flux 475 00:35:50,870 --> 00:35:52,520 function. 476 00:35:52,520 --> 00:35:54,870 And the integral, of course, gives me 477 00:35:54,870 --> 00:35:59,130 f of u at the upper end minus f of u at the lower end. 478 00:35:59,130 --> 00:36:01,880 OK, so that's the conservation law. 479 00:36:01,880 --> 00:36:03,900 Let me write that word down. 480 00:36:03,900 --> 00:36:06,310 This is the conservation law. 481 00:36:06,310 --> 00:36:07,300 What is it saying? 482 00:36:12,970 --> 00:36:20,630 It's saying that the change in the quantity in the integral is 483 00:36:20,630 --> 00:36:24,820 given by -- if there's any change, any time derivative, 484 00:36:24,820 --> 00:36:28,710 the rate of change in the interval -- 485 00:36:28,710 --> 00:36:33,950 it's the conservation of this integral, 486 00:36:33,950 --> 00:36:42,480 meaning that any change in that integral happens because there 487 00:36:42,480 --> 00:36:46,990 is flow going out the right-hand side of the interval 488 00:36:46,990 --> 00:36:50,220 or there's flow coming in the left side of the interval. 489 00:36:50,220 --> 00:36:53,990 So that's why we have a minus sign between them. 490 00:36:53,990 --> 00:37:02,550 So that's a statement of conservation, and -- oh, 491 00:37:02,550 --> 00:37:07,350 there's one more example that I'll mention now. 492 00:37:07,350 --> 00:37:10,840 Let me give this particular equation a name, 493 00:37:10,840 --> 00:37:12,040 just because you see it. 494 00:37:12,040 --> 00:37:16,130 You often see it written -- called Burger's equation. 495 00:37:22,550 --> 00:37:32,650 But actually, the full name is the inviscid, no viscosity, 496 00:37:32,650 --> 00:37:34,550 Burgers' equation because it's a 0. 497 00:37:37,370 --> 00:37:41,280 So just one word about Burger's equation. 498 00:37:41,280 --> 00:37:47,090 So Burger allowed a u_xx term here. 499 00:37:47,090 --> 00:37:52,150 So he had one of our -- nonlinear convection 500 00:37:52,150 --> 00:37:59,710 and a diffusion term, and his convection was this simple 501 00:37:59,710 --> 00:38:02,590 expression u*u_x. 502 00:38:02,590 --> 00:38:05,450 It's the kind of thing you see in the Navier-Stokes equations, 503 00:38:05,450 --> 00:38:06,000 right? 504 00:38:06,000 --> 00:38:08,430 The nonlinearity in Navier-Stokes 505 00:38:08,430 --> 00:38:11,760 is sort of in this term, of this general sort, 506 00:38:11,760 --> 00:38:15,330 but this is a major simplification. 507 00:38:15,330 --> 00:38:19,545 Such a simplification that Burger could solve, 508 00:38:19,545 --> 00:38:21,170 even though the equation was nonlinear. 509 00:38:24,480 --> 00:38:26,140 Well, two people, at the same time, 510 00:38:26,140 --> 00:38:30,240 discovered the change of variable, 511 00:38:30,240 --> 00:38:32,730 a simple little trick. 512 00:38:32,730 --> 00:38:34,420 Hopf and Cole. 513 00:38:34,420 --> 00:38:38,430 I'll put their names down because they 514 00:38:38,430 --> 00:38:41,520 had nice, short last names. 515 00:38:41,520 --> 00:38:46,370 They both discovered -- in the case of a diffusion term, 516 00:38:46,370 --> 00:38:49,660 they both discovered a little -- a change of variables that made 517 00:38:49,660 --> 00:38:51,130 the equation linear. 518 00:38:51,130 --> 00:38:52,870 Then they could solve the equation. 519 00:38:52,870 --> 00:38:58,440 That's in the textbook in the section on conservation laws. 520 00:38:58,440 --> 00:39:02,660 And then the important thing that Burger could do 521 00:39:02,660 --> 00:39:07,380 was let the diffusion coefficient go to zero. 522 00:39:07,380 --> 00:39:11,050 So then in the limit, he got the inviscid Burger's equation, 523 00:39:11,050 --> 00:39:12,950 the one we're looking at. 524 00:39:12,950 --> 00:39:14,950 And he could look at the solution 525 00:39:14,950 --> 00:39:16,480 and let the limit go to zero. 526 00:39:16,480 --> 00:39:24,580 And that's another way to figure out what's the right solution. 527 00:39:24,580 --> 00:39:28,900 That's the viscosity method, a fundamental method, 528 00:39:28,900 --> 00:39:34,340 for finding out when we run into a problem, shocks 529 00:39:34,340 --> 00:39:35,350 run into each other. 530 00:39:35,350 --> 00:39:39,940 We don't know what to choose. 531 00:39:39,940 --> 00:39:44,830 One way to find out, in this 1D problem 532 00:39:44,830 --> 00:39:50,300 and also in much tougher nonlinear multidimensional 533 00:39:50,300 --> 00:39:54,990 problems, is put a little viscosity in, 534 00:39:54,990 --> 00:39:57,230 and let it be small. 535 00:39:57,230 --> 00:39:59,650 Let it even get smaller. 536 00:39:59,650 --> 00:40:07,030 Then -- with a little viscosity, officially this term -- 537 00:40:07,030 --> 00:40:13,150 in some sense, this term, because it's a second 538 00:40:13,150 --> 00:40:17,440 derivative, is officially, mathematically speaking, 539 00:40:17,440 --> 00:40:25,150 sort of dominates this one as far as -- I mean, 540 00:40:25,150 --> 00:40:27,320 this will produce a smooth solution. 541 00:40:30,420 --> 00:40:33,610 This term, as we know doesn't smooth the solution. 542 00:40:33,610 --> 00:40:37,270 It just advects it, just carries it along. 543 00:40:37,270 --> 00:40:39,780 But this term will smooth it. 544 00:40:39,780 --> 00:40:44,880 So this is the dominant term, the second derivative. 545 00:40:44,880 --> 00:40:45,790 No surprise. 546 00:40:50,060 --> 00:40:53,880 So when we introduce that, that smooths everything. 547 00:40:53,880 --> 00:40:59,380 We have no problem to say there is a solution, at least 548 00:40:59,380 --> 00:41:05,890 for simpler models, and then we can 549 00:41:05,890 --> 00:41:09,740 see what happens when that term approaches zero. 550 00:41:09,740 --> 00:41:11,340 OK, so that's another method. 551 00:41:11,340 --> 00:41:13,540 And, of course, that's kind of what we're 552 00:41:13,540 --> 00:41:15,710 doing with finite differences. 553 00:41:15,710 --> 00:41:20,120 Finite differences, if we put in a little viscosity, 554 00:41:20,120 --> 00:41:23,500 a little second-space difference, 555 00:41:23,500 --> 00:41:26,730 and take delta t small enough for stability, 556 00:41:26,730 --> 00:41:29,140 then we can solve it. 557 00:41:29,140 --> 00:41:35,970 And if we take it very small, we should get something close 558 00:41:35,970 --> 00:41:37,660 to the inviscid limit. 559 00:41:37,660 --> 00:41:42,070 OK, I was just giving Burger's name 560 00:41:42,070 --> 00:41:44,970 to that particular example. 561 00:41:44,970 --> 00:41:49,210 So this isn't the only example, but it's the simplest example. 562 00:41:49,210 --> 00:41:55,010 OK, another example of conservation laws 563 00:41:55,010 --> 00:42:00,640 that everybody likes, I think, is the traffic flow, 564 00:42:00,640 --> 00:42:03,000 traffic flow in one direction, say, on the Mass. 565 00:42:03,000 --> 00:42:04,250 Pike. 566 00:42:04,250 --> 00:42:07,470 So what's the equation of traffic flow? 567 00:42:07,470 --> 00:42:11,250 Let me put that on a board here, and then I 568 00:42:11,250 --> 00:42:14,300 have to come back to this. 569 00:42:14,300 --> 00:42:17,370 Well, I'll put it here. 570 00:42:17,370 --> 00:42:21,060 So Example 2 -- well, I don't know if you can call that 571 00:42:21,060 --> 00:42:21,810 Example 1. 572 00:42:21,810 --> 00:42:24,150 We didn't give it a physical meaning. 573 00:42:24,150 --> 00:42:35,080 Example 2 will be traffic flow in 1D, and what's the unknown? 574 00:42:38,960 --> 00:42:43,790 Well, it's the thing I've been calling u, 575 00:42:43,790 --> 00:42:48,200 but for traffic flow, the natural unknown 576 00:42:48,200 --> 00:42:51,430 is the density of the traffic, the density of the cars. 577 00:42:58,300 --> 00:43:02,100 And everybody uses rho for density. 578 00:43:02,100 --> 00:43:04,510 So can I use rho rather than u? 579 00:43:07,850 --> 00:43:16,580 And then, in the equation will be a car -- 580 00:43:16,580 --> 00:43:23,470 a velocity that I'll call v. That's not a new unknown. 581 00:43:26,600 --> 00:43:31,860 In fact, the velocity experimentally depends -- oh, 582 00:43:31,860 --> 00:43:34,230 there's a relation. 583 00:43:34,230 --> 00:43:39,610 The density and velocity are, by experiments, 584 00:43:39,610 --> 00:43:41,430 if you run an experiment on the Mass. 585 00:43:41,430 --> 00:43:45,500 Pike in heavy traffic -- that could be a homework problem, 586 00:43:45,500 --> 00:43:52,390 too -- to find the relation between v and rho. 587 00:43:52,390 --> 00:43:55,660 So I'm going to keep this rho as the unknown 588 00:43:55,660 --> 00:44:02,970 and imagine that we have some experimental velocity v of rho. 589 00:44:02,970 --> 00:44:05,390 And what would we expect? 590 00:44:05,390 --> 00:44:12,890 I guess we expect that as rho increases -- 591 00:44:12,890 --> 00:44:16,600 as the density increases, what will happen? 592 00:44:16,600 --> 00:44:20,680 As the density increases, the traffic is going to slow down, 593 00:44:20,680 --> 00:44:26,830 ultimately come to a stop, where if the density is very small, 594 00:44:26,830 --> 00:44:33,370 so the density 0 -- I mean, we just have a single driver, 595 00:44:33,370 --> 00:44:40,020 the police are not out, so he's going v_max or 2*v_max. 596 00:44:43,180 --> 00:44:51,950 But as the road gets crowded, the velocity 597 00:44:51,950 --> 00:44:53,830 drops and maybe drops linearly. 598 00:44:57,650 --> 00:45:06,990 That would be a possible and not that terrible 599 00:45:06,990 --> 00:45:09,440 relation between v and rho. 600 00:45:09,440 --> 00:45:12,421 So what would be the flux function in this traffic flow 601 00:45:12,421 --> 00:45:12,920 problem? 602 00:45:15,500 --> 00:45:20,200 The conservation law is like conservation of cars, right? 603 00:45:20,200 --> 00:45:30,730 If we look at a unit, a piece of the turnpike, then 604 00:45:30,730 --> 00:45:33,590 the conservation law says that the total number 605 00:45:33,590 --> 00:45:43,060 of cars inside, which is this, changes by cars leaving 606 00:45:43,060 --> 00:45:47,280 and by cars entering. 607 00:45:47,280 --> 00:45:50,480 And so the flux function I think will be -- 608 00:45:50,480 --> 00:45:56,450 so we have to know like sort of how many cars are leaving, 609 00:45:56,450 --> 00:46:01,970 and I think the flux function would be probably v times rho. 610 00:46:01,970 --> 00:46:09,200 I think the flux function of -- rho, now, is my unknown -- 611 00:46:09,200 --> 00:46:13,620 I think would be v of rho times rho, 612 00:46:13,620 --> 00:46:15,370 the velocity times the density. 613 00:46:15,370 --> 00:46:18,400 That tells me how many cars are going out. 614 00:46:18,400 --> 00:46:25,080 OK, so I guess what I want to say is that we will meet -- 615 00:46:25,080 --> 00:46:28,200 that we meet, in this traffic flow example, 616 00:46:28,200 --> 00:46:35,040 exactly these possibilities that -- 617 00:46:35,040 --> 00:46:38,020 we meet them mathematically just the way we meet them in actual 618 00:46:38,020 --> 00:46:45,190 driving -- of a shock. 619 00:46:45,190 --> 00:46:50,020 So this shock corresponds to -- so what that's? 620 00:46:50,020 --> 00:46:56,940 Like stop-go -- this corresponds to, I suppose, 621 00:46:56,940 --> 00:46:58,200 like meeting a red light? 622 00:47:01,350 --> 00:47:06,470 Coming up to a red light, the traffic you're behind normally 623 00:47:06,470 --> 00:47:07,420 stops. 624 00:47:07,420 --> 00:47:12,290 It has to slow way down and stop. 625 00:47:12,290 --> 00:47:16,000 Then the light turns green. 626 00:47:16,000 --> 00:47:18,690 The first car takes off. 627 00:47:18,690 --> 00:47:21,600 The cars begin to spread out. 628 00:47:21,600 --> 00:47:30,410 That's the solution that I don't see 629 00:47:30,410 --> 00:47:35,330 with this starting function, a sort of fan of cars. 630 00:47:35,330 --> 00:47:40,240 So maybe I could write those words down: shock or a fan. 631 00:47:44,620 --> 00:47:49,090 The shock is what happens when characteristics come together, 632 00:47:49,090 --> 00:47:51,110 cars come together. 633 00:47:51,110 --> 00:47:56,840 The fan is when the characteristics fan out. 634 00:47:56,840 --> 00:48:02,140 Actually, I could illustrate a fan here right away. 635 00:48:02,140 --> 00:48:11,650 Suppose I reverse 0 and 1 just to see what -- OK, 636 00:48:11,650 --> 00:48:17,460 so here's my -- this is my x again. 637 00:48:17,460 --> 00:48:26,730 So here I'm going to have u_0 to be 0 now, u_0 to be -- well, 638 00:48:26,730 --> 00:48:30,520 I can even -- it's called a Riemann problem when there are 639 00:48:30,520 --> 00:48:32,620 just two values here. 640 00:48:32,620 --> 00:48:34,230 u_0 is 1. 641 00:48:34,230 --> 00:48:41,660 So the so-called Riemann problem is the cleanest example 642 00:48:41,660 --> 00:48:45,760 of the whole conservation law theory 643 00:48:45,760 --> 00:48:50,340 when there are just two starting values, 0 and 1, or 1 and 0. 644 00:48:50,340 --> 00:48:52,170 And it makes a big difference. 645 00:48:52,170 --> 00:48:57,740 1 and 0, we saw when we reached time 1, we had a 1 and a 0, 646 00:48:57,740 --> 00:49:00,100 and after that, a shock formed. 647 00:49:00,100 --> 00:49:02,870 Here 0 and 1, the characteristics 648 00:49:02,870 --> 00:49:05,840 here are going as always. 649 00:49:05,840 --> 00:49:07,230 So u is 1 here. 650 00:49:07,230 --> 00:49:12,670 The characteristics here, this is staying at 0. 651 00:49:12,670 --> 00:49:14,920 There's no reason to change. 652 00:49:14,920 --> 00:49:22,920 But here we have now a region, just like the other one, 653 00:49:22,920 --> 00:49:24,870 but with a major difference. 654 00:49:24,870 --> 00:49:27,580 That now, it's 0 on the left and 1 655 00:49:27,580 --> 00:49:30,500 on the right, where before, it was 1 on the left 656 00:49:30,500 --> 00:49:32,220 and 0 on the right. 657 00:49:32,220 --> 00:49:35,300 And the question is, what happens in here? 658 00:49:35,300 --> 00:49:41,180 Well, you might think a shock, a step up 659 00:49:41,180 --> 00:49:43,780 from 0 to 1 at some point. 660 00:49:43,780 --> 00:49:51,540 And I would have to agree that you could do that in such a way 661 00:49:51,540 --> 00:49:55,160 that you maintained the conservation law. 662 00:49:58,540 --> 00:50:01,860 But that's not what happens. 663 00:50:01,860 --> 00:50:04,470 This is going to be a fan in here. 664 00:50:08,100 --> 00:50:10,940 We're going to have -- what we'll have is characteristics 665 00:50:10,940 --> 00:50:11,790 in here. 666 00:50:11,790 --> 00:50:14,300 This will -- let me draw the profile. 667 00:50:14,300 --> 00:50:21,000 So the profile is 0 there to 0 at some time. 668 00:50:21,000 --> 00:50:26,580 Then over here, where this characteristic is met, it's 1, 669 00:50:26,580 --> 00:50:34,310 and this is what happens: Instead 670 00:50:34,310 --> 00:50:37,680 of getting worse, less smooth, the solution 671 00:50:37,680 --> 00:50:41,930 gets better, more smooth, when the direction is reversed. 672 00:50:41,930 --> 00:50:45,220 So we see that our problem had to be nonlinear. 673 00:50:45,220 --> 00:50:50,010 A linear problem couldn't do these two opposite things 674 00:50:50,010 --> 00:50:53,110 in the same thing. 675 00:50:53,110 --> 00:50:57,630 So here we would have a fan and the characteristics go there, 676 00:50:57,630 --> 00:50:59,730 and it's reflected there. 677 00:50:59,730 --> 00:51:04,660 OK, so there's a picture of the phenomena that we're looking at 678 00:51:04,660 --> 00:51:09,590 and we would like to see numerically, too. 679 00:51:09,590 --> 00:51:16,720 So next time I want to find -- so I can say now what I have 680 00:51:16,720 --> 00:51:18,250 to do next time. 681 00:51:18,250 --> 00:51:27,800 One is to locate the shock, the shock line, when there is one, 682 00:51:27,800 --> 00:51:36,240 and two, to decide -- how do we decide between shock or fan 683 00:51:36,240 --> 00:51:44,150 when from the point of view of pure conservation, 684 00:51:44,150 --> 00:51:46,400 we could choose. 685 00:51:46,400 --> 00:51:49,290 But nature makes a choice. 686 00:51:49,290 --> 00:51:52,000 And one way to decide, make that decision, 687 00:51:52,000 --> 00:51:55,170 is Burger's way of putting in a little diffusion 688 00:51:55,170 --> 00:52:01,280 and letting it disappear, but we need a direct rule. 689 00:52:01,280 --> 00:52:04,190 So we need a direct entropy condition. 690 00:52:04,190 --> 00:52:09,700 So this is going to be a shock speed -- 691 00:52:09,700 --> 00:52:14,280 a formula for shock speed s -- s equals shock speed -- 692 00:52:14,280 --> 00:52:18,260 we'll find, if there is a shock. 693 00:52:18,260 --> 00:52:21,360 And number two is going to be -- this decision is going to be 694 00:52:21,360 --> 00:52:26,610 based on a quantity called entropy, 695 00:52:26,610 --> 00:52:30,020 which appears only in nonlinear problems, really. 696 00:52:30,020 --> 00:52:36,620 OK, so that's a beginning picture, 697 00:52:36,620 --> 00:52:46,180 which these 16.920 notes are an excellent reference 698 00:52:46,180 --> 00:52:48,800 for, the book also. 699 00:52:48,800 --> 00:52:52,090 And Monday's lecture will pick up there. 700 00:52:52,090 --> 00:52:56,570 So we can see the two situations here, 701 00:52:56,570 --> 00:53:01,420 and the question is what do we do at this point. 702 00:53:01,420 --> 00:53:02,820 OK, see you Monday. 703 00:53:02,820 --> 00:53:04,710 Have a good weekend. 704 00:53:04,710 --> 00:53:08,800 And any email questions about the homework, 705 00:53:08,800 --> 00:53:13,460 just you could send to me or -- and/or to -- maybe and to Mr. 706 00:53:13,460 --> 00:53:17,590 Cho, who will be looking at the homeworks that come in. 707 00:53:17,590 --> 00:53:19,310 Good, thanks.