1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,770 Commons license. 3 00:00:03,770 --> 00:00:06,020 Your support will help MIT OpenCourseWare 4 00:00:06,020 --> 00:00:10,090 continue to offer high quality educational resources for free. 5 00:00:10,090 --> 00:00:12,670 To make a donation or to view additional materials 6 00:00:12,670 --> 00:00:16,580 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,580 --> 00:00:17,235 at ocw.mit.edu. 8 00:00:26,120 --> 00:00:28,280 QIQI WANG: Any questions so far? 9 00:00:28,280 --> 00:00:31,170 Especially if you have any additional questions on the ODE 10 00:00:31,170 --> 00:00:34,070 part, if anything is not clear I'd 11 00:00:34,070 --> 00:00:39,900 like to have them answered before we move on to PDEs. 12 00:00:39,900 --> 00:00:41,830 And you're going to see other things 13 00:00:41,830 --> 00:00:47,940 we do on PDE really depend on ODEs or prerequisite ODEs. 14 00:00:50,771 --> 00:00:51,687 AUDIENCE: [INAUDIBLE]. 15 00:00:56,369 --> 00:00:57,160 QIQI WANG: Oh yeah. 16 00:00:57,160 --> 00:00:58,880 Good idea. 17 00:00:58,880 --> 00:01:01,920 Let me open it using my Chrome. 18 00:01:01,920 --> 00:01:06,991 Because the [INAUDIBLE] didn't work. 19 00:01:06,991 --> 00:01:07,490 OK. 20 00:01:17,261 --> 00:01:17,760 OK. 21 00:01:17,760 --> 00:01:20,170 These are the measurable outcomes. 22 00:01:20,170 --> 00:01:21,610 These also updated. 23 00:01:21,610 --> 00:01:23,870 So you have the ODEs. 24 00:01:23,870 --> 00:01:24,900 Now you scroll down. 25 00:01:24,900 --> 00:01:27,856 You have the Product Differentiated Equations. 26 00:01:27,856 --> 00:01:30,680 So we can look at all the measurable outcomes again. 27 00:01:30,680 --> 00:01:33,350 And for each of them, you can click on them. 28 00:01:33,350 --> 00:01:36,020 And it'll show you all the sections 29 00:01:36,020 --> 00:01:41,550 that you can use by yourself on how 30 00:01:41,550 --> 00:01:43,730 to navigate through the materials 31 00:01:43,730 --> 00:01:47,590 and make sure you have learned what we want you to learn. 32 00:01:47,590 --> 00:01:48,090 All right. 33 00:01:48,090 --> 00:01:52,184 We have an identifier with the PDE using [INAUDIBLE] law. 34 00:01:52,184 --> 00:01:53,600 And that is that one of the things 35 00:01:53,600 --> 00:01:56,230 we are going to be talking about today. 36 00:01:56,230 --> 00:01:58,960 Qualitatively describe the solution of simple ODEs. 37 00:01:58,960 --> 00:02:00,622 And that is also something that we are 38 00:02:00,622 --> 00:02:01,830 going to be describing today. 39 00:02:01,830 --> 00:02:06,024 But you can read all of these also in the readings. 40 00:02:06,024 --> 00:02:07,460 OK. 41 00:02:07,460 --> 00:02:13,670 Now let's dive into PDEs. 42 00:02:13,670 --> 00:02:14,960 OK. 43 00:02:14,960 --> 00:02:17,890 PDEs versus ODEs. 44 00:02:17,890 --> 00:02:21,700 When we solve for ODEs, we are looking at solution, 45 00:02:21,700 --> 00:02:24,908 we are discretizing a function of time. 46 00:02:24,908 --> 00:02:26,310 Right? 47 00:02:26,310 --> 00:02:28,940 So this is when we solve ODEs and when 48 00:02:28,940 --> 00:02:31,630 we want to aproximate-- The difference between an ODE 49 00:02:31,630 --> 00:02:34,680 and [INAUDIBLE] equation is because it compares 50 00:02:34,680 --> 00:02:37,180 a differential operator d/dt. 51 00:02:37,180 --> 00:02:42,090 And a lot of things we'll learn is how to approximate this d/dt 52 00:02:42,090 --> 00:02:48,890 and how we use the approximation to solve the ODE. 53 00:02:48,890 --> 00:02:52,740 Now once we go to PDEs, there are two cases. 54 00:02:52,740 --> 00:02:59,000 One case is now I have a U function of both space 55 00:02:59,000 --> 00:02:59,960 and time. 56 00:02:59,960 --> 00:03:01,740 So we have space and time. 57 00:03:01,740 --> 00:03:05,270 That gives me a PDE. 58 00:03:05,270 --> 00:03:06,700 And now when we take derivatives, 59 00:03:06,700 --> 00:03:09,060 there are two derivatives we can take. 60 00:03:09,060 --> 00:03:12,000 We can take the positive derivative of U 61 00:03:12,000 --> 00:03:14,400 with respect to space. 62 00:03:14,400 --> 00:03:17,740 And the reason it is a positive derivative 63 00:03:17,740 --> 00:03:22,430 is because when you fix all the other independent variables-- 64 00:03:22,430 --> 00:03:26,430 so it is a derivative to x while holding time fixed. 65 00:03:26,430 --> 00:03:29,715 Or you can take partial derivatives to t. 66 00:03:29,715 --> 00:03:32,310 That is, taking derivatives to time 67 00:03:32,310 --> 00:03:36,276 while fixing the spatial location. 68 00:03:36,276 --> 00:03:37,650 This is very important because we 69 00:03:37,650 --> 00:03:41,420 are going to see when we also can take derivatives 70 00:03:41,420 --> 00:03:44,540 of this function of space and time, [INAUDIBLE] 71 00:03:44,540 --> 00:03:46,860 neither hold space nor time fixed. 72 00:03:46,860 --> 00:03:51,430 So we can also take something like a directional derivative 73 00:03:51,430 --> 00:03:54,240 when we are both moving in space and time. 74 00:03:54,240 --> 00:03:55,700 And that is something we are going 75 00:03:55,700 --> 00:03:59,000 to see in the characteristics. 76 00:03:59,000 --> 00:04:01,020 We can take the derivative in interaction 77 00:04:01,020 --> 00:04:03,100 and call it a characteristic reaction. 78 00:04:03,100 --> 00:04:06,650 And that is going to give us some interesting results. 79 00:04:06,650 --> 00:04:09,720 Another possible case for PDE is when 80 00:04:09,720 --> 00:04:12,220 there are multiple spatial locations, 81 00:04:12,220 --> 00:04:17,040 we can have U of x and y or U of x, y, and z. 82 00:04:17,040 --> 00:04:18,760 And this is also a PDE because we 83 00:04:18,760 --> 00:04:20,570 can take the derivative to x. 84 00:04:20,570 --> 00:04:27,420 We can take-- and of course, in most cases you can 85 00:04:27,420 --> 00:04:29,730 have a function of x, y, and z. 86 00:04:29,730 --> 00:04:30,310 And t. 87 00:04:30,310 --> 00:04:32,710 That gives you a four-dimensional solution 88 00:04:32,710 --> 00:04:35,480 you can solve numerically. 89 00:04:35,480 --> 00:04:38,300 And a lot of the simulations, the aerodynamics simulations 90 00:04:38,300 --> 00:04:41,560 if you want to simulate the flow around an airplane, 91 00:04:41,560 --> 00:04:44,000 it's actually the unsteady flow air on an airplane, 92 00:04:44,000 --> 00:04:45,730 you do have to look at this case. 93 00:04:45,730 --> 00:04:48,370 You have three spatial dimensions and time. 94 00:04:48,370 --> 00:04:51,078 And you need to solve partial differential equations 95 00:04:51,078 --> 00:04:51,744 involving these. 96 00:04:54,530 --> 00:04:58,760 And the mass per volume is what? 97 00:04:58,760 --> 00:05:00,300 It's density, right? 98 00:05:00,300 --> 00:05:04,390 The unit amount of momentum per volume is momentum density. 99 00:05:04,390 --> 00:05:07,600 The unit amount of energy per volume is energy density. 100 00:05:07,600 --> 00:05:10,100 So these are the quantities we solve. 101 00:05:10,100 --> 00:05:14,145 When these quantities can be spatially dependent-- 102 00:05:14,145 --> 00:05:18,510 the density of air, for example, in a compatible flow 103 00:05:18,510 --> 00:05:22,530 can be different from one spatial location to another. 104 00:05:22,530 --> 00:05:24,240 In an unsteady compressible flow, 105 00:05:24,240 --> 00:05:27,310 it can be different at different times. 106 00:05:27,310 --> 00:05:29,090 So this kind of view is going to be 107 00:05:29,090 --> 00:05:34,380 a function of space and time. 108 00:05:34,380 --> 00:05:34,880 OK. 109 00:05:34,880 --> 00:05:38,980 So how do we express this kind of conservation law 110 00:05:38,980 --> 00:05:43,820 in terms of partial differential equations? 111 00:05:43,820 --> 00:05:46,950 First of all, what does it mean by, let's say, 112 00:05:46,950 --> 00:05:49,610 mass being conserved? 113 00:05:49,610 --> 00:05:57,310 So let's take a special case where U is the density. 114 00:05:57,310 --> 00:06:00,973 So let's say the unit amount of mass, the volume. 115 00:06:04,270 --> 00:06:06,030 So can somebody tell me what does 116 00:06:06,030 --> 00:06:10,109 it mean by mass being conserved in the mathematical sense? 117 00:06:10,109 --> 00:06:10,984 AUDIENCE: [INAUDIBLE] 118 00:06:21,167 --> 00:06:21,750 QIQI WANG: OK. 119 00:06:21,750 --> 00:06:23,650 So yeah. 120 00:06:23,650 --> 00:06:26,240 That's a very good answer. 121 00:06:26,240 --> 00:06:29,070 So first of all, you are choosing a controlled volume, 122 00:06:29,070 --> 00:06:29,850 right? 123 00:06:29,850 --> 00:06:32,960 So first of all, you are choosing a control volume. 124 00:06:32,960 --> 00:06:36,090 Let's say do my control volume, let's actually 125 00:06:36,090 --> 00:06:37,510 do the control volume like this. 126 00:06:37,510 --> 00:06:39,590 It can be an arbitrary control volume. 127 00:06:39,590 --> 00:06:42,220 Let's call it omega. 128 00:06:42,220 --> 00:06:46,080 So mass being conserved means the amount 129 00:06:46,080 --> 00:06:50,680 of mass inside the control volume 130 00:06:50,680 --> 00:06:55,910 is not going to magically appear or magically disappear. 131 00:06:55,910 --> 00:06:58,620 The increase has to be coming from outside. 132 00:06:58,620 --> 00:07:02,500 And the decrease has to go outside. 133 00:07:02,500 --> 00:07:03,660 Right? 134 00:07:03,660 --> 00:07:05,870 That means-- 135 00:07:05,870 --> 00:07:08,200 And also let me express what the mass is. 136 00:07:08,200 --> 00:07:13,430 The mass is the integral of the density integrated 137 00:07:13,430 --> 00:07:14,300 over the volume. 138 00:07:14,300 --> 00:07:17,510 Let's express it as this integral 139 00:07:17,510 --> 00:07:20,040 so that it makes more sense. 140 00:07:20,040 --> 00:07:24,535 Now this is the mass inside the control 141 00:07:24,535 --> 00:07:28,060 volume at a certain time. 142 00:07:28,060 --> 00:07:34,320 The time derivative of this mass which 143 00:07:34,320 --> 00:07:39,540 represents in a very small amount of time 144 00:07:39,540 --> 00:07:41,900 how much mass increased or how much mass has decreased 145 00:07:41,900 --> 00:07:43,660 inside the control volume. 146 00:07:43,660 --> 00:07:47,890 It has to be equal to the rate of mass coming 147 00:07:47,890 --> 00:07:50,240 into the volume minus the rate of mass 148 00:07:50,240 --> 00:07:52,160 going out of the volume. 149 00:07:52,160 --> 00:07:53,630 Right? 150 00:07:53,630 --> 00:07:56,760 And the mass going in and out can only 151 00:07:56,760 --> 00:08:00,140 happen on the surface of the control volume, 152 00:08:00,140 --> 00:08:01,910 on a boundary of the control volume. 153 00:08:01,910 --> 00:08:04,320 And let me write like this. 154 00:08:04,320 --> 00:08:09,200 Plus the rate over the surface of control volume. 155 00:08:09,200 --> 00:08:12,840 Let me call it partial omega. 156 00:08:12,840 --> 00:08:16,847 This is a notation of the boundary of the control volume 157 00:08:16,847 --> 00:08:18,340 omega. 158 00:08:18,340 --> 00:08:21,170 It has to be equal to an integral 159 00:08:21,170 --> 00:08:25,990 at the boundary of the control volume times the flux of row. 160 00:08:25,990 --> 00:08:33,280 Let me call it f of row d S. That will be equal to zero. 161 00:08:33,280 --> 00:08:34,039 OK. 162 00:08:34,039 --> 00:08:36,919 Let me expand this a little bit more. 163 00:08:36,919 --> 00:08:39,485 So this [INAUDIBLE]. 164 00:08:39,485 --> 00:08:47,780 It represents for a unit amount of time at a unit surface. 165 00:08:47,780 --> 00:08:54,630 How much mass has gone through that surface? 166 00:08:54,630 --> 00:08:57,176 It is directional because there is a direction 167 00:08:57,176 --> 00:09:00,535 of the flow of mass, right? 168 00:09:00,535 --> 00:09:14,403 For example, if you know it's a fluid flow, then in fluid flow 169 00:09:14,403 --> 00:09:22,221 f of row is going to be equal to row times the local velocity, 170 00:09:22,221 --> 00:09:22,720 right? 171 00:09:22,720 --> 00:09:27,360 So if you have a velocity vector and you know the density, 172 00:09:27,360 --> 00:09:33,630 then the unit among the mass going through a surface 173 00:09:33,630 --> 00:09:37,260 is equal to the row U times the area of the surface 174 00:09:37,260 --> 00:09:39,830 normal to the velocity. 175 00:09:39,830 --> 00:09:40,900 Right? 176 00:09:40,900 --> 00:09:46,250 So that basically is what we mean 177 00:09:46,250 --> 00:09:49,210 by a quantity being conserved. 178 00:09:52,500 --> 00:09:55,400 All Right. 179 00:09:55,400 --> 00:09:58,670 Now this is not a differential equation, right? 180 00:09:58,670 --> 00:10:00,860 Anybody have a question? 181 00:10:00,860 --> 00:10:03,951 Question on this conservation law? 182 00:10:03,951 --> 00:10:04,450 OK. 183 00:10:04,450 --> 00:10:07,350 This is not a differential question yet. 184 00:10:07,350 --> 00:10:08,800 Right? 185 00:10:08,800 --> 00:10:12,980 And honestly I don't really know how to solve it. 186 00:10:12,980 --> 00:10:15,960 So how do we convert this into a differential equation 187 00:10:15,960 --> 00:10:17,197 that we can solve? 188 00:10:17,197 --> 00:10:19,414 AUDIENCE: [INAUDIBLE]. 189 00:10:19,414 --> 00:10:20,330 QIQI WANG: Good point. 190 00:10:20,330 --> 00:10:24,590 We can shrink this omega to something very small. 191 00:10:24,590 --> 00:10:27,630 And this is exactly what we are going to see 192 00:10:27,630 --> 00:10:30,430 in the finite volume approach. 193 00:10:30,430 --> 00:10:32,260 So in the finite volume approach, 194 00:10:32,260 --> 00:10:35,430 we are actually going to shrink this omega to something 195 00:10:35,430 --> 00:10:41,650 very small and use this form to solve this equation. 196 00:10:41,650 --> 00:10:44,967 Another method we're going to discuss is finite difference. 197 00:10:44,967 --> 00:10:46,800 So we're going to be teaching three methods. 198 00:10:46,800 --> 00:10:50,520 Finite difference, finite volume, finite element. 199 00:10:50,520 --> 00:10:53,210 So let's leave finite element to the last 200 00:10:53,210 --> 00:10:55,880 because it's the most advanced method. 201 00:10:55,880 --> 00:10:56,680 Finite difference. 202 00:10:56,680 --> 00:10:59,310 I'm trying to avoid, we're going to be discussing first. 203 00:10:59,310 --> 00:11:01,860 And finite volume is exactly taking that equation 204 00:11:01,860 --> 00:11:05,550 and shrinking omega to very small. 205 00:11:05,550 --> 00:11:09,170 The finite difference approach, on the other hand, 206 00:11:09,170 --> 00:11:12,185 has to first convert this equation 207 00:11:12,185 --> 00:11:15,470 into a differential equation. 208 00:11:15,470 --> 00:11:20,090 In finite difference, we do not allow any integrals. 209 00:11:20,090 --> 00:11:22,650 We have to express everything through derivatives. 210 00:11:22,650 --> 00:11:25,996 Now how to do that? 211 00:11:25,996 --> 00:11:29,330 [INAUDIBLE] theorem or divergent theorem, right? 212 00:11:29,330 --> 00:11:43,730 With divergent theorem, we can convert the surface integral 213 00:11:43,730 --> 00:11:48,600 into a volume integral inside omega 214 00:11:48,600 --> 00:11:50,833 of the divergence of the flux. 215 00:11:54,460 --> 00:11:56,800 Right? 216 00:11:56,800 --> 00:12:00,520 So the conservation equation becomes like this, it becomes-- 217 00:12:00,520 --> 00:12:04,610 and we can also take the timed derivative inside of the fixed 218 00:12:04,610 --> 00:12:05,800 control volume. 219 00:12:05,800 --> 00:12:10,720 So we have integral over a really arbitrary domain, omega. 220 00:12:10,720 --> 00:12:12,780 Because this conservation law has 221 00:12:12,780 --> 00:12:16,240 to satisfy for any control volume. 222 00:12:16,240 --> 00:12:19,120 So this is the time derivative then. 223 00:12:19,120 --> 00:12:22,310 And we copy the divergence term into here. 224 00:12:26,230 --> 00:12:28,720 The x has to be zero. 225 00:12:28,720 --> 00:12:32,782 And this has to be zero for any control volume. 226 00:12:35,580 --> 00:12:39,680 For any omega. 227 00:12:39,680 --> 00:12:45,280 That means what is inside the integral has to be zero. 228 00:12:45,280 --> 00:12:50,090 Because if the integral is non-zero anywhere, 229 00:12:50,090 --> 00:12:52,960 then we can make a very small control volume around it. 230 00:12:52,960 --> 00:12:57,490 And that integral is not going to be zero. 231 00:12:57,490 --> 00:12:59,440 So the integral has to be zero everywhere. 232 00:13:02,680 --> 00:13:06,377 Now this is a differential equation, right? 233 00:13:06,377 --> 00:13:07,710 This is a differential equation. 234 00:13:07,710 --> 00:13:11,210 And it is a partial differential equation 235 00:13:11,210 --> 00:13:14,880 because we not only have derivatives in time, 236 00:13:14,880 --> 00:13:17,840 we also have divergence. 237 00:13:17,840 --> 00:13:21,610 And divergence is a derivative in space. 238 00:13:21,610 --> 00:13:27,460 So this is my PDE in the differential form. 239 00:13:30,390 --> 00:13:30,890 Right. 240 00:13:30,890 --> 00:13:33,080 This is my PDE in the differential form. 241 00:13:33,080 --> 00:13:35,650 And that is what we have over here, differential 242 00:13:35,650 --> 00:13:37,990 consideration law of U. 243 00:13:37,990 --> 00:13:41,320 So a differential conservation law of U. 244 00:13:41,320 --> 00:13:46,160 In general, we have derived the conservation of mass, right? 245 00:13:46,160 --> 00:13:50,220 In general, the differential form of the conversion law 246 00:13:50,220 --> 00:13:55,940 is dU/dt plus the divergence of some flux which 247 00:13:55,940 --> 00:13:59,990 is a function of U is equal to a fourth term. 248 00:13:59,990 --> 00:14:03,880 And the fourth term is going to be non-zero 249 00:14:03,880 --> 00:14:08,340 for things like, for example, if you have energy-- 250 00:14:08,340 --> 00:14:09,960 if you have momentum-- for example, 251 00:14:09,960 --> 00:14:13,030 if you use momentum, momentum density, 252 00:14:13,030 --> 00:14:17,304 then the fourth term will be external force like gravity. 253 00:14:17,304 --> 00:14:18,260 Right? 254 00:14:18,260 --> 00:14:20,990 So you can also have a [INAUDIBLE]. 255 00:14:20,990 --> 00:14:23,300 So when we talk about a differential 256 00:14:23,300 --> 00:14:25,020 form of a conservation law, we are 257 00:14:25,020 --> 00:14:27,692 talking about equation [INAUDIBLE] like this. 258 00:14:30,631 --> 00:14:31,130 All right. 259 00:14:31,130 --> 00:14:33,770 Let's take a look at a few examples. 260 00:14:33,770 --> 00:14:40,070 So the first example is what we do a lot of research on. 261 00:14:40,070 --> 00:14:44,420 The conservation of mass, momentum, and energy 262 00:14:44,420 --> 00:14:47,120 with flows. 263 00:14:47,120 --> 00:14:49,310 And by solving differential equations, 264 00:14:49,310 --> 00:14:52,690 that governs the [INAUDIBLE] conservation of three things-- 265 00:14:52,690 --> 00:14:56,720 mass, momentum, and energy-- we can really 266 00:14:56,720 --> 00:15:00,310 explain a lot of phenomenon that we 267 00:15:00,310 --> 00:15:02,960 are interested in in aerospace engineering. 268 00:15:02,960 --> 00:15:11,000 Like the flow around airplanes, rockets, and missiles. 269 00:15:11,000 --> 00:15:14,740 And also flow inside [INAUDIBLE] engines. 270 00:15:14,740 --> 00:15:18,200 We can all solve these problem by solving 271 00:15:18,200 --> 00:15:21,720 differential equations that just governs the conservation 272 00:15:21,720 --> 00:15:25,461 of mass, momentum, and energy. 273 00:15:25,461 --> 00:15:25,960 All right. 274 00:15:25,960 --> 00:15:30,330 So this is one of the prime examples 275 00:15:30,330 --> 00:15:34,560 of conservation loss expressing partial differential equations. 276 00:15:37,680 --> 00:15:42,830 Another example of conservation law is wave propagation. 277 00:15:42,830 --> 00:15:46,412 So this is like surface waves. 278 00:15:46,412 --> 00:15:49,370 And many of the simulations you can see actually 279 00:15:49,370 --> 00:15:53,658 inside [INAUDIBLE] where we are solving partial differential 280 00:15:53,658 --> 00:15:58,080 equations that govern really the conservation of mass 281 00:15:58,080 --> 00:16:05,351 and momentum of water, that is under the force of gravity. 282 00:16:08,240 --> 00:16:10,130 Another type of conservation law is 283 00:16:10,130 --> 00:16:15,190 what you find when you open a website 284 00:16:15,190 --> 00:16:17,970 and look at the weather forecast. 285 00:16:17,970 --> 00:16:21,450 So how do people predict the weather? 286 00:16:21,450 --> 00:16:24,630 They solve differential equations. 287 00:16:24,630 --> 00:16:26,190 They solve differential equations 288 00:16:26,190 --> 00:16:32,560 that govern the conservation of mass and momentum, weather 289 00:16:32,560 --> 00:16:35,260 molecules, energy. 290 00:16:35,260 --> 00:16:37,892 Of course, you have [INAUDIBLE] from the sun. 291 00:16:37,892 --> 00:16:41,000 And you have things from the ocean, things like that. 292 00:16:41,000 --> 00:16:44,310 But essentially it is a differential equation 293 00:16:44,310 --> 00:16:48,450 governing the conservation of stuff that happens 294 00:16:48,450 --> 00:16:50,270 on the Earth's atmosphere. 295 00:16:50,270 --> 00:16:53,238 Of course, it's a very large scale PDE. 296 00:16:53,238 --> 00:16:55,220 But you look at the weather forecast 297 00:16:55,220 --> 00:16:59,870 because people are able to solve PDEs. 298 00:16:59,870 --> 00:17:05,910 So this is what we are going to be looking at in the PDE 299 00:17:05,910 --> 00:17:09,329 section of this class. 300 00:17:09,329 --> 00:17:11,970 We look at how to numerically solve partial differential 301 00:17:11,970 --> 00:17:14,569 equations. 302 00:17:14,569 --> 00:17:16,490 OK. 303 00:17:16,490 --> 00:17:18,310 So first of all, we are going to be 304 00:17:18,310 --> 00:17:23,770 looking at some examples of conservation laws. 305 00:17:23,770 --> 00:17:27,713 And these examples are the very elementary type 306 00:17:27,713 --> 00:17:31,630 of conservation loss that a lot of them, 307 00:17:31,630 --> 00:17:35,990 you should be able to qualitatively describe 308 00:17:35,990 --> 00:17:38,970 the solution, how the behaves. 309 00:17:38,970 --> 00:17:41,070 But really in complex differential equations 310 00:17:41,070 --> 00:17:43,560 like this, it's really the combination 311 00:17:43,560 --> 00:17:48,450 of multiple of these elementary conservation laws. 312 00:17:48,450 --> 00:17:50,460 And if you are able to solve each 313 00:17:50,460 --> 00:17:52,390 of these elementary conservation laws 314 00:17:52,390 --> 00:17:54,220 and you know how to combine them together 315 00:17:54,220 --> 00:17:57,360 and how to do that in complex geometry, 316 00:17:57,360 --> 00:18:01,720 we can really solve these very complex differential equations. 317 00:18:01,720 --> 00:18:04,800 So the first conservation law is advection. 318 00:18:04,800 --> 00:18:09,730 Vachon And also many people in this field called convection. 319 00:18:14,080 --> 00:18:16,370 So convection is a physical phenomenon 320 00:18:16,370 --> 00:18:23,910 that is usually created by the bulk movement of molecules 321 00:18:23,910 --> 00:18:26,440 inside some fluid. 322 00:18:26,440 --> 00:18:29,400 Like, for example, the generation of the hurricane 323 00:18:29,400 --> 00:18:31,882 is really a convective phenomenon. 324 00:18:31,882 --> 00:18:35,510 So let's look at-- at the most elementary form-- what 325 00:18:35,510 --> 00:18:38,815 convection is like. 326 00:18:38,815 --> 00:18:40,929 So let me go over here. 327 00:18:40,929 --> 00:18:42,925 I have some MATLAB [INAUDIBLE]. 328 00:18:55,410 --> 00:18:55,990 OK. 329 00:18:55,990 --> 00:18:57,948 And I want you to look at the first code there. 330 00:18:57,948 --> 00:19:01,020 But I'm going to give the source code to you via Dropbox. 331 00:19:03,970 --> 00:19:07,620 So here, what I'm doing is-- OK. 332 00:19:07,620 --> 00:19:11,860 I'm opening in MATLAB a window that has nothing in it. 333 00:19:11,860 --> 00:19:16,180 I like some of you to come up and design 334 00:19:16,180 --> 00:19:22,250 what's called an initial condition for this [INAUDIBLE]. 335 00:19:22,250 --> 00:19:23,600 All right? 336 00:19:23,600 --> 00:19:27,900 And let me first describe what the advection equation does. 337 00:19:27,900 --> 00:19:32,720 The advection equation describes the evolution 338 00:19:32,720 --> 00:19:34,660 of a concentration. 339 00:19:34,660 --> 00:19:40,880 Let's say mass density of a certain species. 340 00:19:40,880 --> 00:19:45,404 Density of, let's say, a pollutant or [INAUDIBLE]. 341 00:19:45,404 --> 00:19:49,400 Then how does the density evolve under the bulk movement 342 00:19:49,400 --> 00:19:53,090 of a fluid? 343 00:19:53,090 --> 00:19:56,470 So convection, that is one field we 344 00:19:56,470 --> 00:19:58,410 are going to be going over here, describes 345 00:19:58,410 --> 00:20:04,230 the initial distribution of that species. 346 00:20:04,230 --> 00:20:07,905 Or the initial density of the species. 347 00:20:07,905 --> 00:20:09,530 And after [INAUDIBLE] is in a position, 348 00:20:09,530 --> 00:20:14,022 we are all going to be observing how does the distribution go 349 00:20:14,022 --> 00:20:17,690 under the bulk movement of some fluid? 350 00:20:17,690 --> 00:20:20,150 There's somebody here [INAUDIBLE]. 351 00:20:20,150 --> 00:20:21,626 Somebody is speaking at the back? 352 00:20:21,626 --> 00:20:24,279 AUDIENCE: [INAUDIBLE]. 353 00:20:24,279 --> 00:20:25,070 QIQI WANG: Come on. 354 00:20:31,958 --> 00:20:37,140 [LAUGHTER] 355 00:20:37,140 --> 00:20:37,640 OK. 356 00:20:37,640 --> 00:20:38,843 Somebody-- 357 00:20:38,843 --> 00:20:39,676 [INTERPOSING VOICES] 358 00:20:42,570 --> 00:20:44,734 QIQI WANG: [INAUDIBLE], people. 359 00:20:44,734 --> 00:20:46,019 AUDIENCE: [INAUDIBLE]. 360 00:20:46,019 --> 00:20:46,602 QIQI WANG: OK. 361 00:20:46,602 --> 00:20:48,937 Thank you. 362 00:20:48,937 --> 00:20:49,859 AUDIENCE: [INAUDIBLE]. 363 00:20:49,859 --> 00:20:51,150 QIQI WANG: Come up and draw it. 364 00:20:55,150 --> 00:20:56,067 AUDIENCE: [INAUDIBLE]. 365 00:20:56,067 --> 00:20:58,066 QIQI WANG: You want to move [INAUDIBLE] starting 366 00:20:58,066 --> 00:20:59,485 from the left to the right. 367 00:20:59,485 --> 00:21:00,370 AUDIENCE: OK. 368 00:21:00,370 --> 00:21:01,745 QIQI WANG: And it's going to be-- 369 00:21:01,745 --> 00:21:05,816 try to make it periodic. [INAUDIBLE] on the screen. 370 00:21:05,816 --> 00:21:06,740 AUDIENCE: [INAUDIBLE]. 371 00:21:06,740 --> 00:21:09,512 [LAUGHTER] 372 00:21:09,512 --> 00:21:11,810 AUDIENCE: You want it to be periodic? 373 00:21:11,810 --> 00:21:13,029 QIQI WANG: Yeah. 374 00:21:13,029 --> 00:21:13,570 AUDIENCE: OK. 375 00:21:16,480 --> 00:21:19,390 QIQI WANG: Going from the left to right, we want [INAUDIBLE]. 376 00:21:19,390 --> 00:21:20,789 Yeah. 377 00:21:20,789 --> 00:21:21,330 AUDIENCE: OK. 378 00:21:24,522 --> 00:21:25,355 [INTERPOSING VOICES] 379 00:21:25,355 --> 00:21:26,450 QIQI WANG: Let me try it again. 380 00:21:26,450 --> 00:21:27,400 Let me try it again. 381 00:21:33,100 --> 00:21:35,550 AUDIENCE: Is it going to freak out if it's negative? 382 00:21:35,550 --> 00:21:36,300 QIQI WANG: No, no. 383 00:21:36,300 --> 00:21:39,630 It's not going to freak out. 384 00:21:39,630 --> 00:21:42,547 This [INAUDIBLE] with most [INAUDIBLE]. 385 00:21:42,547 --> 00:21:43,088 AUDIENCE: OK. 386 00:21:46,896 --> 00:21:48,802 QIQI WANG: Good. 387 00:21:48,802 --> 00:21:49,760 AUDIENCE: That wasn't-- 388 00:21:49,760 --> 00:21:50,232 QIQI WANG: Keep going. 389 00:21:50,232 --> 00:21:50,704 Keep going. 390 00:21:50,704 --> 00:21:51,204 Keep going. 391 00:21:51,204 --> 00:21:51,898 Keep going. 392 00:21:51,898 --> 00:21:53,064 AUDIENCE: It's not periodic. 393 00:21:53,064 --> 00:21:53,536 QIQI WANG: It's fine, it's fine. 394 00:21:53,536 --> 00:21:54,480 AUDIENCE: OK. 395 00:21:54,480 --> 00:21:56,550 QIQI WANG: All right. 396 00:21:56,550 --> 00:21:57,610 OK. 397 00:21:57,610 --> 00:21:59,070 We actually have two windows. 398 00:21:59,070 --> 00:22:05,210 That's why it's-- so let's look at the right window first. 399 00:22:05,210 --> 00:22:07,230 Thank you. 400 00:22:07,230 --> 00:22:08,350 OK. 401 00:22:08,350 --> 00:22:09,940 Let's look at this window first. 402 00:22:09,940 --> 00:22:14,060 So this is the evolution of the equation after a certain time, 403 00:22:14,060 --> 00:22:14,980 right? 404 00:22:14,980 --> 00:22:17,292 It's accelerating, right? 405 00:22:17,292 --> 00:22:19,302 This is the solution to the partial differential 406 00:22:19,302 --> 00:22:22,680 equation at a certain point. 407 00:22:22,680 --> 00:22:25,840 So this is the spatial coordinate x. 408 00:22:25,840 --> 00:22:28,010 And the time is not typical. 409 00:22:28,010 --> 00:22:30,124 This is the [INAUDIBLE]. 410 00:22:30,124 --> 00:22:32,730 And this [INAUDIBLE] over here. 411 00:22:32,730 --> 00:22:35,516 [INAUDIBLE] is how the solution evolved. 412 00:22:41,280 --> 00:22:43,190 What is this solution doing? 413 00:22:43,190 --> 00:22:43,897 Yeah? 414 00:22:43,897 --> 00:22:45,765 AUDIENCE: It's moving all the [INAUDIBLE]. 415 00:22:53,960 --> 00:22:55,577 QIQI WANG: OK. 416 00:22:55,577 --> 00:22:59,260 The answer is-- the solution, kind of, moves while also 417 00:22:59,260 --> 00:23:00,574 [INAUDIBLE]. 418 00:23:00,574 --> 00:23:03,576 But here the template [INAUDIBLE]. 419 00:23:03,576 --> 00:23:04,548 AUDIENCE: [INAUDIBLE] 420 00:23:09,275 --> 00:23:09,900 QIQI WANG: Yes. 421 00:23:09,900 --> 00:23:10,608 Good observation. 422 00:23:10,608 --> 00:23:12,620 The amplitude has to be the same. 423 00:23:12,620 --> 00:23:13,920 Right? 424 00:23:13,920 --> 00:23:16,230 0.6 minus [INAUDIBLE]. 425 00:23:20,070 --> 00:23:22,590 If x equals stays the same and the whole thing 426 00:23:22,590 --> 00:23:26,558 just moves towards the right at a fixed velocity. 427 00:23:30,494 --> 00:23:37,720 So this is a solution to the pure advection equation. 428 00:23:37,720 --> 00:23:42,200 And it is the solution that keeps moving towards the right 429 00:23:42,200 --> 00:23:43,860 without diffusing. 430 00:23:43,860 --> 00:23:48,740 So the question is partial U-- let 431 00:23:48,740 --> 00:23:56,510 me call it small u-- partial small u, partial t plus a big U 432 00:23:56,510 --> 00:24:01,160 partial U-partial x equal to zero. 433 00:24:01,160 --> 00:24:04,830 Here there is a big distinction between the small u which 434 00:24:04,830 --> 00:24:13,080 is a function of space and time, is the unknown 435 00:24:13,080 --> 00:24:15,690 which is also what we plotted. 436 00:24:15,690 --> 00:24:21,160 And the big U is neither function of space nor time. 437 00:24:21,160 --> 00:24:21,890 It's a constant. 438 00:24:25,070 --> 00:24:25,570 OK? 439 00:24:25,570 --> 00:24:29,370 So big U is a constant advective velocity. 440 00:24:29,370 --> 00:24:31,760 And small u is the unknown. 441 00:24:35,440 --> 00:24:37,370 So two things. 442 00:24:37,370 --> 00:24:41,180 One is, is this equation a conservation law? 443 00:24:45,680 --> 00:24:48,960 Do you think, just from guessing from the solution you saw, 444 00:24:48,960 --> 00:24:50,540 is that a conservation law? 445 00:24:53,931 --> 00:24:54,430 Yes. 446 00:24:54,430 --> 00:24:58,300 Because it looks like the density or the solution 447 00:24:58,300 --> 00:24:59,380 is conserved, right? 448 00:24:59,380 --> 00:25:00,510 It just moves around. 449 00:25:00,510 --> 00:25:08,535 Whenever that goes into-- if you look at a control volume, 450 00:25:08,535 --> 00:25:09,940 that's because [INAUDIBLE]. 451 00:25:09,940 --> 00:25:15,274 Whatever that [INAUDIBLE] right? 452 00:25:15,274 --> 00:25:18,160 So it is a conservation law. 453 00:25:18,160 --> 00:25:23,560 Mathematically, how does that come into a conversation law? 454 00:25:23,560 --> 00:25:27,590 Mathematically if I write the equation a little bit 455 00:25:27,590 --> 00:25:33,430 differently, if I write the equation differently-- 456 00:25:33,430 --> 00:25:41,750 partial u-partial t plus partial F of u partial x 457 00:25:41,750 --> 00:25:48,270 is equal to zero where the F of u, the flux of u 458 00:25:48,270 --> 00:25:55,195 is just equal to big U times small u. 459 00:25:55,195 --> 00:25:57,111 Right? 460 00:25:57,111 --> 00:25:59,550 Does that look more like a conversation law? 461 00:26:02,570 --> 00:26:03,310 Why? 462 00:26:03,310 --> 00:26:10,470 Because we already mentioned that derivative in x is-- Yeah. 463 00:26:10,470 --> 00:26:17,390 This is what the divergence is having a multiple dimension, 464 00:26:17,390 --> 00:26:19,990 right? 465 00:26:19,990 --> 00:26:26,690 The divergence of [INAUDIBLE] is e x component of the method dx 466 00:26:26,690 --> 00:26:29,446 plus the y component of the method dy. 467 00:26:29,446 --> 00:26:30,946 And in one dimension there is no dy. 468 00:26:30,946 --> 00:26:35,280 So it would be dx is the divergence. 469 00:26:35,280 --> 00:26:37,100 That's what happens in one dimension. 470 00:26:37,100 --> 00:26:40,710 The divergence is [INAUDIBLE]. 471 00:26:40,710 --> 00:26:42,870 And now the use of flux. 472 00:26:42,870 --> 00:26:46,865 The flux is equal to big U times small u. 473 00:26:46,865 --> 00:26:48,170 That makes sense, right? 474 00:26:48,170 --> 00:26:52,720 Because if the small u is the solution [INAUDIBLE], 475 00:26:52,720 --> 00:26:54,280 then the [INAUDIBLE] of the density 476 00:26:54,280 --> 00:26:58,300 should not be the velocity times the local density. 477 00:26:58,300 --> 00:27:00,280 Right? 478 00:27:00,280 --> 00:27:03,302 The higher the velocity is, the faster the mass 479 00:27:03,302 --> 00:27:06,850 goes through [INAUDIBLE]. 480 00:27:06,850 --> 00:27:09,983 The higher the density is, also the faster the mass 481 00:27:09,983 --> 00:27:12,256 goes through [INAUDIBLE]. 482 00:27:12,256 --> 00:27:15,788 The flux indicated is velocity times the [INAUDIBLE]. 483 00:27:19,060 --> 00:27:21,240 Right? 484 00:27:21,240 --> 00:27:22,960 OK. 485 00:27:22,960 --> 00:27:25,950 Now there is a conservation law, right? 486 00:27:25,950 --> 00:27:27,380 It is a conservation law. 487 00:27:27,380 --> 00:27:31,170 And it seems it has an analytic solution of everything that's 488 00:27:31,170 --> 00:27:34,640 moving towards the right. 489 00:27:34,640 --> 00:27:35,720 Is that true? 490 00:27:35,720 --> 00:27:36,240 Yes. 491 00:27:36,240 --> 00:27:39,330 We do have an analytic solution. 492 00:27:39,330 --> 00:27:43,530 And the analytic solution looks like this. 493 00:27:43,530 --> 00:27:45,730 Let's take a look at this again. 494 00:27:45,730 --> 00:27:50,673 So this is the solution at a particular time-- in this case, 495 00:27:50,673 --> 00:27:55,110 it is at time 0.5. 496 00:27:55,110 --> 00:27:57,300 But if you don't know the solution 497 00:27:57,300 --> 00:28:01,860 at a certain time, the entire solution as a function of space 498 00:28:01,860 --> 00:28:03,220 and time, it looks like this. 499 00:28:06,426 --> 00:28:08,090 It looks like this. 500 00:28:08,090 --> 00:28:14,620 Now over here I already have this-- and [INAUDIBLE]. 501 00:28:14,620 --> 00:28:16,930 The solution as [INAUDIBLE]. 502 00:28:16,930 --> 00:28:20,370 But other solutions [INAUDIBLE]. 503 00:28:20,370 --> 00:28:23,250 OK. 504 00:28:23,250 --> 00:28:27,599 Blue is negative value and red is positive value. 505 00:28:27,599 --> 00:28:28,432 [INTERPOSING VOICES] 506 00:28:31,680 --> 00:28:34,850 QIQI WANG: Here you can see that at time equal to zero, 507 00:28:34,850 --> 00:28:36,203 this is [INAUDIBLE]. 508 00:28:38,911 --> 00:28:40,160 This is the initial condition. 509 00:28:42,715 --> 00:28:50,260 As time increases, let's say, at 0.1, the peak 510 00:28:50,260 --> 00:28:54,169 has gone from here instead of here at the initial condition. 511 00:28:54,169 --> 00:28:55,960 At the initial condition, the peak is here. 512 00:28:55,960 --> 00:28:58,555 The bottom is here. [INAUDIBLE] the peak 513 00:28:58,555 --> 00:29:00,096 has moved towards the right. 514 00:29:00,096 --> 00:29:01,720 The bottom has moved towards the right. 515 00:29:01,720 --> 00:29:04,890 Everything has moved towards the right. 516 00:29:04,890 --> 00:29:06,605 And it goes over the [INAUDIBLE]. 517 00:29:06,605 --> 00:29:07,980 And we're going to discuss later, 518 00:29:07,980 --> 00:29:09,688 it depends on the conditions [INAUDIBLE]. 519 00:29:15,395 --> 00:29:18,720 And keep moving towards the right. 520 00:29:18,720 --> 00:29:20,280 All right. 521 00:29:20,280 --> 00:29:25,950 And you'll see the lines over here, the lines which really 522 00:29:25,950 --> 00:29:30,530 is the contour of the solution for space and time-- 523 00:29:30,530 --> 00:29:35,490 these lines are called the characteristic lines 524 00:29:35,490 --> 00:29:39,160 in the solution of a partial differential equation. 525 00:29:41,635 --> 00:29:42,135 OK. 526 00:29:42,135 --> 00:29:43,620 So why are they important? 527 00:29:43,620 --> 00:29:45,580 Why does it have a name? 528 00:29:49,830 --> 00:29:52,550 Do we see something particular about these lines? 529 00:29:52,550 --> 00:29:54,975 AUDIENCE: [INAUDIBLE]. 530 00:29:54,975 --> 00:29:57,410 QIQI WANG: Huh? 531 00:29:57,410 --> 00:30:00,780 They all have the same flow or-- first of all, 532 00:30:00,780 --> 00:30:03,216 they are straight, right? 533 00:30:03,216 --> 00:30:05,090 We all see the contour of the random function 534 00:30:05,090 --> 00:30:08,740 of space and time, you would expect the [INAUDIBLE] 535 00:30:08,740 --> 00:30:10,532 to be a curve. 536 00:30:10,532 --> 00:30:11,880 Right? 537 00:30:11,880 --> 00:30:14,030 But these contours are straight. 538 00:30:14,030 --> 00:30:19,050 And we're going to see later on that [INAUDIBLE] 539 00:30:19,050 --> 00:30:22,086 another equation we happen to be looking at. 540 00:30:22,086 --> 00:30:26,210 The second is, the solution is [INAUDIBLE] along these lines. 541 00:30:26,210 --> 00:30:28,040 And these are really the characteristics 542 00:30:28,040 --> 00:30:30,190 of characteristic lines. 543 00:30:30,190 --> 00:30:33,120 And in this case, we can analytically 544 00:30:33,120 --> 00:30:37,780 derive what they are because there is an analytical solution 545 00:30:37,780 --> 00:30:40,470 of this differential equation. 546 00:30:40,470 --> 00:30:45,460 U of x and t is just equal to whatever initial condition is-- 547 00:30:45,460 --> 00:30:53,587 let me call u naught the initial condition-- x minus U times t. 548 00:30:56,401 --> 00:30:57,810 OK. 549 00:30:57,810 --> 00:30:59,125 What does this mean? 550 00:30:59,125 --> 00:31:05,470 It means-- remember u zero of x minus Ut. 551 00:31:05,470 --> 00:31:08,870 It means at t little zero, it makes 552 00:31:08,870 --> 00:31:13,956 sense that at t little zero, my solution U of x, 553 00:31:13,956 --> 00:31:16,366 t-- that is [INAUDIBLE]. 554 00:31:16,366 --> 00:31:18,620 That is my initial condition. 555 00:31:18,620 --> 00:31:26,352 When t increases, the solution at a particular x 556 00:31:26,352 --> 00:31:30,780 is the initial condition at a smaller x. 557 00:31:30,780 --> 00:31:32,780 Let's assume U is positive. 558 00:31:32,780 --> 00:31:35,910 If U is positive, then at a positive t, 559 00:31:35,910 --> 00:31:40,060 my solution at a certain x is equal to the initial condition 560 00:31:40,060 --> 00:31:42,115 at a smaller x. 561 00:31:42,115 --> 00:31:44,156 That make sense? 562 00:31:44,156 --> 00:31:49,110 At a particular time, my solution at one 563 00:31:49,110 --> 00:31:54,350 x is equal to the initial condition at a smaller x. 564 00:31:54,350 --> 00:32:00,250 And how much smaller it is really equals U times 565 00:32:00,250 --> 00:32:05,572 t [INAUDIBLE] is [INAUDIBLE] of how fast these lines go 566 00:32:05,572 --> 00:32:06,740 towards the right. 567 00:32:06,740 --> 00:32:08,710 And how fast those lines go towards the right 568 00:32:08,710 --> 00:32:11,770 is determined by U. It's determined 569 00:32:11,770 --> 00:32:15,870 by how fast the waves are convecting towards the right. 570 00:32:21,290 --> 00:32:25,410 Any questions on this equation? 571 00:32:25,410 --> 00:32:29,890 So this is good example of a partial difference equation. 572 00:32:29,890 --> 00:32:33,640 Because we know its analytic solution. 573 00:32:33,640 --> 00:32:37,130 And when we use a numerical MATLAB 574 00:32:37,130 --> 00:32:39,164 to solve the differential equation, 575 00:32:39,164 --> 00:32:42,410 we can compare against this analytic solution. 576 00:32:42,410 --> 00:32:45,270 It is like when we look at all these, 577 00:32:45,270 --> 00:32:50,220 we want to try our numerical methods on du/dt for the lambda 578 00:32:50,220 --> 00:32:56,340 U. Not because it is the most useful equation 579 00:32:56,340 --> 00:32:59,840 to solve-- I mean, it is useful but the reason 580 00:32:59,840 --> 00:33:02,800 we want numerical methods is because we want to apply this 581 00:33:02,800 --> 00:33:06,005 to more complex equations-- the reason we want 582 00:33:06,005 --> 00:33:08,750 numerical methods for PDEs is because we want 583 00:33:08,750 --> 00:33:12,600 to solve free-flow equations. 584 00:33:12,600 --> 00:33:17,440 But in order to have our methods and see how our methods behave, 585 00:33:17,440 --> 00:33:21,932 to analyze what method is useful to apply our methods, 586 00:33:21,932 --> 00:33:24,480 we're not going to have equations like this. 587 00:33:24,480 --> 00:33:28,020 This one of the elementary types of differential 588 00:33:28,020 --> 00:33:30,974 equations-- the convection equation or advection equation. 589 00:33:35,614 --> 00:33:37,220 All right. 590 00:33:37,220 --> 00:33:42,260 And so the characteristic lines are x 591 00:33:42,260 --> 00:33:48,000 equals to some x zero plus U t. 592 00:33:48,000 --> 00:33:52,010 So this is the characteristic line. 593 00:33:54,750 --> 00:33:58,686 And we are going to also see similar characteristic lines 594 00:33:58,686 --> 00:34:01,650 later on. 595 00:34:01,650 --> 00:34:03,430 OK. 596 00:34:03,430 --> 00:34:09,120 Let me give you another example of conservation law. 597 00:34:09,120 --> 00:34:11,840 Now this time, the equation we are solving 598 00:34:11,840 --> 00:34:13,420 is no longer linear. 599 00:34:13,420 --> 00:34:16,070 It's called the Burgers equation. 600 00:34:16,070 --> 00:34:26,389 It is the equation people use to model fluid flows in the higher 601 00:34:26,389 --> 00:34:30,310 fidelity than the advection, than the linear advection 602 00:34:30,310 --> 00:34:31,179 equation. 603 00:34:31,179 --> 00:34:33,219 And one of the key features we're 604 00:34:33,219 --> 00:34:37,920 going to see in this equation is shock waves. 605 00:34:37,920 --> 00:34:42,474 Shock waves as we see in supersonic [INAUDIBLE] 606 00:34:42,474 --> 00:34:44,929 that develop shock waves. 607 00:34:44,929 --> 00:34:47,130 OK. 608 00:34:47,130 --> 00:34:51,190 So the bulk of the equation looks like this. 609 00:34:51,190 --> 00:34:57,960 Partial t plus u partial u-partial x equal to 0. 610 00:35:00,930 --> 00:35:05,760 It looks the same as what we did before. 611 00:35:05,760 --> 00:35:08,361 But there is a one key difference 612 00:35:08,361 --> 00:35:09,701 in what we did before. 613 00:35:13,390 --> 00:35:15,809 This U is a constant that has nothing 614 00:35:15,809 --> 00:35:16,850 to do with this solution. 615 00:35:19,650 --> 00:35:24,960 In both of the equations, this U was not really a solution. 616 00:35:24,960 --> 00:35:30,160 So the bulk of the equation had a quadratic [INAUDIBLE] 617 00:35:30,160 --> 00:35:32,340 because this one is a [INAUDIBLE] which 618 00:35:32,340 --> 00:35:34,330 will make U twice as large. 619 00:35:34,330 --> 00:35:37,635 That [INAUDIBLE] is going to be four times as large. 620 00:35:37,635 --> 00:35:40,330 All right? 621 00:35:40,330 --> 00:35:41,380 OK. 622 00:35:41,380 --> 00:35:43,865 Is this equation a conservation law at all? 623 00:35:46,720 --> 00:35:47,540 Yes. 624 00:35:47,540 --> 00:35:53,320 Can we write the equation into a form that looks like this? 625 00:35:53,320 --> 00:35:56,980 Remember in order to write in conservation form, 626 00:35:56,980 --> 00:35:58,800 we have to have an equation like this. 627 00:35:58,800 --> 00:36:00,190 Right? 628 00:36:00,190 --> 00:36:05,400 Can we write this equation in some form 629 00:36:05,400 --> 00:36:07,170 with somehow defined F? 630 00:36:11,280 --> 00:36:13,194 One half of U squared exactly. 631 00:36:15,871 --> 00:36:16,370 OK. 632 00:36:16,370 --> 00:36:22,284 Because if you take this and wrap this into the [INAUDIBLE] 633 00:36:22,284 --> 00:36:24,868 the derivative of U squared, it is going 634 00:36:24,868 --> 00:36:28,258 to be two times U times the u. 635 00:36:28,258 --> 00:36:32,090 And the [INAUDIBLE] is [INAUDIBLE] here. 636 00:36:32,090 --> 00:36:35,338 You get back to the first equation. 637 00:36:40,730 --> 00:36:41,800 OK. 638 00:36:41,800 --> 00:36:45,290 Now let's see how this equation behaves differently 639 00:36:45,290 --> 00:36:52,690 from the linear advection equation. 640 00:36:52,690 --> 00:36:55,006 Can I get somebody else to draw me another solution? 641 00:36:59,440 --> 00:37:00,160 Thank you. 642 00:37:03,744 --> 00:37:04,660 AUDIENCE: [INAUDIBLE]. 643 00:37:13,150 --> 00:37:26,242 QIQI WANG: Start on the left to right to the [INAUDIBLE] 644 00:37:26,242 --> 00:37:27,472 All right. 645 00:37:27,472 --> 00:37:28,620 Thank you. 646 00:37:28,620 --> 00:37:31,160 So let's see how this solution goes. 647 00:37:31,160 --> 00:37:34,430 And [INAUDIBLE]. 648 00:37:38,264 --> 00:37:39,180 AUDIENCE: [INAUDIBLE]. 649 00:37:41,932 --> 00:37:42,640 QIQI WANG: Sorry? 650 00:37:48,107 --> 00:37:49,023 AUDIENCE: [INAUDIBLE]. 651 00:37:52,825 --> 00:37:54,200 QIQI WANG: What do we see that is 652 00:37:54,200 --> 00:38:02,325 different from what we saw in the linear advection equation? 653 00:38:02,325 --> 00:38:04,462 AUDIENCE: [INAUDIBLE]. 654 00:38:04,462 --> 00:38:05,295 [INTERPOSING VOICES] 655 00:38:08,265 --> 00:38:09,255 QIQI WANG: Huh? 656 00:38:09,255 --> 00:38:10,633 AUDIENCE: The magnitude. 657 00:38:10,633 --> 00:38:12,299 QIQI WANG: The magnitude is [INAUDIBLE]. 658 00:38:12,299 --> 00:38:14,794 But [INAUDIBLE]. 659 00:38:14,794 --> 00:38:16,291 AUDIENCE: [INAUDIBLE]. 660 00:38:16,291 --> 00:38:18,716 QIQI WANG: The bottom is coming up a little bit. 661 00:38:18,716 --> 00:38:20,995 And the top is going down a little bit. 662 00:38:20,995 --> 00:38:21,494 Right? 663 00:38:21,494 --> 00:38:25,060 So there is something that [INAUDIBLE]. 664 00:38:25,060 --> 00:38:25,560 Right? 665 00:38:25,560 --> 00:38:27,420 This is one of the things we didn't 666 00:38:27,420 --> 00:38:30,410 see in the previous equation. 667 00:38:30,410 --> 00:38:34,900 We have developed a [INAUDIBLE] right? 668 00:38:34,900 --> 00:38:48,722 And [INAUDIBLE] What else do we see? 669 00:38:48,722 --> 00:38:52,753 Does the wave go towards the right at a uniform speed? 670 00:38:57,190 --> 00:38:59,683 [INTERPOSING VOICES] 671 00:38:59,683 --> 00:39:00,641 QIQI WANG: [INAUDIBLE]. 672 00:39:03,920 --> 00:39:06,830 Normally we have something that moves toward the right, we 673 00:39:06,830 --> 00:39:10,242 [INAUDIBLE] the solution has been [INAUDIBLE]. 674 00:39:14,960 --> 00:39:15,460 OK. 675 00:39:15,460 --> 00:39:18,606 And if we go back to this one, looking 676 00:39:18,606 --> 00:39:23,014 at the solution of the function of space and time, 677 00:39:23,014 --> 00:39:25,690 we can see that we no longer have 678 00:39:25,690 --> 00:39:31,050 parallel characteristic lines. 679 00:39:31,050 --> 00:39:35,002 But we still have these characteristic lines. 680 00:39:35,002 --> 00:39:36,930 Right? 681 00:39:36,930 --> 00:39:41,690 And there you will see other things [INAUDIBLE]. 682 00:39:41,690 --> 00:39:44,202 This is initially not a shock wave, 683 00:39:44,202 --> 00:39:45,660 but now it has become a shock wave. 684 00:39:45,660 --> 00:39:46,993 A shock wave is not [INAUDIBLE]. 685 00:39:49,770 --> 00:39:53,186 All the characteristic lines that are not shock waves 686 00:39:53,186 --> 00:39:53,686 [INAUDIBLE]. 687 00:39:58,715 --> 00:40:04,050 [INAUDIBLE] characteristics are [INAUDIBLE]. 688 00:40:04,050 --> 00:40:05,410 All right. 689 00:40:05,410 --> 00:40:06,346 So OK. 690 00:40:06,346 --> 00:40:09,000 Let's let the animation go. 691 00:40:09,000 --> 00:40:11,700 And let's take a five minute break. 692 00:40:11,700 --> 00:40:17,160 And after the break, we're going to be looking at a more, 693 00:40:17,160 --> 00:40:20,335 we're going to be sort of animating these solutions all 694 00:40:20,335 --> 00:40:21,690 ourselves. 695 00:40:21,690 --> 00:40:26,620 And we'll see what we get in terms of the solution. 696 00:40:26,620 --> 00:40:27,760 OK. 697 00:40:27,760 --> 00:40:31,860 So let's-- we have seen some solutions of partial 698 00:40:31,860 --> 00:40:33,690 differential equations. 699 00:40:33,690 --> 00:40:34,190 All right. 700 00:40:34,190 --> 00:40:36,590 So right now what I'd like you to do 701 00:40:36,590 --> 00:40:41,100 is have everybody come to the front of the classroom. 702 00:40:41,100 --> 00:40:43,940 And we are actually going to be emulating the solution 703 00:40:43,940 --> 00:40:45,478 of the differential equation. 704 00:40:45,478 --> 00:40:46,972 AUDIENCE: [INAUDIBLE]. 705 00:40:46,972 --> 00:40:49,462 [LAUGHTER] 706 00:40:49,462 --> 00:40:52,948 [INTERPOSING VOICES] 707 00:41:08,031 --> 00:41:13,185 [INTERPOSING VOICES] 708 00:41:21,332 --> 00:41:21,915 QIQI WANG: OK. 709 00:41:21,915 --> 00:41:26,200 Let's stand so that the two doors are boundaries to this. 710 00:41:26,200 --> 00:41:28,451 So let's [INAUDIBLE]. 711 00:41:28,451 --> 00:41:31,193 AUDIENCE: [INAUDIBLE]. 712 00:41:31,193 --> 00:41:31,818 QIQI WANG: Yes. 713 00:41:31,818 --> 00:41:32,780 Sure. 714 00:41:32,780 --> 00:41:35,233 Can somebody there just push the off button? 715 00:41:35,233 --> 00:41:36,320 All right. 716 00:41:36,320 --> 00:41:36,820 OK. 717 00:41:36,820 --> 00:41:40,690 So let's [INAUDIBLE] ourselves a little bit? 718 00:41:40,690 --> 00:41:45,140 [INAUDIBLE] so that we can go in and out of the door. 719 00:41:45,140 --> 00:41:49,595 [INTERPOSING VOICES] 720 00:41:59,990 --> 00:42:01,300 QIQI WANG: Let me just have-- 721 00:42:01,300 --> 00:42:02,850 AUDIENCE: [INAUDIBLE]. 722 00:42:02,850 --> 00:42:03,850 QIQI WANG: Just in case. 723 00:42:03,850 --> 00:42:05,600 AUDIENCE: We're worried about [INAUDIBLE]. 724 00:42:05,600 --> 00:42:06,530 AUDIENCE: Yeah. 725 00:42:06,530 --> 00:42:08,030 [LAUGHTER] 726 00:42:08,030 --> 00:42:09,688 QIQI WANG: Good point. 727 00:42:09,688 --> 00:42:11,624 [INTERPOSING VOICES] 728 00:42:14,913 --> 00:42:15,496 QIQI WANG: OK. 729 00:42:18,240 --> 00:42:19,310 So let's [INAUDIBLE]. 730 00:42:19,310 --> 00:42:25,737 So let me get one student here that's going to be an observer. 731 00:42:25,737 --> 00:42:27,168 OK. 732 00:42:27,168 --> 00:42:29,053 [LAUGHTER] 733 00:42:29,053 --> 00:42:29,553 OK. 734 00:42:29,553 --> 00:42:32,994 So if you stand over here, you can really [INAUDIBLE]. 735 00:42:32,994 --> 00:42:40,022 Let's say the height [INAUDIBLE] is the solution of my PDE. 736 00:42:40,022 --> 00:42:41,014 OK. 737 00:42:41,014 --> 00:42:42,502 [LAUGHTER] 738 00:42:42,502 --> 00:42:45,237 [INTERPOSING VOICES] 739 00:42:45,237 --> 00:42:45,820 QIQI WANG: OK. 740 00:42:45,820 --> 00:42:48,500 So I have a distribution like this. 741 00:42:48,500 --> 00:42:49,880 This is my solution. 742 00:42:49,880 --> 00:42:50,900 [INAUDIBLE] space. 743 00:42:50,900 --> 00:42:52,050 Right? 744 00:42:52,050 --> 00:42:54,810 And imagine I have this [INAUDIBLE]. 745 00:42:58,980 --> 00:43:03,910 So what I want to do is I want to simulate 746 00:43:03,910 --> 00:43:07,890 the solution of an advective differential equation. 747 00:43:07,890 --> 00:43:11,510 So when I say I'm going to the next concept, 748 00:43:11,510 --> 00:43:14,052 let's think of-- we won't move all at one time. 749 00:43:14,052 --> 00:43:15,465 Right? 750 00:43:15,465 --> 00:43:17,385 Let's just move our position. 751 00:43:17,385 --> 00:43:20,940 What do you think we should do when we evolve to the next time 752 00:43:20,940 --> 00:43:22,290 slot? 753 00:43:22,290 --> 00:43:23,629 [INTERPOSING VOICES] 754 00:43:23,629 --> 00:43:24,670 QIQI WANG: Move that way. 755 00:43:24,670 --> 00:43:27,050 Somebody has to be emulate the periodic-- 756 00:43:27,050 --> 00:43:29,330 [LAUGHTER] 757 00:43:29,330 --> 00:43:30,850 When I go, let's go. 758 00:43:30,850 --> 00:43:31,940 Go. 759 00:43:31,940 --> 00:43:32,440 One step. 760 00:43:32,440 --> 00:43:33,542 OK. 761 00:43:33,542 --> 00:43:35,584 Step one step. 762 00:43:35,584 --> 00:43:36,921 Oh it doesn't [INAUDIBLE]. 763 00:43:36,921 --> 00:43:37,420 OK. 764 00:43:37,420 --> 00:43:38,981 Two steps. 765 00:43:38,981 --> 00:43:39,480 OK. 766 00:43:39,480 --> 00:43:41,520 So let's go there for two more steps. 767 00:43:46,660 --> 00:43:47,160 All right. 768 00:43:47,160 --> 00:43:48,570 [LAUGHTER] 769 00:43:48,570 --> 00:43:50,400 OK. 770 00:43:50,400 --> 00:43:52,700 Now it's a perfectly good solution 771 00:43:52,700 --> 00:43:56,390 because everybody moves there at a constant speed, right? 772 00:43:56,390 --> 00:43:59,510 Let's stop for a moment. 773 00:43:59,510 --> 00:44:04,040 Let's think of emulating the solution of a Burgers equation. 774 00:44:04,040 --> 00:44:05,540 [INTERPOSING VOICES] 775 00:44:05,540 --> 00:44:07,956 AUDIENCE: Does anybody feel like they're on a [INAUDIBLE]? 776 00:44:07,956 --> 00:44:09,394 [LAUGHTER] 777 00:44:09,394 --> 00:44:11,560 QIQI WANG: So we haven't talked about diffusion yet. 778 00:44:11,560 --> 00:44:13,660 We're going to be talking about that later. 779 00:44:13,660 --> 00:44:15,550 But in Burgers equation, what happens 780 00:44:15,550 --> 00:44:19,660 is that when we look at the characteristic lines-- 781 00:44:19,660 --> 00:44:23,720 [INAUDIBLE] actually following the characteristic lines. 782 00:44:23,720 --> 00:44:26,960 Because your height is going to stay constant, right? 783 00:44:26,960 --> 00:44:29,074 As long as [INAUDIBLE] your height 784 00:44:29,074 --> 00:44:30,540 is going to stay constant. 785 00:44:30,540 --> 00:44:38,730 Now what is different is that in the purely advective equation, 786 00:44:38,730 --> 00:44:41,450 the characteristic lines stay parallel. 787 00:44:41,450 --> 00:44:43,400 That means all of you-- no matter 788 00:44:43,400 --> 00:44:48,080 of your height or no matter what solution value you are at-- 789 00:44:48,080 --> 00:44:51,545 are going to be moving at the same speed. 790 00:44:51,545 --> 00:44:52,880 Right? 791 00:44:52,880 --> 00:44:55,220 In a Burgers equation, what is difference? 792 00:44:55,220 --> 00:44:59,530 The solution-- remember we have U times du/dt. 793 00:44:59,530 --> 00:45:03,140 And the U in front of the du/dt is no longer constant. 794 00:45:03,140 --> 00:45:06,830 It is actually the solution itself. 795 00:45:06,830 --> 00:45:08,490 So what does it mean? 796 00:45:08,490 --> 00:45:11,450 How fast should you move when you are emulating 797 00:45:11,450 --> 00:45:13,588 the [INAUDIBLE] equation? 798 00:45:13,588 --> 00:45:14,712 AUDIENCE: Four [INAUDIBLE]? 799 00:45:17,980 --> 00:45:18,796 QIQI WANG: Exactly. 800 00:45:18,796 --> 00:45:20,420 The [INAUDIBLE] are going to be moving. 801 00:45:20,420 --> 00:45:23,652 It's going to be proportional to the solution itself. 802 00:45:23,652 --> 00:45:26,330 So in other words, if you're [INAUDIBLE] for the solution, 803 00:45:26,330 --> 00:45:29,125 then the speed you move is going to be actually proportional 804 00:45:29,125 --> 00:45:32,994 to how tall you are. 805 00:45:32,994 --> 00:45:34,840 [LAUGHTER] 806 00:45:34,840 --> 00:45:36,060 OK. 807 00:45:36,060 --> 00:45:36,860 So let me do this. 808 00:45:36,860 --> 00:45:39,900 When I say-- when I say step, let's 809 00:45:39,900 --> 00:45:41,910 just move for a single step. 810 00:45:41,910 --> 00:45:49,770 And let's say the speed, the step size we're going to take, 811 00:45:49,770 --> 00:45:54,140 let's say, is-- let's estimate-- is like a third of your height. 812 00:45:54,140 --> 00:45:55,348 Let's try to see if you can-- 813 00:45:55,348 --> 00:45:55,889 AUDIENCE: OK. 814 00:45:55,889 --> 00:45:59,180 --estimate how to do that. 815 00:45:59,180 --> 00:46:01,760 And whoever goes out of the door has to come back 816 00:46:01,760 --> 00:46:04,442 as a periodic [INAUDIBLE]. 817 00:46:04,442 --> 00:46:06,147 [INTERPOSING VOICES] 818 00:46:06,147 --> 00:46:06,730 QIQI WANG: OK. 819 00:46:06,730 --> 00:46:09,424 Everybody ready? 820 00:46:09,424 --> 00:46:10,380 One step. 821 00:46:13,547 --> 00:46:14,380 [INTERPOSING VOICES] 822 00:46:17,540 --> 00:46:18,456 QIQI WANG: OK. 823 00:46:18,456 --> 00:46:19,944 [INAUDIBLE] 824 00:46:19,944 --> 00:46:22,391 [LAUGHTER] 825 00:46:22,391 --> 00:46:22,890 OK. 826 00:46:22,890 --> 00:46:25,760 Another step. 827 00:46:25,760 --> 00:46:29,068 [LAUGHTER] 828 00:46:30,280 --> 00:46:31,612 OK. 829 00:46:31,612 --> 00:46:32,153 AUDIENCE: Um? 830 00:46:32,153 --> 00:46:33,075 QIQI WANG: Yeah? 831 00:46:33,075 --> 00:46:34,920 AUDIENCE: Can characteristic lines cross each other? 832 00:46:34,920 --> 00:46:36,360 Or when they're right in front of each other? 833 00:46:36,360 --> 00:46:36,760 QIQI WANG: OK. 834 00:46:36,760 --> 00:46:37,968 We have a good question here. 835 00:46:37,968 --> 00:46:40,670 Can characteristic lines cross each other? 836 00:46:40,670 --> 00:46:42,140 Because look at here. 837 00:46:42,140 --> 00:46:43,702 What happened here? 838 00:46:43,702 --> 00:46:46,490 [LAUGHTER] 839 00:46:46,490 --> 00:46:48,670 So what is the physical phenomenon that 840 00:46:48,670 --> 00:46:50,181 is happening at this point? 841 00:46:50,181 --> 00:46:51,014 [INTERPOSING VOICES] 842 00:46:55,684 --> 00:46:56,620 QIQI WANG: OK. 843 00:46:56,620 --> 00:46:59,201 Well what physical phenomenon is happening here? 844 00:46:59,201 --> 00:47:00,492 AUDIENCE: We have a shock wave. 845 00:47:00,492 --> 00:47:01,960 [LAUGHTER] 846 00:47:01,960 --> 00:47:03,930 QIQI WANG: We have a shock wave. 847 00:47:03,930 --> 00:47:04,510 Right? 848 00:47:04,510 --> 00:47:06,770 We have a shock wave. 849 00:47:06,770 --> 00:47:12,010 Because its characteristics in front is moving slower than 850 00:47:12,010 --> 00:47:14,370 the characteristics at the back. 851 00:47:14,370 --> 00:47:17,350 So instead the molecules at the back catch up. 852 00:47:17,350 --> 00:47:21,270 And you get a shock wave forming. 853 00:47:21,270 --> 00:47:25,610 So actually once you have a shock wave forming, [INAUDIBLE] 854 00:47:25,610 --> 00:47:27,852 you all should move at the same time. 855 00:47:27,852 --> 00:47:29,710 You also move at the same speed that 856 00:47:29,710 --> 00:47:32,710 is equal to the average of your speed. 857 00:47:32,710 --> 00:47:33,445 OK. 858 00:47:33,445 --> 00:47:34,616 So another step. 859 00:47:34,616 --> 00:47:37,592 [INTERPOSING VOICES] 860 00:47:41,560 --> 00:47:45,361 QIQI WANG: Oh there is another shock wave forming over here. 861 00:47:45,361 --> 00:47:45,860 OK. 862 00:47:45,860 --> 00:47:49,040 Another step. 863 00:47:49,040 --> 00:47:51,347 [INTERPOSING VOICES] 864 00:47:51,347 --> 00:47:51,930 QIQI WANG: OK. 865 00:47:51,930 --> 00:47:52,720 Another step. 866 00:47:56,665 --> 00:47:59,520 Another step. 867 00:47:59,520 --> 00:48:01,355 [INTERPOSING VOICES] 868 00:48:01,355 --> 00:48:02,230 QIQI WANG: All right. 869 00:48:02,230 --> 00:48:04,665 So we have a lot of shock waves, right? 870 00:48:04,665 --> 00:48:05,540 A lot of shock waves. 871 00:48:05,540 --> 00:48:07,795 And we actually get either shock waves or a solution 872 00:48:07,795 --> 00:48:10,370 that is very smooth, right? 873 00:48:10,370 --> 00:48:13,190 So we end up getting a solution not unlike what we 874 00:48:13,190 --> 00:48:16,620 saw previously on the screen. 875 00:48:16,620 --> 00:48:22,000 So when the characteristic in front moves back up, 876 00:48:22,000 --> 00:48:23,420 they spread out. 877 00:48:23,420 --> 00:48:26,570 And the result of the solution becomes smooth. 878 00:48:26,570 --> 00:48:28,443 That is like an anti-shock wave. 879 00:48:28,443 --> 00:48:31,596 It's a verification wave. 880 00:48:31,596 --> 00:48:34,756 It helps when you have supersonic expansion. 881 00:48:34,756 --> 00:48:36,660 [INAUDIBLE] smoother. 882 00:48:36,660 --> 00:48:40,150 On the other hand, if the solution at the back is faster, 883 00:48:40,150 --> 00:48:41,250 then it catches up. 884 00:48:41,250 --> 00:48:45,320 And the characteristic lines converse each other. 885 00:48:45,320 --> 00:48:47,860 And [INAUDIBLE] form these continuities 886 00:48:47,860 --> 00:48:51,610 which is shock waves. 887 00:48:51,610 --> 00:48:53,030 OK. 888 00:48:53,030 --> 00:48:53,530 OK. 889 00:48:53,530 --> 00:48:54,821 Do we want to move more? 890 00:48:54,821 --> 00:48:55,320 Or-- 891 00:48:55,320 --> 00:48:56,250 [LAUGHTER] 892 00:48:56,250 --> 00:48:59,598 Or do we understand how the differential equations behave? 893 00:48:59,598 --> 00:49:01,277 [INTERPOSING VOICES] 894 00:49:01,277 --> 00:49:01,860 QIQI WANG: OK. 895 00:49:01,860 --> 00:49:02,360 Propagate. 896 00:49:02,360 --> 00:49:03,837 Let's go-- 897 00:49:03,837 --> 00:49:04,670 [INTERPOSING VOICES] 898 00:49:10,430 --> 00:49:11,400 QIQI WANG: Traffic jam. 899 00:49:11,400 --> 00:49:11,900 OK. 900 00:49:11,900 --> 00:49:12,450 That's fine. 901 00:49:12,450 --> 00:49:14,426 That's good enough. 902 00:49:14,426 --> 00:49:18,637 Let's go back to our seats. 903 00:49:18,637 --> 00:49:23,547 [INTERPOSING VOICES] 904 00:49:55,840 --> 00:49:58,560 QIQI WANG: And interestingly the bulk of the equation 905 00:49:58,560 --> 00:50:01,380 looks something really similar to the Burgers equation. 906 00:50:03,900 --> 00:50:06,760 If we use the model of the flow of cars on highways, 907 00:50:06,760 --> 00:50:14,686 or freeways, so imagine the lots of cars-- [INAUDIBLE]. 908 00:50:14,686 --> 00:50:16,130 Cars are conserved, right? 909 00:50:16,130 --> 00:50:18,240 Think of a highway that has no entrance and exit. 910 00:50:18,240 --> 00:50:20,892 The The cars are conserved. 911 00:50:20,892 --> 00:50:24,850 No car disappears, no car is generated like spontaneously. 912 00:50:24,850 --> 00:50:26,600 So the number of cars is conserved. 913 00:50:26,600 --> 00:50:28,500 And the flux, which is proportional 914 00:50:28,500 --> 00:50:31,790 to the density of the cars and also proportional 915 00:50:31,790 --> 00:50:35,317 to the speed of the cars-- and the speed of the cars 916 00:50:35,317 --> 00:50:37,025 can usually depend on the density, right? 917 00:50:37,025 --> 00:50:39,740 If you have really a lot of traffic, the cars move slowly. 918 00:50:39,740 --> 00:50:41,336 If you have not a lot of traffic, 919 00:50:41,336 --> 00:50:46,720 they will move at slightly faster than the speed limit. 920 00:50:46,720 --> 00:50:49,240 So really the flux depends on density. 921 00:50:49,240 --> 00:50:52,360 So you really have conservation laws. 922 00:50:52,360 --> 00:50:55,761 And you have behavior similar to this. 923 00:50:55,761 --> 00:50:59,610 But in the case of traffic flow, the shock waves 924 00:50:59,610 --> 00:51:01,510 actually move backwards. 925 00:51:01,510 --> 00:51:04,810 So think you have a red light, all the cars 926 00:51:04,810 --> 00:51:06,840 kind of bumping into each other and stopping. 927 00:51:06,840 --> 00:51:09,090 We have a shock wave propagating backwards. 928 00:51:09,090 --> 00:51:14,110 So this kind of equation is not exactly the Burgers equation 929 00:51:14,110 --> 00:51:17,380 but something pretty similar can model 930 00:51:17,380 --> 00:51:22,960 the evolution of the density of cars on a highway. 931 00:51:22,960 --> 00:51:27,020 And maybe we can take a look at the Burgers equation again. 932 00:51:27,020 --> 00:51:30,621 And who was the observer? 933 00:51:30,621 --> 00:51:31,120 OK. 934 00:51:31,120 --> 00:51:31,730 You observed. 935 00:51:31,730 --> 00:51:34,313 Can you tell us what [INAUDIBLE] what the initial distribution 936 00:51:34,313 --> 00:51:35,768 of the height was? 937 00:51:35,768 --> 00:51:38,140 [LAUGHTER] 938 00:51:38,140 --> 00:51:41,210 And we can kind of a look at the solution evolution 939 00:51:41,210 --> 00:51:46,638 again, just try to recall where you were. 940 00:51:46,638 --> 00:51:50,702 And was the solution kind of reproducing 941 00:51:50,702 --> 00:51:54,040 the kind of shock wave [INAUDIBLE] a little bit 942 00:51:54,040 --> 00:51:56,490 as well? 943 00:51:56,490 --> 00:51:59,430 AUDIENCE: OK. 944 00:51:59,430 --> 00:52:03,250 There was a group of tall people over here. 945 00:52:03,250 --> 00:52:04,850 And some shorter ones. 946 00:52:04,850 --> 00:52:06,350 A few tall ones. 947 00:52:06,350 --> 00:52:07,559 [LAUGHTER] 948 00:52:07,559 --> 00:52:08,350 A few shorter ones. 949 00:52:11,610 --> 00:52:14,670 QIQI WANG: All right. 950 00:52:14,670 --> 00:52:18,654 [CHATTER] 951 00:52:24,104 --> 00:52:24,770 QIQI WANG: Yeah. 952 00:52:24,770 --> 00:52:29,469 I see kind of a-- these [INAUDIBLE] 953 00:52:29,469 --> 00:52:31,010 performed shock waves really quickly. 954 00:52:37,330 --> 00:52:39,670 Oh, these I don't have [INAUDIBLE]. 955 00:52:39,670 --> 00:52:40,799 Sorry. 956 00:52:40,799 --> 00:52:42,674 So there are those people coming to the door. 957 00:52:42,674 --> 00:52:44,630 I think those are [INAUDIBLE]. 958 00:52:44,630 --> 00:52:48,334 [LAUGHTER] 959 00:52:48,334 --> 00:52:49,682 So yeah. 960 00:52:49,682 --> 00:52:51,610 [INAUDIBLE] 961 00:52:51,610 --> 00:52:55,160 There is the replication wave over here. 962 00:52:55,160 --> 00:52:56,829 And you can see the [INAUDIBLE] shock 963 00:52:56,829 --> 00:52:58,120 waves already formed over here. 964 00:52:58,120 --> 00:53:00,620 So there are already people [INAUDIBLE]. 965 00:53:00,620 --> 00:53:04,810 And here because there is a continuous [INAUDIBLE] 966 00:53:04,810 --> 00:53:11,600 over here, you can already see the [INAUDIBLE] are converging. 967 00:53:11,600 --> 00:53:13,020 Initially they just spread out. 968 00:53:13,020 --> 00:53:15,000 Now they are closer with each other. 969 00:53:15,000 --> 00:53:25,950 And [INAUDIBLE] that have already formed. 970 00:53:30,710 --> 00:53:31,210 OK. 971 00:53:31,210 --> 00:53:36,599 Let's wait until this shock wave forms [INAUDIBLE]. 972 00:53:36,599 --> 00:53:38,140 AUDIENCE: I think that's [INAUDIBLE]. 973 00:53:41,996 --> 00:53:43,940 QIQI WANG: OK. 974 00:53:43,940 --> 00:53:48,020 So let's see the response and we can move onto the diffusion 975 00:53:48,020 --> 00:53:48,520 equations. 976 00:53:55,870 --> 00:53:56,860 Yeah. 977 00:53:56,860 --> 00:54:00,460 The diffusion equation, I think, is much more difficult to act. 978 00:54:00,460 --> 00:54:02,280 Because we can't really diffuse our height 979 00:54:02,280 --> 00:54:04,826 into people around you. 980 00:54:04,826 --> 00:54:06,960 [LAUGHTER] 981 00:54:06,960 --> 00:54:11,330 So if you think of a way to do that, I'll do it next year. 982 00:54:11,330 --> 00:54:12,690 [LAUGHTER] 983 00:54:12,690 --> 00:54:13,190 OK. 984 00:54:13,190 --> 00:54:21,030 Now the shock waves are converging each other. 985 00:54:21,030 --> 00:54:24,090 And now these three guys are moving at the same speed 986 00:54:24,090 --> 00:54:26,281 as each other. 987 00:54:26,281 --> 00:54:26,780 OK. 988 00:54:30,508 --> 00:54:31,910 All right. 989 00:54:31,910 --> 00:54:34,404 And sorry for not having periodic on the conditions. 990 00:54:42,580 --> 00:54:46,410 So now let's look at the Burgers equation. 991 00:54:46,410 --> 00:54:49,240 The Burgers equation has characteristics 992 00:54:49,240 --> 00:54:55,540 that are x equal to some x zero plus my U times t. 993 00:54:55,540 --> 00:55:02,320 And U is the local solution that stays the same 994 00:55:02,320 --> 00:55:05,330 along the characteristic. 995 00:55:05,330 --> 00:55:06,300 OK. 996 00:55:06,300 --> 00:55:09,160 Why is it like this? 997 00:55:09,160 --> 00:55:11,540 Because if you look at the derivative 998 00:55:11,540 --> 00:55:16,602 of the solution along the characteristic line, 999 00:55:16,602 --> 00:55:19,950 so du/dt-- let's look at this. 1000 00:55:19,950 --> 00:55:25,120 Let's look at the solution that we have like this, right? 1001 00:55:25,120 --> 00:55:28,940 Let's look at the derivative of the solution 1002 00:55:28,940 --> 00:55:32,910 along this characteristic line. 1003 00:55:32,910 --> 00:55:35,620 So we're looking at du/dt. 1004 00:55:35,620 --> 00:55:38,980 Notice I'm using the total derivative because I'm 1005 00:55:38,980 --> 00:55:42,640 looking at the solution along the characteristic 1006 00:55:42,640 --> 00:55:50,410 which is at spatial location x zero plus Ut and at time t. 1007 00:55:50,410 --> 00:55:53,998 So I'm allowing my space and time 1008 00:55:53,998 --> 00:55:58,360 to change together as t increases. 1009 00:55:58,360 --> 00:56:01,476 I'm not fixing the space and taking the derivative 1010 00:56:01,476 --> 00:56:02,277 through time. 1011 00:56:02,277 --> 00:56:03,860 I'm taking the derivative through time 1012 00:56:03,860 --> 00:56:08,740 while the space depends on time as I go along this curve. 1013 00:56:08,740 --> 00:56:12,260 So this derivative is like taking 1014 00:56:12,260 --> 00:56:16,290 the derivative of yourself as you move in, right? 1015 00:56:16,290 --> 00:56:22,920 And it's like taking the time derivative of a moving 1016 00:56:22,920 --> 00:56:25,600 point in the solution. 1017 00:56:25,600 --> 00:56:29,220 So now what is it? 1018 00:56:29,220 --> 00:56:30,678 Anybody remember the chain rule? 1019 00:56:35,070 --> 00:56:40,164 How do you apply the chain rule to expand the total derivative 1020 00:56:40,164 --> 00:56:41,330 into the partial derivative? 1021 00:56:45,875 --> 00:56:48,210 Sorry? 1022 00:56:48,210 --> 00:56:50,280 x naught is a constant, yes. 1023 00:56:50,280 --> 00:56:56,080 And looking at one particular characteristic. 1024 00:56:59,492 --> 00:56:59,992 Hm? 1025 00:57:03,540 --> 00:57:04,456 AUDIENCE: [INAUDIBLE]. 1026 00:57:14,872 --> 00:57:17,352 QIQI WANG: Uh huh. 1027 00:57:17,352 --> 00:57:17,860 Du/dx? 1028 00:57:20,540 --> 00:57:21,400 dx dt. 1029 00:57:21,400 --> 00:57:23,280 And here my is x is this. 1030 00:57:23,280 --> 00:57:27,190 So dx dt is U, right? 1031 00:57:27,190 --> 00:57:31,100 So this is equal to U. And plus partial u-partial t. 1032 00:57:31,100 --> 00:57:34,730 Because this is also a function of time. 1033 00:57:34,730 --> 00:57:36,800 OK. 1034 00:57:36,800 --> 00:57:41,120 Now if you look at this, du dt, du dt. 1035 00:57:44,520 --> 00:57:50,050 du dx times U. U times du dx. 1036 00:57:50,050 --> 00:57:53,870 So this is exactly equal to the left hand 1037 00:57:53,870 --> 00:57:55,906 side of the partial differential equation. 1038 00:57:55,906 --> 00:57:58,030 And according to the partial differential equation, 1039 00:57:58,030 --> 00:58:00,820 it is equal to zero, right? 1040 00:58:04,820 --> 00:58:05,850 What does this mean? 1041 00:58:05,850 --> 00:58:08,140 That means the derivative of this solution 1042 00:58:08,140 --> 00:58:11,400 as we go along the characteristic 1043 00:58:11,400 --> 00:58:12,372 is equal to zero. 1044 00:58:12,372 --> 00:58:13,830 Or the solution doesn't [INAUDIBLE] 1045 00:58:13,830 --> 00:58:15,662 along the characteristic. 1046 00:58:15,662 --> 00:58:21,810 That is exactly what we saw in the behavior of the solution 1047 00:58:21,810 --> 00:58:23,310 and exactly what we tried to emulate 1048 00:58:23,310 --> 00:58:27,610 as a group along the classroom. 1049 00:58:27,610 --> 00:58:32,060 And the behavior is derived from the analytic form 1050 00:58:32,060 --> 00:58:35,020 of the differential question. 1051 00:58:35,020 --> 00:58:36,730 And you can use the same math here 1052 00:58:36,730 --> 00:58:40,990 to derive it in the advection equation. 1053 00:58:40,990 --> 00:58:44,520 You can use the same math to derive similar things 1054 00:58:44,520 --> 00:58:47,180 for all non-linear conservation laws, 1055 00:58:47,180 --> 00:58:49,822 all scalar non-linear conservation laws. 1056 00:58:53,260 --> 00:58:55,550 You're going to get-- all of the characteristics 1057 00:58:55,550 --> 00:58:57,370 are going to be different. 1058 00:58:57,370 --> 00:59:02,390 The U is-- and you need to convert the conservation 1059 00:59:02,390 --> 00:59:04,250 law into something like this. 1060 00:59:04,250 --> 00:59:09,860 You have to say-- you have to take this out, which is really 1061 00:59:09,860 --> 00:59:13,380 partial df du, you have to take this out. 1062 00:59:13,380 --> 00:59:16,148 But [INAUDIBLE] as long as you have this df du, 1063 00:59:16,148 --> 00:59:18,470 df du is the speed of your characteristic line. 1064 00:59:18,470 --> 00:59:22,130 And you can derive the same fact that the solution doesn't 1065 00:59:22,130 --> 00:59:23,516 change along the characteristics. 1066 00:59:26,110 --> 00:59:26,610 OK. 1067 00:59:26,610 --> 00:59:28,500 So I only have a few minutes. 1068 00:59:28,500 --> 00:59:30,660 Let me do another example. 1069 00:59:30,660 --> 00:59:35,380 That is, what if we have advection and diffusion 1070 00:59:35,380 --> 00:59:36,340 at the same time? 1071 00:59:45,390 --> 00:59:49,640 So when I have advection and diffusion at the same time, 1072 00:59:49,640 --> 00:59:54,410 then I need to specify how much convection I have 1073 00:59:54,410 --> 00:59:58,170 and how much diffusion I have. 1074 00:59:58,170 --> 01:00:02,650 So let me start to build a case where I only have-- so 1075 01:00:02,650 --> 01:00:04,320 let me say, first of all, I only have 1076 01:00:04,320 --> 01:00:09,580 a case of-- so u is the coefficient on the convection. 1077 01:00:09,580 --> 01:00:14,340 This is the coefficient on the-- kappa 1078 01:00:14,340 --> 01:00:17,440 is the coefficient on the diffusion. 1079 01:00:17,440 --> 01:00:24,631 Let's take a large-- just one convection and one diffusion. 1080 01:00:24,631 --> 01:00:25,130 OK. 1081 01:00:25,130 --> 01:00:27,004 Can somebody help me draw initial conditions? 1082 01:00:31,901 --> 01:00:32,400 Come on. 1083 01:00:32,400 --> 01:00:36,250 We just get less than five minutes left. 1084 01:00:36,250 --> 01:00:36,750 All right. 1085 01:00:36,750 --> 01:00:37,250 Thanks. 1086 01:00:42,040 --> 01:00:44,415 AUDIENCE: Are the [INAUDIBLE] periodic or does it matter? 1087 01:00:44,415 --> 01:00:45,623 QIQI WANG: It doesn't matter. 1088 01:00:53,820 --> 01:00:55,305 All right. 1089 01:00:55,305 --> 01:00:57,285 Ooh. 1090 01:00:57,285 --> 01:00:57,790 OK. 1091 01:00:57,790 --> 01:01:01,470 Whether you draw a pretty [INAUDIBLE], 1092 01:01:01,470 --> 01:01:04,020 then some very [INAUDIBLE]. 1093 01:01:04,020 --> 01:01:07,055 That is the color of the diffusion. 1094 01:01:07,055 --> 01:01:07,555 OK. 1095 01:01:07,555 --> 01:01:12,671 So you see that kappa equal to one is very strong diffusion. 1096 01:01:12,671 --> 01:01:13,170 OK. 1097 01:01:13,170 --> 01:01:14,220 Thank you. 1098 01:01:14,220 --> 01:01:14,720 OK. 1099 01:01:14,720 --> 01:01:18,240 Let me try another one that is-- let 1100 01:01:18,240 --> 01:01:20,446 me make kappa equal to 0.01. 1101 01:01:20,446 --> 01:01:24,820 Can somebody come down to draw the last initial condition 1102 01:01:24,820 --> 01:01:28,410 before we go home? 1103 01:01:28,410 --> 01:01:28,910 Thank you. 1104 01:01:43,010 --> 01:01:43,590 All right. 1105 01:01:43,590 --> 01:01:46,560 So now because we have a much smaller kappa tendency, 1106 01:01:46,560 --> 01:01:47,990 the solution is smoothed out. 1107 01:01:47,990 --> 01:01:49,760 Right? 1108 01:01:49,760 --> 01:01:51,750 The initial discontinuity is down. 1109 01:01:51,750 --> 01:01:54,015 And the solution becomes smoother 1110 01:01:54,015 --> 01:01:56,341 and smoother and smoother. 1111 01:01:56,341 --> 01:01:58,180 At the same time, its [INAUDIBLE] 1112 01:01:58,180 --> 01:01:59,540 was the right periodically. 1113 01:02:02,120 --> 01:02:06,540 But because the coefficient is small, 1114 01:02:06,540 --> 01:02:09,646 the [INAUDIBLE] is pretty apparent. 1115 01:02:09,646 --> 01:02:11,700 But the rate is much smaller. 1116 01:02:11,700 --> 01:02:15,542 And the main impact is like [INAUDIBLE] 1117 01:02:15,542 --> 01:02:17,843 and there is a smaller diffusion, 1118 01:02:17,843 --> 01:02:21,091 but still it is visible. 1119 01:02:21,091 --> 01:02:21,590 All right. 1120 01:02:21,590 --> 01:02:28,061 So this is the behavior of the equation with [INAUDIBLE] flux. 1121 01:02:28,061 --> 01:02:29,467 All right? 1122 01:02:29,467 --> 01:02:30,050 Any questions? 1123 01:02:34,460 --> 01:02:35,440 No? 1124 01:02:35,440 --> 01:02:36,268 All right. 1125 01:02:36,268 --> 01:02:39,110 Then I will see you next Monday then. 1126 01:02:39,110 --> 01:02:40,589 Or I'll see you on Friday. 1127 01:02:40,589 --> 01:02:43,547 I'm going to announce the office hours. 1128 01:02:43,547 --> 01:02:45,519 Yeah? 1129 01:02:45,519 --> 01:02:48,477 [INTERPOSING VOICES] 1130 01:02:48,477 --> 01:02:50,784 AUDIENCE: [INAUDIBLE]. 1131 01:02:50,784 --> 01:02:51,450 QIQI WANG: Yeah. 1132 01:02:51,450 --> 01:02:54,070 The project now only has two questions 1133 01:02:54,070 --> 01:02:56,490 that is relating to ODEs. 1134 01:02:56,490 --> 01:02:58,920 And I'm going to add another question that 1135 01:02:58,920 --> 01:03:01,880 is going to be related to PDEs. 1136 01:03:01,880 --> 01:03:03,430 All right.