1 00:00:00,000 --> 00:00:01,458 INTRODUCTION: The following content 2 00:00:01,458 --> 00:00:04,420 is provided by MIT OpenCourseWare under a Creative 3 00:00:04,420 --> 00:00:06,090 Commons license. 4 00:00:06,090 --> 00:00:08,200 Additional information about our license 5 00:00:08,200 --> 00:00:10,490 and MIT OpenCourseWare in general 6 00:00:10,490 --> 00:00:11,930 is available at OCW.MIT.edu. 7 00:00:17,840 --> 00:00:18,880 PROFESSOR: OK. 8 00:00:18,880 --> 00:00:23,740 So I'll start now with the third lecture. 9 00:00:23,740 --> 00:00:26,600 And we're into partial differential equations. 10 00:00:26,600 --> 00:00:29,590 So we had two on ordinary differential equations, 11 00:00:29,590 --> 00:00:33,560 and the idea of stability came up, and the order of accuracy 12 00:00:33,560 --> 00:00:34,420 came up. 13 00:00:34,420 --> 00:00:39,160 And those will be the key questions today -- 14 00:00:39,160 --> 00:00:42,990 accuracy and stability. 15 00:00:42,990 --> 00:00:46,740 We do get beyond that, actually. 16 00:00:46,740 --> 00:00:54,030 I mean, to look at the contour, at the output from a difference 17 00:00:54,030 --> 00:00:54,530 method. 18 00:00:54,530 --> 00:00:58,390 Once you've decided, yes, it's stable, yes it's accurate, 19 00:00:58,390 --> 00:00:59,890 you still have to look at the output 20 00:00:59,890 --> 00:01:02,930 and see, you know, how good is it, where is it weak, 21 00:01:02,930 --> 00:01:06,440 where is it on target. 22 00:01:06,440 --> 00:01:11,830 But first come the accuracy and stability questions. 23 00:01:11,830 --> 00:01:15,620 And beginning with this equation. 24 00:01:15,620 --> 00:01:20,160 So this will be the simplest initial value 25 00:01:20,160 --> 00:01:23,950 problem we can think of. 26 00:01:23,950 --> 00:01:26,230 I call it a one-way wave equation. 27 00:01:26,230 --> 00:01:30,720 Notice the two-way wave equation, which we'll get too, 28 00:01:30,720 --> 00:01:33,890 would have second derivatives there. 29 00:01:33,890 --> 00:01:38,250 And that will send waves in both directions. 30 00:01:38,250 --> 00:01:41,350 This, we'll see, sends a wave only in one direction, 31 00:01:41,350 --> 00:01:43,530 so it's a nice scalar problem. 32 00:01:43,530 --> 00:01:46,100 First-order, constant coefficient, 33 00:01:46,100 --> 00:01:47,610 that's the velocity. 34 00:01:47,610 --> 00:01:49,670 Perfect. 35 00:01:49,670 --> 00:01:55,840 The model problem for wave equations. 36 00:01:59,440 --> 00:02:02,730 Because it's just first order, I just 37 00:02:02,730 --> 00:02:10,270 begin it with the function at time 0. 38 00:02:10,270 --> 00:02:16,050 So I'm looking for the solution at time t to this equation. 39 00:02:16,050 --> 00:02:17,270 OK. 40 00:02:17,270 --> 00:02:19,770 So that will not be hard to find. 41 00:02:19,770 --> 00:02:26,640 And I want to do it first for pure exponentials. 42 00:02:26,640 --> 00:02:30,230 When u of x_0, the initial function, 43 00:02:30,230 --> 00:02:32,650 is a pure exponential e to the i*k*x. 44 00:02:32,650 --> 00:02:40,070 Because it really pays, to -- Fourier is always around. 45 00:02:40,070 --> 00:02:46,380 And to see what happens, suppose this is given, for example -- 46 00:02:46,380 --> 00:02:51,390 so this will be my example, e to the i*k*x. 47 00:02:51,390 --> 00:02:52,020 OK. 48 00:02:52,020 --> 00:02:53,760 Now what's special about e to the i*k*x? 49 00:02:56,600 --> 00:02:59,830 The special thing is that with constant coefficients and no 50 00:02:59,830 --> 00:03:05,710 boundaries, the solution will be a multiple of e to the i*k*x. 51 00:03:05,710 --> 00:03:11,000 In other words, we can separate variables. 52 00:03:11,000 --> 00:03:12,340 Maybe I'll put it here. 53 00:03:12,340 --> 00:03:22,880 I can then look for the solution u to be a function times -- 54 00:03:22,880 --> 00:03:26,990 a number really, but it will depend on the time, 55 00:03:26,990 --> 00:03:34,430 on the frequency k, and the time t -- times e to the i*k*x. 56 00:03:34,430 --> 00:03:36,580 You see how that separated variables? 57 00:03:36,580 --> 00:03:44,960 x is separated from t, and the frequency controls what 58 00:03:44,960 --> 00:03:48,970 this growth factor will be. 59 00:03:48,970 --> 00:03:53,410 So this growth factors is a key quantity to compute. 60 00:03:53,410 --> 00:03:56,330 And we compute it just by substituting 61 00:03:56,330 --> 00:04:00,130 that hope for a solution into the equation. 62 00:04:00,130 --> 00:04:04,720 And getting an equation for G, because e to the i*k*x will 63 00:04:04,720 --> 00:04:05,250 cancel. 64 00:04:05,250 --> 00:04:08,950 That's the key point, that everything 65 00:04:08,950 --> 00:04:14,660 remains a pure harmonic with that single frequency k. 66 00:04:14,660 --> 00:04:21,710 So if I plug it in, I get dG/dt -- 67 00:04:21,710 --> 00:04:26,270 I'm taking the time derivative of u -- times e to the i*k*x, 68 00:04:26,270 --> 00:04:27,290 right? 69 00:04:27,290 --> 00:04:28,630 That's du/dt. 70 00:04:28,630 --> 00:04:33,100 Because t was separate from x, that's what 71 00:04:33,100 --> 00:04:35,400 I get when I plug in this u. 72 00:04:35,400 --> 00:04:38,380 So I'm always plugging in this u. 73 00:04:38,380 --> 00:04:42,810 What do I get on the right-hand side? c, the x derivative. 74 00:04:42,810 --> 00:04:47,240 So that's just G. The x derivative will bring down i*k, 75 00:04:47,240 --> 00:04:49,120 e to the i*k*x. 76 00:04:53,000 --> 00:04:58,010 No surprise that I can cancel e to the i*k*x, which is never 0. 77 00:04:58,010 --> 00:05:03,010 And I have a simple equation for G. Linear, 78 00:05:03,010 --> 00:05:05,650 constant coefficients, but the coefficient 79 00:05:05,650 --> 00:05:07,700 depends on k, of course. 80 00:05:07,700 --> 00:05:14,300 The coefficient is i*c*k, and of course the solution is G is 81 00:05:14,300 --> 00:05:19,860 equal to -- we're back to our simplest model of an ordinary 82 00:05:19,860 --> 00:05:21,760 differential equation. 83 00:05:21,760 --> 00:05:25,590 Now the coefficient a is now i*k*c, 84 00:05:25,590 --> 00:05:30,070 so it's e to the i*k*c*t, right? 85 00:05:32,910 --> 00:05:38,340 That shows us what happens to the e to the i*k*x. 86 00:05:38,340 --> 00:05:42,020 So now I'll put that here, because now we know it. 87 00:05:42,020 --> 00:05:47,790 So we have altogether e to the i*k*c*t, and here's an x. 88 00:05:47,790 --> 00:05:51,340 So let's put those together as x plus c*t, 89 00:05:51,340 --> 00:05:57,030 because that's a guide to what's coming. 90 00:05:57,030 --> 00:06:01,360 So that's the solution: separated variables, looked 91 00:06:01,360 --> 00:06:05,630 for the growth factor, tried an exponential. 92 00:06:05,630 --> 00:06:08,770 And we'll do exactly the same thing for difference method. 93 00:06:08,770 --> 00:06:11,870 So this was von Neumann's brilliant idea, 94 00:06:11,870 --> 00:06:15,590 watch exponentials, watch every frequency, 95 00:06:15,590 --> 00:06:20,850 and see what happens to the multiple of e to the i*k*x. 96 00:06:20,850 --> 00:06:25,950 Now here is very special, because all frequencies are 97 00:06:25,950 --> 00:06:31,600 producing this combination x plus c*t. 98 00:06:31,600 --> 00:06:36,670 So when we combine frequencies -- that's what Fourier said, 99 00:06:36,670 --> 00:06:40,920 take combinations of these things -- 100 00:06:40,920 --> 00:06:45,010 the solution will be a combination of these things. 101 00:06:45,010 --> 00:06:48,350 And what do we get? 102 00:06:48,350 --> 00:06:51,910 If our original function was u of x and 0, 103 00:06:51,910 --> 00:06:55,860 it's a combination of these, e to the i*k*x's. 104 00:06:55,860 --> 00:07:01,740 And at later time, we have the same combination, 105 00:07:01,740 --> 00:07:03,930 by linearity of these guys. 106 00:07:06,720 --> 00:07:12,560 All that's happening is x is getting changed to x plus c*t. 107 00:07:15,320 --> 00:07:17,300 So, I'm getting the answer. 108 00:07:21,120 --> 00:07:26,440 Here was the answer for one exponential. 109 00:07:26,440 --> 00:07:29,030 But now I'm going to do the answer for a general u, 110 00:07:29,030 --> 00:07:31,810 and it's going to be easy to pick off. 111 00:07:31,810 --> 00:07:36,080 It's just like picking off this x plus c*t. 112 00:07:36,080 --> 00:07:41,710 It will be u at x plus c*t at time 0. 113 00:07:41,710 --> 00:07:44,950 That's the solution. 114 00:07:44,950 --> 00:07:47,990 For every u, not just for the exponentials, 115 00:07:47,990 --> 00:07:49,960 but for all combinations of them, 116 00:07:49,960 --> 00:07:52,480 which give us any u of x and 0. 117 00:07:52,480 --> 00:07:56,535 So u of x and 0 was a combination of e 118 00:07:56,535 --> 00:08:01,350 to the i*k*x's, and then the solution is the same 119 00:08:01,350 --> 00:08:04,200 combination of these guys. 120 00:08:04,200 --> 00:08:10,420 Well, we can understand what that solution is. 121 00:08:10,420 --> 00:08:13,920 It's what I call a one-way wave. 122 00:08:13,920 --> 00:08:17,520 Let me understand what's happening here. 123 00:08:17,520 --> 00:08:19,480 So here's the key picture now. 124 00:08:25,370 --> 00:08:29,460 I want to a draw a picture in the xt-plane. 125 00:08:29,460 --> 00:08:31,960 So I'm not graphing u here. 126 00:08:31,960 --> 00:08:39,710 This is a picture in the xt-plane that will show us 127 00:08:39,710 --> 00:08:41,510 what this formula means. 128 00:08:41,510 --> 00:08:45,630 Because this is the solution that we then 129 00:08:45,630 --> 00:08:48,561 want to approximate by difference method. 130 00:08:48,561 --> 00:08:49,060 OK. 131 00:08:49,060 --> 00:08:51,490 So what does this mean? 132 00:08:51,490 --> 00:08:56,620 This means that the value, for example, u of 0, 0 -- 133 00:08:56,620 --> 00:08:57,990 so there's the point (0, 0). 134 00:09:05,830 --> 00:09:10,870 Here's what's happening, at a later time, say up here, 135 00:09:10,870 --> 00:09:14,210 some later time, the value of u -- 136 00:09:14,210 --> 00:09:21,820 this is the line x plus ct equals 0; 137 00:09:21,820 --> 00:09:25,480 x plus ct equals 0 along that line. 138 00:09:25,480 --> 00:09:27,510 And what's the point? 139 00:09:27,510 --> 00:09:32,300 On that line, the solution is the same all the way along. 140 00:09:32,300 --> 00:09:35,120 So whatever the initial value is here, 141 00:09:35,120 --> 00:09:42,150 it travels, so that u at this point is u at this point. 142 00:09:42,150 --> 00:09:47,020 So let me to call that point P, and say what I mean now. 143 00:09:47,020 --> 00:09:52,750 I mean that u of P is the same as u at 0, 0. 144 00:09:52,750 --> 00:10:00,790 Because x plus c*t is the same here and here. 145 00:10:00,790 --> 00:10:09,240 Let me take another point, let's say x_0, another initial value. 146 00:10:09,240 --> 00:10:12,610 Then, again, along the line like this -- 147 00:10:12,610 --> 00:10:17,880 this will be the line x plus c*t equals capital X, 148 00:10:17,880 --> 00:10:23,360 because it's the line where x plus c*t is a constant, 149 00:10:23,360 --> 00:10:30,340 and the constant is chosen so that it starts at this point, 150 00:10:30,340 --> 00:10:35,490 at t equals 0, X is big X. So now -- let's see -- 151 00:10:35,490 --> 00:10:42,570 I better make clear -- that this line carries this initial 152 00:10:42,570 --> 00:10:43,370 value. 153 00:10:43,370 --> 00:10:47,500 This line carries this initial value. 154 00:10:47,500 --> 00:10:48,810 You see it? 155 00:10:48,810 --> 00:10:51,730 The value of u all along this line 156 00:10:51,730 --> 00:10:54,950 is the value it started with right there. 157 00:10:54,950 --> 00:10:58,500 And you know the name for those lines? 158 00:10:58,500 --> 00:11:03,100 So these are lines, they happen to be straight here 159 00:11:03,100 --> 00:11:07,290 because we got constant coefficients, straight 160 00:11:07,290 --> 00:11:09,840 and even parallel. 161 00:11:09,840 --> 00:11:13,090 And they're the lines on which the information travels. 162 00:11:13,090 --> 00:11:17,640 Whatever the information was at times 0. 163 00:11:17,640 --> 00:11:20,460 By information, I mean the value of u 164 00:11:20,460 --> 00:11:25,120 at 0, 0, or the value of u at this point. 165 00:11:25,120 --> 00:11:27,520 That information travels along that line 166 00:11:27,520 --> 00:11:30,030 to give this solution all along the line. 167 00:11:30,030 --> 00:11:33,180 And the name of the line, anybody know it? 168 00:11:33,180 --> 00:11:34,820 Characteristic. 169 00:11:34,820 --> 00:11:45,005 So this is a characteristic line, they all are. 170 00:11:45,005 --> 00:11:45,660 This too. 171 00:11:45,660 --> 00:11:49,370 All these line are characteristic lines. 172 00:11:49,370 --> 00:11:50,655 Can you read characteristic? 173 00:11:50,655 --> 00:11:54,830 I should of have written it a little better. 174 00:11:54,830 --> 00:12:02,580 So this is like a key feature of wave equations. 175 00:12:02,580 --> 00:12:04,460 It's a key feature of wave equations 176 00:12:04,460 --> 00:12:09,200 that we will not see for heat equations. 177 00:12:09,200 --> 00:12:17,810 And so let me just say it again, because it will bear also 178 00:12:17,810 --> 00:12:21,430 on the stability question for difference equations. 179 00:12:21,430 --> 00:12:23,630 The information, the true information, 180 00:12:23,630 --> 00:12:27,410 for the true solution, travels along characteristic lines. 181 00:12:30,370 --> 00:12:33,480 We're in one dimension here, which 182 00:12:33,480 --> 00:12:36,590 is a major simplification. 183 00:12:36,590 --> 00:12:42,970 In two dimensions we'll have characteristic cones. 184 00:12:42,970 --> 00:12:46,450 In three dimension certainly we'd call it a cone. 185 00:12:46,450 --> 00:12:48,170 And actually, I guess what I'm saying 186 00:12:48,170 --> 00:12:54,540 is that if I speak a word, you know, 187 00:12:54,540 --> 00:13:00,270 my voice or the sound waves solve the 3D wave equation. 188 00:13:00,270 --> 00:13:04,710 So when I snap my fingers, that sound 189 00:13:04,710 --> 00:13:14,450 travels from this point out along characteristics. 190 00:13:17,670 --> 00:13:19,930 And of course it travels in all directions, 191 00:13:19,930 --> 00:13:25,250 so we've got a more interesting, more complicated picture 192 00:13:25,250 --> 00:13:28,220 in higher dimensions, but we really 193 00:13:28,220 --> 00:13:30,370 see it here in one dimension. 194 00:13:30,370 --> 00:13:31,630 OK. 195 00:13:31,630 --> 00:13:39,120 Now, I should also include a graph of the solution. 196 00:13:39,120 --> 00:13:41,000 What does the solution look like? 197 00:13:41,000 --> 00:13:46,010 Maybe I'll find a spot here for that graph. 198 00:13:49,210 --> 00:13:51,140 Now I'm going to graph u. 199 00:13:54,180 --> 00:13:56,270 This is still x. 200 00:13:56,270 --> 00:13:59,950 So I'll graph u at maybe two different times. 201 00:13:59,950 --> 00:14:01,650 So the starting time. 202 00:14:01,650 --> 00:14:04,280 Suppose we have a wall of water. 203 00:14:07,270 --> 00:14:14,710 So this will be u of x and 0, a step function, maybe at 0. 204 00:14:14,710 --> 00:14:20,970 So u of x and 0 is 1 and then 0. 205 00:14:20,970 --> 00:14:21,500 OK. 206 00:14:24,150 --> 00:14:29,060 So there's a typical starting initial value. 207 00:14:32,260 --> 00:14:34,220 Well, I don't know if it's typical. 208 00:14:34,220 --> 00:14:36,640 Pretty special to have 1 and then 0, 209 00:14:36,640 --> 00:14:38,650 a perfect step function. 210 00:14:38,650 --> 00:14:40,530 But this will illustrate -- 211 00:14:40,530 --> 00:14:43,660 Now I want to ask, what's the solution to the equation? 212 00:14:43,660 --> 00:14:47,600 What is u of x and t? 213 00:14:47,600 --> 00:14:50,370 I want to graph u of x and t. 214 00:14:50,370 --> 00:14:52,580 What what happens? 215 00:14:52,580 --> 00:14:57,640 You have to assemble, sort of, this picture, which 216 00:14:57,640 --> 00:15:03,910 is in the xt-plane, into a graph here of u of x and t. 217 00:15:03,910 --> 00:15:05,390 What happens? 218 00:15:05,390 --> 00:15:10,980 Well all these zero starting values travel along 219 00:15:10,980 --> 00:15:15,520 characteristics, and tell us that u is 0 along all 220 00:15:15,520 --> 00:15:16,450 these characteristics. 221 00:15:16,450 --> 00:15:20,650 So along all these characteristics, u is 0. 222 00:15:20,650 --> 00:15:23,740 This here, the starting value is 1. 223 00:15:23,740 --> 00:15:31,180 So along these characteristics we're starting with a 1, 224 00:15:31,180 --> 00:15:32,290 we end with a 1. 225 00:15:32,290 --> 00:15:33,780 Do you see what happens? 226 00:15:37,340 --> 00:15:38,580 How do I graph it? 227 00:15:38,580 --> 00:15:40,530 The wave is moving. 228 00:15:40,530 --> 00:15:44,930 The wall of water is moving that way, moving to the left. 229 00:15:44,930 --> 00:15:48,080 Because I'm taking c to be a positive number. 230 00:15:48,080 --> 00:15:54,690 So u of x and t is a wall, or a wave you could say, 231 00:15:54,690 --> 00:16:04,270 moving left with speed c. 232 00:16:06,780 --> 00:16:10,960 Every solution to the equation is doing this. 233 00:16:10,960 --> 00:16:16,050 And in this particular case it's a wall of water 234 00:16:16,050 --> 00:16:18,290 and it stays a wall of water. 235 00:16:18,290 --> 00:16:19,350 That the point. 236 00:16:19,350 --> 00:16:22,370 Shapes are not changed as we go. 237 00:16:22,370 --> 00:16:29,370 There's no -- dispersion would be the right word, I think. 238 00:16:29,370 --> 00:16:32,510 We would expect, in a general equation, 239 00:16:32,510 --> 00:16:41,060 that the shape of the wave would change a little as it travels. 240 00:16:41,060 --> 00:16:43,920 In this equation it doesn't change. 241 00:16:43,920 --> 00:16:47,680 The shape stays the same. 242 00:16:47,680 --> 00:16:51,820 See, again I'm thinking of this wave as a combination 243 00:16:51,820 --> 00:16:53,660 of pure exponentials. 244 00:16:53,660 --> 00:16:56,240 And they all travel with the same speed c, 245 00:16:56,240 --> 00:16:58,520 so the whole wave travels with speed c. 246 00:16:58,520 --> 00:17:05,980 And at a later time, that's the graph of the solution. 247 00:17:05,980 --> 00:17:08,280 OK. 248 00:17:08,280 --> 00:17:15,900 So that much is all we have to do, all we can do really, 249 00:17:15,900 --> 00:17:18,310 in solving the differential equations. 250 00:17:21,140 --> 00:17:22,530 It's a simple model, certainly. 251 00:17:22,530 --> 00:17:25,680 But now I come to the difference equation. 252 00:17:25,680 --> 00:17:27,710 Well, when I say the difference equation, 253 00:17:27,710 --> 00:17:30,720 I should say equations plural. 254 00:17:30,720 --> 00:17:35,800 Because here I've written four possibilities. 255 00:17:40,990 --> 00:17:45,870 Three of them are actually used, and one is a disaster. 256 00:17:52,640 --> 00:17:56,860 But none of the four is perfect. 257 00:17:56,860 --> 00:17:59,460 Most people would say Lax-Wendroff 258 00:17:59,460 --> 00:18:04,330 is the best of these four, because it has 259 00:18:04,330 --> 00:18:06,250 one extra order of accuracy. 260 00:18:09,500 --> 00:18:11,100 Oh yeah, yeah. 261 00:18:11,100 --> 00:18:14,350 I guess these all have only first-order accuracy. 262 00:18:17,160 --> 00:18:19,200 But Lax-Wendroff moves up to second. 263 00:18:19,200 --> 00:18:23,930 So that's a method that everybody knows, 264 00:18:23,930 --> 00:18:25,650 because it gets up to second. 265 00:18:25,650 --> 00:18:29,860 But actually upwind is the first one to start with. 266 00:18:29,860 --> 00:18:32,120 Let me start with that one. 267 00:18:32,120 --> 00:18:33,060 OK. 268 00:18:33,060 --> 00:18:37,850 So I've multiplied up by the delta t that should be 269 00:18:37,850 --> 00:18:42,410 in the denominator there, especially so that you would 270 00:18:42,410 --> 00:18:48,620 see -- so upwind, or forward difference -- 271 00:18:48,620 --> 00:18:51,580 I'm taking the standard forward difference in time -- 272 00:18:51,580 --> 00:18:55,330 all these methods are going to be explicit, 273 00:18:55,330 --> 00:19:01,370 so they're all going to be solvable in a simple way for u 274 00:19:01,370 --> 00:19:03,400 at the new time. 275 00:19:03,400 --> 00:19:07,780 So a forward difference in time for du/dt, 276 00:19:07,780 --> 00:19:12,450 and c times a forward difference in space for du/dx. 277 00:19:12,450 --> 00:19:15,180 So that's the first idea that would occur to you, 278 00:19:15,180 --> 00:19:19,980 and it's pretty reasonable. 279 00:19:19,980 --> 00:19:24,480 We can check its accuracy, but you can guess what it would be. 280 00:19:24,480 --> 00:19:29,070 What's your guess on the accuracy of this method? 281 00:19:29,070 --> 00:19:31,220 You're seeing a forward difference 282 00:19:31,220 --> 00:19:34,360 in time, a forward difference in space. 283 00:19:34,360 --> 00:19:37,680 The odds are those finite differences give you 284 00:19:37,680 --> 00:19:43,400 an accuracy p -- if we use p as the measure of accuracy -- 285 00:19:43,400 --> 00:19:50,580 which would be probably 1, first-order accurate. 286 00:19:50,580 --> 00:19:55,500 So we'll see the accuracy as only p equal to 1. 287 00:19:55,500 --> 00:19:57,680 OK. 288 00:19:57,680 --> 00:20:00,290 What about the stability? 289 00:20:00,290 --> 00:20:02,090 I'm looking at this first method. 290 00:20:05,130 --> 00:20:07,800 So one comment about the method is 291 00:20:07,800 --> 00:20:12,180 that this ratio c delta t over delta x, 292 00:20:12,180 --> 00:20:20,830 let me call it r, that's a key number. 293 00:20:20,830 --> 00:20:28,350 it's called the Courant number by some. 294 00:20:32,010 --> 00:20:35,290 So in the literature you might see the phrase Courant 295 00:20:35,290 --> 00:20:38,000 number for that ratio r. 296 00:20:38,000 --> 00:20:43,420 And maybe my first point is, stability 297 00:20:43,420 --> 00:20:47,020 is going to put a bound on r. 298 00:20:47,020 --> 00:20:50,580 We can tell right away, for this explicit method 299 00:20:50,580 --> 00:20:58,030 and for any explicit method, that r can't be unlimited. 300 00:20:58,030 --> 00:21:02,230 There's going to be a bound on r, which 301 00:21:02,230 --> 00:21:04,180 it means a bound on delta t. 302 00:21:04,180 --> 00:21:09,700 It's telling us that delta t has to be less than some constant 303 00:21:09,700 --> 00:21:12,820 times delta x. 304 00:21:12,820 --> 00:21:17,240 And stability is the job find that constant. 305 00:21:17,240 --> 00:21:20,060 How small does delta t have to be? 306 00:21:20,060 --> 00:21:25,780 Now let me sort of understand this forward difference method 307 00:21:25,780 --> 00:21:28,210 in this picture. 308 00:21:30,740 --> 00:21:37,180 So I'm going to do the forward difference, delta t of u 309 00:21:37,180 --> 00:21:44,630 equals that ratio times the forward difference of the space 310 00:21:44,630 --> 00:21:47,900 difference, so time differences is that ratio times this space 311 00:21:47,900 --> 00:21:48,620 difference. 312 00:21:48,620 --> 00:21:49,440 OK. 313 00:21:49,440 --> 00:21:51,850 Method 1. 314 00:21:51,850 --> 00:21:53,520 OK. 315 00:21:53,520 --> 00:22:01,510 So I guess somehow, you want to see that the -- 316 00:22:01,510 --> 00:22:04,200 you want to see a picture like this for the difference 317 00:22:04,200 --> 00:22:07,880 equation, So the difference equation, 318 00:22:07,880 --> 00:22:16,965 we imagine we've got steps delta x, all these steps are delta x. 319 00:22:16,965 --> 00:22:19,420 Let's suppose that's 0. 320 00:22:19,420 --> 00:22:20,710 OK. 321 00:22:20,710 --> 00:22:22,390 So what happening now? 322 00:22:22,390 --> 00:22:27,460 And time going this way and I'm solving this equation. 323 00:22:27,460 --> 00:22:28,460 What's happening here? 324 00:22:36,390 --> 00:22:42,650 Let me copy this method, u at a position j -- 325 00:22:42,650 --> 00:22:47,950 so this might be the point j delta x -- 326 00:22:47,950 --> 00:22:54,590 at time n plus 1 is -- I want to write this equation, as always, 327 00:22:54,590 --> 00:22:56,640 in the most convenient way. 328 00:22:56,640 --> 00:22:58,120 So what am I going to do? 329 00:22:58,120 --> 00:23:00,370 I'm going to take this u of x and t 330 00:23:00,370 --> 00:23:03,450 and put it on the right-hand side. 331 00:23:03,450 --> 00:23:09,650 So then my equation looks like that ratio r times u 332 00:23:09,650 --> 00:23:13,790 at j plus 1 at time n. 333 00:23:13,790 --> 00:23:16,640 That's this. 334 00:23:16,640 --> 00:23:24,120 I'm one position over, but I'm at the given time. 335 00:23:24,120 --> 00:23:30,160 And then what's the coefficient of u_(j, n)? 336 00:23:30,160 --> 00:23:33,470 So can you just tell me, what do I put in there? 337 00:23:33,470 --> 00:23:36,750 So this will be that nice way to look at our difference 338 00:23:36,750 --> 00:23:38,070 equation. 339 00:23:38,070 --> 00:23:43,510 Each new value is a combination of two old values, r 340 00:23:43,510 --> 00:23:47,570 times the guy here. 341 00:23:47,570 --> 00:23:50,200 So in other words, here's my little molecule. 342 00:23:52,720 --> 00:23:54,920 This is the new value. 343 00:23:54,920 --> 00:23:58,700 It comes from r times this plus what, 344 00:23:58,700 --> 00:24:01,600 you have to tell me what multiplies 345 00:24:01,600 --> 00:24:03,350 this u at this point. 346 00:24:03,350 --> 00:24:10,910 So here's u_(j, n+1), and here's u_(j+1, n), and here is u_(j, 347 00:24:10,910 --> 00:24:11,410 n). 348 00:24:11,410 --> 00:24:14,340 And what goes in parentheses? 349 00:24:14,340 --> 00:24:16,660 1 minus r. 350 00:24:16,660 --> 00:24:23,020 Because I get the 1 when that flips over to the other side. 351 00:24:23,020 --> 00:24:24,120 Good. 352 00:24:24,120 --> 00:24:26,560 So it's just a combination. 353 00:24:26,560 --> 00:24:29,030 The new value is a combination of those two. 354 00:24:31,790 --> 00:24:41,230 And I guess what I want to show is that it could not possibly 355 00:24:41,230 --> 00:24:48,360 be stable unless r is less or equal to 1. 356 00:24:48,360 --> 00:24:51,250 So this will be the Courant condition. 357 00:24:51,250 --> 00:24:54,310 The condition on the Courant number. 358 00:24:54,310 --> 00:25:00,110 So the Courant condition or the CFL -- 359 00:25:00,110 --> 00:25:07,230 that's a really a better condition -- will be -- 360 00:25:07,230 --> 00:25:08,890 let me say what it is. 361 00:25:08,890 --> 00:25:14,980 The CFL condition is r less or equal to 1. 362 00:25:14,980 --> 00:25:20,770 That's a condition for stability or convergence 363 00:25:20,770 --> 00:25:27,490 that comes to us from comparing the real characteristics 364 00:25:27,490 --> 00:25:31,461 with the finite difference picture. 365 00:25:31,461 --> 00:25:31,960 OK. 366 00:25:31,960 --> 00:25:33,920 So what's CFL? 367 00:25:33,920 --> 00:25:35,260 C is Courant still. 368 00:25:35,260 --> 00:25:40,450 So this came from a really early paper, 369 00:25:40,450 --> 00:25:43,850 and I think actually Lewy. 370 00:25:43,850 --> 00:25:46,170 That's Courant, that's Friedrichs, 371 00:25:46,170 --> 00:25:47,930 they were close friends. 372 00:25:47,930 --> 00:25:51,330 That's Lewy, all three were friends actually. 373 00:25:51,330 --> 00:25:58,400 And I think it was actually Lewy who spotted what we're just 374 00:25:58,400 --> 00:26:02,450 going to do right now, the requirement that r 375 00:26:02,450 --> 00:26:03,870 had to be less or equal 1. 376 00:26:06,700 --> 00:26:13,690 Now let me just make clear that this Courant-Friedrichs-Lewy 377 00:26:13,690 --> 00:26:16,770 condition is going to be a necessary condition. 378 00:26:16,770 --> 00:26:21,050 It has to hold, or there is no hope. 379 00:26:21,050 --> 00:26:27,780 But it's not enough, it's not a sufficient condition. 380 00:26:27,780 --> 00:26:33,830 We can't guarantee that a method stable just because the Courant 381 00:26:33,830 --> 00:26:35,670 condition is satisfied. 382 00:26:35,670 --> 00:26:37,220 The Courant condition will tell us 383 00:26:37,220 --> 00:26:39,830 something about the position of characteristics, 384 00:26:39,830 --> 00:26:43,220 we'll see it in a second. 385 00:26:43,220 --> 00:26:50,130 For example, this centered one is unstable, this guy, 386 00:26:50,130 --> 00:26:52,180 when the right-hand side -- you might say, 387 00:26:52,180 --> 00:26:56,070 oh this is a better method; always better to take 388 00:26:56,070 --> 00:26:59,680 a centered difference for du/dx than a one-sided. 389 00:26:59,680 --> 00:27:02,680 But not now. 390 00:27:02,680 --> 00:27:06,790 The one-sided one is OK. 391 00:27:06,790 --> 00:27:14,690 The centered one is going to be totally unstable, 392 00:27:14,690 --> 00:27:19,470 for all r, this centered one. 393 00:27:19,470 --> 00:27:25,600 And we'll identify that not from Courant-Friedrichs-Lewy -- 394 00:27:25,600 --> 00:27:28,540 their paper wouldn't have spotted the difference between 395 00:27:28,540 --> 00:27:31,870 one and two, but von Neumann did; 396 00:27:31,870 --> 00:27:38,130 von Neumann quickly noticed that if you use an exponential -- 397 00:27:38,130 --> 00:27:40,360 so we'll do one van Neumann in a moment -- 398 00:27:40,360 --> 00:27:44,280 if you plug in an exponential to number 2, 399 00:27:44,280 --> 00:27:47,440 this centered difference, it takes off. 400 00:27:47,440 --> 00:27:51,670 And it's unstable even if delta t is small. 401 00:27:51,670 --> 00:27:52,400 OK. 402 00:27:52,400 --> 00:27:56,000 Now what is this CFL, Courant-Friedrichs-Lewy 403 00:27:56,000 --> 00:27:56,500 condition? 404 00:27:56,500 --> 00:27:59,960 Where does that come from? 405 00:27:59,960 --> 00:28:04,790 It comes from just thinking about the information. 406 00:28:08,850 --> 00:28:10,450 I have to write down the thinking now 407 00:28:10,450 --> 00:28:13,260 that goes into the Courant-Friedrichs-Lewy 408 00:28:13,260 --> 00:28:17,930 condition, the CFL condition. 409 00:28:24,920 --> 00:28:26,640 OK. 410 00:28:26,640 --> 00:28:29,710 It's straightforward. 411 00:28:29,710 --> 00:28:42,110 I have to think, at a time t, let's say t equals n delta t. 412 00:28:42,110 --> 00:28:47,710 Suppose I've taken n times steps. 413 00:28:47,710 --> 00:28:58,780 Then the true u at, let's say, 0 and that time t is -- 414 00:28:58,780 --> 00:29:02,720 or x and t, because we know the true solution. 415 00:29:02,720 --> 00:29:08,020 The true u at x and t is, as we know, 416 00:29:08,020 --> 00:29:17,080 the initial function at the point x plus c*t. 417 00:29:17,080 --> 00:29:22,780 So that's the correct solution. 418 00:29:22,780 --> 00:29:25,920 And Courant-Friedrichs-Lewy are just 419 00:29:25,920 --> 00:29:33,500 saying that you better use this number in finite difference 420 00:29:33,500 --> 00:29:38,710 method or you don't have a chance. 421 00:29:38,710 --> 00:29:39,210 Right. 422 00:29:39,210 --> 00:29:42,240 That makes sense. 423 00:29:42,240 --> 00:29:46,291 So what do I mean by use this number? 424 00:29:46,291 --> 00:29:46,790 OK. 425 00:29:46,790 --> 00:29:48,450 So let me draw. 426 00:29:48,450 --> 00:29:52,110 So u at x and t is some point here. 427 00:29:52,110 --> 00:29:54,540 This is at the point x and t. 428 00:29:54,540 --> 00:29:57,010 This used these two values. 429 00:29:57,010 --> 00:30:00,350 They use these three values. 430 00:30:00,350 --> 00:30:02,620 This one used that one, that one, that one. 431 00:30:02,620 --> 00:30:04,800 This one used those two. 432 00:30:04,800 --> 00:30:08,750 Those four used these five values. 433 00:30:08,750 --> 00:30:09,250 Right. 434 00:30:15,470 --> 00:30:20,180 See, the difference method is not propagating the information 435 00:30:20,180 --> 00:30:26,370 entirely on that line, it's really taking values 436 00:30:26,370 --> 00:30:30,020 over that whole interval with some combinations of r 437 00:30:30,020 --> 00:30:31,520 and 1 minus r. 438 00:30:31,520 --> 00:30:33,840 And in the end, after five times steps, 439 00:30:33,840 --> 00:30:37,950 it's giving us this value up there. 440 00:30:37,950 --> 00:30:46,700 But the point is that if this point x plus c*t were out here, 441 00:30:46,700 --> 00:30:50,890 then refining delta x and refining delta t, 442 00:30:50,890 --> 00:30:54,440 and using more and more points, and filling in and using more 443 00:30:54,440 --> 00:30:58,900 of these, but not getting out here, would get you nowhere. 444 00:30:58,900 --> 00:31:00,410 You wouldn't have a chance. 445 00:31:00,410 --> 00:31:02,090 So that's the Courant-Friedrichs-Lewy 446 00:31:02,090 --> 00:31:02,870 condition. 447 00:31:02,870 --> 00:31:05,010 How far out is this? 448 00:31:05,010 --> 00:31:09,240 This is n delta x inside this. 449 00:31:09,240 --> 00:31:14,870 And if this is x, this is n delta x's away, 450 00:31:14,870 --> 00:31:18,520 and this is x plus c*t. 451 00:31:18,520 --> 00:31:20,470 And c is n delta t's. 452 00:31:24,890 --> 00:31:26,390 Right? 453 00:31:26,390 --> 00:31:33,150 t is n delta t. 454 00:31:33,150 --> 00:31:33,840 Are you with me? 455 00:31:38,690 --> 00:31:43,280 As I keep delta t over delta x, the ratio r, fixed 456 00:31:43,280 --> 00:31:49,130 and use more and more points, I'm filling up this interval 457 00:31:49,130 --> 00:31:53,100 with points that I've used. 458 00:31:53,100 --> 00:31:56,590 But I'm never going to get to a point there. 459 00:31:56,590 --> 00:32:02,890 So the idea is then, unstable, couldn't converge, 460 00:32:02,890 --> 00:32:16,120 couldn't work, if the distance reached here, n delta x, 461 00:32:16,120 --> 00:32:24,410 is smaller than c*n delta t, it couldn't work. 462 00:32:24,410 --> 00:32:26,910 OK. 463 00:32:26,910 --> 00:32:28,830 So I'm taking a negative point of view here. 464 00:32:28,830 --> 00:32:32,860 It can't work in this case. 465 00:32:32,860 --> 00:32:35,510 The reason I take that negative point of view is I'm 466 00:32:35,510 --> 00:32:41,630 not able to say it does work in the case when 467 00:32:41,630 --> 00:32:44,770 delta x is big enough compared to delta t, 468 00:32:44,770 --> 00:32:46,060 and I include that point. 469 00:32:46,060 --> 00:32:47,330 I can't be sure. 470 00:32:47,330 --> 00:32:50,390 But I can be sure that if I don't include 471 00:32:50,390 --> 00:32:56,490 that neighborhood of where the true u is produced, 472 00:32:56,490 --> 00:32:57,530 it can't have a chance. 473 00:32:57,530 --> 00:33:02,100 So let me just cancel n, divide by delta x. 474 00:33:02,100 --> 00:33:09,470 And I say fail if 1 is smaller then c delta t 475 00:33:09,470 --> 00:33:11,120 over delta x, which is r. 476 00:33:11,120 --> 00:33:20,210 It fails if r is bigger than 1, by this reasoning 477 00:33:20,210 --> 00:33:21,290 of characteristics. 478 00:33:25,880 --> 00:33:29,090 There are two more cases that are worth thinking about. 479 00:33:29,090 --> 00:33:37,100 Suppose the method was downwind. 480 00:33:37,100 --> 00:33:41,320 So this is upwind, because I take a forward difference. 481 00:33:41,320 --> 00:33:44,580 The wind is blowing this way. 482 00:33:44,580 --> 00:33:52,200 Forgive me for incomplete use of the word wind. 483 00:33:52,200 --> 00:33:53,640 I don't know what it means. 484 00:33:53,640 --> 00:33:56,330 But one way or another, whatever, the wind 485 00:33:56,330 --> 00:33:57,980 is blowing this way. 486 00:33:57,980 --> 00:34:02,040 The values are coming from the upwind direction. 487 00:34:02,040 --> 00:34:04,790 And that's good, because the true value comes 488 00:34:04,790 --> 00:34:06,640 from the upwind direction. 489 00:34:06,640 --> 00:34:09,570 That's what we saw in characteristics, 490 00:34:09,570 --> 00:34:12,160 the values are blowing downwind. 491 00:34:12,160 --> 00:34:17,410 So, what would happen if I use a backward difference here? 492 00:34:17,410 --> 00:34:20,920 So I didn't write that bad idea, but it's 493 00:34:20,920 --> 00:34:28,310 important to realizes that's method 1a, a bad idea, 494 00:34:28,310 --> 00:34:31,060 is to use a backward difference here. 495 00:34:31,060 --> 00:34:35,880 I should maybe call it 1b, for backward. 496 00:34:35,880 --> 00:34:38,000 You see that it would fail? 497 00:34:38,000 --> 00:34:40,350 If I used a backward difference, what will 498 00:34:40,350 --> 00:34:42,000 what happen to this picture? 499 00:34:42,000 --> 00:34:45,110 This value would come from stuff on the left. 500 00:34:45,110 --> 00:34:47,350 It would totally have no chance to use 501 00:34:47,350 --> 00:34:51,730 the correct initial value. 502 00:34:51,730 --> 00:34:55,330 So a backward difference is an immediate failure. 503 00:34:55,330 --> 00:35:05,500 So r has to be -- It will also fail if -- 504 00:35:05,500 --> 00:35:09,360 well I was going to say r less than 0. 505 00:35:09,360 --> 00:35:12,460 I don't know, can I say that? 506 00:35:12,460 --> 00:35:16,030 The way r could be less than 0 would be the delta x being 507 00:35:16,030 --> 00:35:21,360 negative, and that's what I mean by the downwind method, 508 00:35:21,360 --> 00:35:25,960 where I'm looking for looking for the value here, 509 00:35:25,960 --> 00:35:28,590 and of course I'm not finding it. 510 00:35:28,590 --> 00:35:30,510 OK. 511 00:35:30,510 --> 00:35:35,850 So I guess I'm saying that Courant-Friedrichs-Lewy tells 512 00:35:35,850 --> 00:35:42,700 us right away that this method will only have a chance -- 513 00:35:42,700 --> 00:35:48,700 it only has a chance if r is -- so, possible, 514 00:35:48,700 --> 00:35:52,250 and in fact this is the stability condition. 515 00:35:52,250 --> 00:35:56,290 So let me write that fact, but we haven't proved it yet. 516 00:35:56,290 --> 00:36:00,970 It's stable for r between 0 and 1. 517 00:36:00,970 --> 00:36:07,250 That's the stability condition, which limits delta t. 518 00:36:07,250 --> 00:36:12,650 Because r is delta t over delta x. 519 00:36:12,650 --> 00:36:16,710 And this says that delta t can't be larger 520 00:36:16,710 --> 00:36:19,011 than the multiple of delta x. 521 00:36:19,011 --> 00:36:19,510 OK. 522 00:36:19,510 --> 00:36:25,910 You might say, that sounds like delta t is too small. 523 00:36:25,910 --> 00:36:28,360 Is stability too restrictive? 524 00:36:28,360 --> 00:36:32,560 Well it's restrictive certainly, because it limits delta t. 525 00:36:32,560 --> 00:36:40,980 But actually, for these methods, accuracy also limits delta t. 526 00:36:40,980 --> 00:36:49,130 If I look for an implicit method, that would allow me -- 527 00:36:49,130 --> 00:36:54,450 if I looked for a method that allowed a bigger delta t, 528 00:36:54,450 --> 00:36:59,780 then the error of that, in that method, 529 00:36:59,780 --> 00:37:01,960 proportional to delta t, would be too big. 530 00:37:01,960 --> 00:37:05,470 In other words, accuracy as well as 531 00:37:05,470 --> 00:37:10,470 stability is keeping delta t of the order of delta x. 532 00:37:10,470 --> 00:37:17,430 Accuracy as well as stability is keeping r within a bound, 533 00:37:17,430 --> 00:37:21,230 and stability is keeping it within those bounds. 534 00:37:21,230 --> 00:37:22,490 OK. 535 00:37:22,490 --> 00:37:26,700 So I still have to show that the method is stable. 536 00:37:26,700 --> 00:37:28,840 The Courant-Friedrichs-Lewy condition 537 00:37:28,840 --> 00:37:35,760 just helps to eliminate impossible ratios. 538 00:37:35,760 --> 00:37:36,680 OK. 539 00:37:36,680 --> 00:37:42,490 One more point, which of course everybody notices, in solving 540 00:37:42,490 --> 00:37:46,870 the difference equation. 541 00:37:46,870 --> 00:37:50,960 Suppose r is exactly 1. 542 00:37:50,960 --> 00:37:53,710 Think about the case where r is exactly 1. 543 00:37:58,080 --> 00:38:00,550 Remember that's c delta t over delta x. 544 00:38:06,650 --> 00:38:12,400 Suppose r is exactly 1, then what does the method do? 545 00:38:12,400 --> 00:38:16,190 The method takes the new value, r is 1. 546 00:38:16,190 --> 00:38:18,140 This is 0 now. 547 00:38:18,140 --> 00:38:22,840 So the new value is then, when r is 1, is that old value. 548 00:38:22,840 --> 00:38:28,880 In this picture, when r is 1 this value is this one. 549 00:38:28,880 --> 00:38:34,300 The values are traveling along a line. 550 00:38:34,300 --> 00:38:39,150 I'm not using these you see, because if r is exactly 1, 551 00:38:39,150 --> 00:38:42,990 these are exactly 0. 552 00:38:42,990 --> 00:38:45,100 Now is that good? 553 00:38:45,100 --> 00:38:50,780 Actually it's terrific, because that line, when r is 1, 554 00:38:50,780 --> 00:38:55,210 is the correct characteristic line. 555 00:38:55,210 --> 00:39:00,020 If r is 1, if c delta t equals delta x, 556 00:39:00,020 --> 00:39:06,040 I'm going up exactly with that slope that the true line does. 557 00:39:06,040 --> 00:39:11,710 In other words, the difference equation 558 00:39:11,710 --> 00:39:14,900 gives exactly the solution to the differential equation. 559 00:39:14,900 --> 00:39:17,510 Because it's so simple of course. 560 00:39:17,510 --> 00:39:19,970 Couldn't do it -- well, that's the point, 561 00:39:19,970 --> 00:39:27,230 we can't do it for big problems, because in a big problem c is 562 00:39:27,230 --> 00:39:30,900 going to be variable. 563 00:39:30,900 --> 00:39:35,640 Maybe the problem, c may depend on the position or the time, 564 00:39:35,640 --> 00:39:37,940 or it might depend on the solution 565 00:39:37,940 --> 00:39:40,420 u in a non-linear problem. 566 00:39:40,420 --> 00:39:42,460 Those are ahead of us. 567 00:39:42,460 --> 00:39:47,100 But this possibility in 1D -- it's only, really, 568 00:39:47,100 --> 00:39:51,960 a one-dimensional possibility, where the true information 569 00:39:51,960 --> 00:39:56,240 travels on a line, and we could just hope to stay close to that 570 00:39:56,240 --> 00:39:57,491 characteristic line. 571 00:39:57,491 --> 00:39:57,990 OK. 572 00:40:01,340 --> 00:40:08,770 Now I'm ready to tackle, for any method, the -- 573 00:40:08,770 --> 00:40:10,750 let me get these guys up again. 574 00:40:13,980 --> 00:40:18,090 So first, I want to see, why is upwind stable, 575 00:40:18,090 --> 00:40:22,070 and why is centered unstable. 576 00:40:22,070 --> 00:40:25,480 Should I start with the stability question, rather than 577 00:40:25,480 --> 00:40:27,360 the accuracy question? 578 00:40:27,360 --> 00:40:30,230 The accuracy we could guess was first order, 579 00:40:30,230 --> 00:40:33,440 until we get to Lax-Wendroff. 580 00:40:33,440 --> 00:40:36,900 So accuracy is going to come next time. 581 00:40:40,410 --> 00:40:46,450 We've had Courant's take on stability, the CFL condition, 582 00:40:46,450 --> 00:40:52,060 but now I'm ready for von Neumann's deeper insight. 583 00:40:52,060 --> 00:40:52,560 OK. 584 00:40:52,560 --> 00:40:55,940 So how does von Neumann study stability? 585 00:40:55,940 --> 00:40:58,580 He watches every e to the i*k*x. 586 00:41:01,400 --> 00:41:05,950 So von Neumann's test is going to be each e to the i*k*x 587 00:41:05,950 --> 00:41:11,830 should have a growth factor, in the difference equation, 588 00:41:11,830 --> 00:41:14,840 smaller then 1. 589 00:41:14,840 --> 00:41:18,350 The growth factor in the differential equation, 590 00:41:18,350 --> 00:41:22,670 of course, was right on. 591 00:41:22,670 --> 00:41:27,960 This has magnitude exactly 1. 592 00:41:27,960 --> 00:41:34,130 So wave equations are not giving us any space to work in. 593 00:41:34,130 --> 00:41:36,500 Because we want -- the growth factor and the difference 594 00:41:36,500 --> 00:41:39,560 equation, we want it to be close to the real one. 595 00:41:39,560 --> 00:41:45,930 And at the same time, it better be not -- the real one, 596 00:41:45,930 --> 00:41:50,710 having absolute value exactly 1 on the unit circle. 597 00:41:50,710 --> 00:41:53,680 But the difference equation is allowed 598 00:41:53,680 --> 00:41:57,030 to go into the unit circle, but not to go out. 599 00:41:57,030 --> 00:41:57,540 OK. 600 00:41:57,540 --> 00:42:01,790 That will separate the good ones from the bad. 601 00:42:01,790 --> 00:42:02,440 OK. 602 00:42:02,440 --> 00:42:12,525 Let me work here on -- Can I copy the equation u_(j, 603 00:42:12,525 --> 00:42:23,510 n+1) is equal to r*u_(j+1, n), and 1 minus r u_(j, n). 604 00:42:23,510 --> 00:42:24,710 OK. 605 00:42:24,710 --> 00:42:29,030 Now I'm going to do an exponential again. 606 00:42:29,030 --> 00:42:34,290 So I'm going to have u, say, 0, n. 607 00:42:39,610 --> 00:42:43,590 I'm going to start from e to the i*k*x. 608 00:42:48,470 --> 00:43:01,700 Sorry, let me say -- So, time 0 and position n will be e 609 00:43:01,700 --> 00:43:06,891 to the i*k*n delta x. 610 00:43:06,891 --> 00:43:07,390 Right. 611 00:43:07,390 --> 00:43:11,320 That's my initial, e to the i*k*x, 612 00:43:11,320 --> 00:43:14,100 that's my pure frequency. 613 00:43:14,100 --> 00:43:20,100 I plug that in to the -- oh, not n but j; 614 00:43:20,100 --> 00:43:25,270 j is measuring how many -- sorry, let's get it right -- 615 00:43:25,270 --> 00:43:29,970 j is measuring how many delta x's I'm going across; 616 00:43:29,970 --> 00:43:32,960 n is measuring how many delta t's I'm going up. 617 00:43:32,960 --> 00:43:34,900 So this is the key, here's von Neumann. 618 00:43:38,520 --> 00:43:43,000 Try a pure exponential, plug it in. 619 00:43:43,000 --> 00:43:45,800 Again, it's going to work, because we 620 00:43:45,800 --> 00:43:47,610 have constant coefficients. 621 00:43:47,610 --> 00:43:49,980 So what do I get on the right-hand side? 622 00:43:49,980 --> 00:43:55,550 I get r times this guy at level j plus 1, 623 00:43:55,550 --> 00:44:06,510 e to the i*k j plus 1 delta x times the exponential. 624 00:44:06,510 --> 00:44:08,500 Can I save the exponential, because it's 625 00:44:08,500 --> 00:44:11,970 going to factor out? 626 00:44:11,970 --> 00:44:16,760 Plus a 1 minus 4 times the exponential itself, 627 00:44:16,760 --> 00:44:22,170 and here's the exponential: e to the i*k*j delta x. 628 00:44:22,170 --> 00:44:24,840 This will be one step. 629 00:44:24,840 --> 00:44:37,900 I could say u_(j, 1) coming from level 0 to level 1. 630 00:44:37,900 --> 00:44:45,290 Sorry, e to the i*k*j delta x, so I don't need it here. 631 00:44:45,290 --> 00:44:46,780 Sorry, I did it wrong. 632 00:44:46,780 --> 00:44:54,100 That e to the i*k j plus 1 delta x factors into e to the i*k 633 00:44:54,100 --> 00:44:58,810 delta x from the 1, and e to the i*k*j delta x, 634 00:44:58,810 --> 00:44:59,720 which is over there. 635 00:44:59,720 --> 00:45:01,670 So I only wanted this much. 636 00:45:07,091 --> 00:45:07,590 OK. 637 00:45:12,530 --> 00:45:17,340 Sorry, that's on the permanent tape but maybe it's OK. 638 00:45:17,340 --> 00:45:20,110 Because it makes us think through, 639 00:45:20,110 --> 00:45:24,820 what's the result from a pure exponential? 640 00:45:24,820 --> 00:45:27,780 And again, the result is pure exponential 641 00:45:27,780 --> 00:45:31,810 in, pure exponential out, but multiplied 642 00:45:31,810 --> 00:45:36,380 by some finite difference growth factor G, that 643 00:45:36,380 --> 00:45:38,750 depends on the frequency. 644 00:45:38,750 --> 00:45:44,470 And it depends on delta x and it depends on r. 645 00:45:50,400 --> 00:45:53,520 Well actually it depends on k delta x together. 646 00:45:53,520 --> 00:45:54,880 So I could put that together. 647 00:46:01,530 --> 00:46:05,120 What's von Neumann's question now? 648 00:46:05,120 --> 00:46:09,030 He wants to know is this number -- it's a complex number, 649 00:46:09,030 --> 00:46:13,970 of course, because it's got this cosine of k delta x plus i sine 650 00:46:13,970 --> 00:46:20,120 of k delta x -- he wants to know is it -- magnitude, 651 00:46:20,120 --> 00:46:23,460 does the magnitude get bigger than 1? 652 00:46:23,460 --> 00:46:28,890 If so, n steps will give the n-th power of the thing, 653 00:46:28,890 --> 00:46:30,860 and it will blow up. 654 00:46:30,860 --> 00:46:33,410 Or does the magnitude stay lesser 655 00:46:33,410 --> 00:46:39,790 equal to 1, in which case it won't blow up, in which case 656 00:46:39,790 --> 00:46:42,770 we have stability. 657 00:46:42,770 --> 00:46:46,950 Of course, it would be great if the magnitude was always 658 00:46:46,950 --> 00:46:49,100 exactly 1. 659 00:46:49,100 --> 00:46:53,520 Because that's what the true solution has. 660 00:46:53,520 --> 00:46:59,170 But that's only going to happen in this special, special case 661 00:46:59,170 --> 00:47:02,850 when r equals 1, and I'm going just right 662 00:47:02,850 --> 00:47:04,510 on the characteristic. 663 00:47:04,510 --> 00:47:08,310 When r is exactly 1, that's the case when that's gone, 664 00:47:08,310 --> 00:47:11,730 this is a 1, and that has magnitude exactly 1. 665 00:47:11,730 --> 00:47:15,050 But normally, r is going to be, I'm 666 00:47:15,050 --> 00:47:18,110 going to be on the safe side, r will be less than 1. 667 00:47:18,110 --> 00:47:26,000 And I just want to draw this von Neumann amplification factor. 668 00:47:26,000 --> 00:47:28,000 I'll often use the word growth factor, 669 00:47:28,000 --> 00:47:30,140 just because it's shorter, but another word 670 00:47:30,140 --> 00:47:32,360 is amplification factor. 671 00:47:32,360 --> 00:47:35,630 What does an exponential get amplified by? 672 00:47:35,630 --> 00:47:41,300 Can you identify this number in the complex plane? 673 00:47:41,300 --> 00:47:44,470 Of course, the big question is, where is it 674 00:47:44,470 --> 00:47:47,300 with respect to the unit circle? 675 00:47:47,300 --> 00:47:48,240 OK. 676 00:47:48,240 --> 00:47:52,470 Take k equals 0 -- zero frequency, the DC term, 677 00:47:52,470 --> 00:47:57,340 the constant start -- what is that number equal when k is 0? 678 00:48:00,320 --> 00:48:01,570 1. 679 00:48:01,570 --> 00:48:04,350 So that's normal. 680 00:48:04,350 --> 00:48:10,050 At k equals 0, we're right at the position 1. 681 00:48:10,050 --> 00:48:13,620 That's just telling us that a constant initial value 682 00:48:13,620 --> 00:48:17,060 stays unchanged. 683 00:48:17,060 --> 00:48:20,420 The true growth factor and the von Neumann amplification 684 00:48:20,420 --> 00:48:22,400 factor both 1. 685 00:48:22,400 --> 00:48:27,680 But now let k be non-zero. 686 00:48:27,680 --> 00:48:30,200 Where is this complex number? 687 00:48:30,200 --> 00:48:33,170 And now, I'm going to impose the condition 688 00:48:33,170 --> 00:48:45,140 that r is between 0 and 1, because I know from Courant 689 00:48:45,140 --> 00:48:47,920 that is going to be required. 690 00:48:47,920 --> 00:48:49,040 What's the point here? 691 00:48:49,040 --> 00:48:49,750 OK. 692 00:48:49,750 --> 00:48:53,050 Then, you see, here is a 1 minus r. 693 00:48:56,050 --> 00:49:00,320 So there's 1, 1 minus r will be a little bit in here, 694 00:49:00,320 --> 00:49:03,330 and then what happens when I add on this number? 695 00:49:03,330 --> 00:49:05,750 So the 1 minus r I've done. 696 00:49:05,750 --> 00:49:07,080 It put me here. 697 00:49:07,080 --> 00:49:08,610 It was real. 698 00:49:08,610 --> 00:49:10,250 But this number is not real. 699 00:49:10,250 --> 00:49:16,259 This number is r times some complex number, 700 00:49:16,259 --> 00:49:17,050 but what do I know? 701 00:49:17,050 --> 00:49:19,360 I do something critical here. 702 00:49:19,360 --> 00:49:22,930 What do I know about this number? 703 00:49:22,930 --> 00:49:25,920 I know its absolute value is 1. 704 00:49:25,920 --> 00:49:28,600 So that it gets multiplied by an r. 705 00:49:28,600 --> 00:49:31,860 Look, look, look, look. 706 00:49:31,860 --> 00:49:33,490 I start with a 1 minus r. 707 00:49:33,490 --> 00:49:36,210 So I go to 1, back to r, and now I 708 00:49:36,210 --> 00:49:41,420 add in this part, which is somewhere on a circle of radius 709 00:49:41,420 --> 00:49:46,360 r that's everything in this problem. 710 00:49:46,360 --> 00:49:53,140 It's a circle of radius r around the point 1 minus r. 711 00:49:58,170 --> 00:50:00,720 I'm inside the circle, and you could say, well, 712 00:50:00,720 --> 00:50:04,410 you could have seen that clearly by just using the triangle 713 00:50:04,410 --> 00:50:05,350 inequality. 714 00:50:05,350 --> 00:50:08,980 The magnitude of this can't be bigger than the magnitude 715 00:50:08,980 --> 00:50:12,170 of that, which is what? 716 00:50:12,170 --> 00:50:13,210 r. 717 00:50:13,210 --> 00:50:17,920 Plus the magnitude of that, which is what? 718 00:50:17,920 --> 00:50:19,530 1 minus r. 719 00:50:19,530 --> 00:50:21,840 So the magnitude, by the triangle inequality, 720 00:50:21,840 --> 00:50:25,850 can't be more than r plus 1 minus r which is 1. 721 00:50:25,850 --> 00:50:34,740 But now wait a minute, where did I use the Courant condition? 722 00:50:34,740 --> 00:50:36,790 The way I said that there sounded 723 00:50:36,790 --> 00:50:38,710 like it would always work. 724 00:50:38,710 --> 00:50:40,360 This has magnitude r. 725 00:50:43,800 --> 00:50:45,580 What's going on here? 726 00:50:45,580 --> 00:50:49,320 Suppose r is bigger than 1, what's going wrong? 727 00:50:49,320 --> 00:50:53,220 In fact, what's the picture if r is bigger than 1? 728 00:50:53,220 --> 00:50:59,600 If r is bigger than 1, then my 1 minus r is out here somewhere. 729 00:50:59,600 --> 00:51:02,980 So this is the unstable case, 1 minus r, 730 00:51:02,980 --> 00:51:08,450 and then a circle of radius r, bad news, right? 731 00:51:08,450 --> 00:51:11,670 Every frequency is unstable in fact. 732 00:51:11,670 --> 00:51:15,640 When I'm reaching too far, and that's just telling me 733 00:51:15,640 --> 00:51:18,810 I'm not using the information I need. 734 00:51:18,810 --> 00:51:24,880 I don't have a chance of keeping, controlling 735 00:51:24,880 --> 00:51:27,580 any frequency. 736 00:51:27,580 --> 00:51:28,330 OK. 737 00:51:28,330 --> 00:51:31,440 I've got just enough time to show 738 00:51:31,440 --> 00:51:33,890 how bad this second method is. 739 00:51:33,890 --> 00:51:37,400 So nobody wanted his name associated 740 00:51:37,400 --> 00:51:39,920 with that second method. 741 00:51:39,920 --> 00:51:41,070 Why is that? 742 00:51:41,070 --> 00:51:46,830 So what's the von Neumann quantity for the second method? 743 00:51:46,830 --> 00:51:49,420 OK. 744 00:51:49,420 --> 00:51:53,490 Just space here to write -- so this is now -- 745 00:51:53,490 --> 00:51:56,330 I want to know the number for that second method. 746 00:51:56,330 --> 00:52:04,970 So now the method is u_(j, n+1) is u_(j, n), 747 00:52:04,970 --> 00:52:12,365 from the time difference, plus r over 2, u_(j+1, 748 00:52:12,365 --> 00:52:22,070 n) minus u_(j-1, n). 749 00:52:25,780 --> 00:52:28,950 This is going to be the bad method. 750 00:52:28,950 --> 00:52:33,010 It looks good because the centered difference 751 00:52:33,010 --> 00:52:36,440 is more accurate than a one-sided difference, 752 00:52:36,440 --> 00:52:38,400 but it's also unstable here. 753 00:52:38,400 --> 00:52:42,630 Can you look at that and see what the von Neumann number 754 00:52:42,630 --> 00:52:45,430 G is going to be? 755 00:52:45,430 --> 00:52:48,390 Think of an exponential going in. 756 00:52:48,390 --> 00:52:51,220 What exponential comes out? 757 00:52:51,220 --> 00:52:56,730 So G is a 1 from this. 758 00:52:56,730 --> 00:53:00,120 Now this is our guys that's shifted over, 759 00:53:00,120 --> 00:53:04,940 so that would be in r over 2 e to the i*k delta x, 760 00:53:04,940 --> 00:53:07,410 just as it was before. 761 00:53:07,410 --> 00:53:16,880 And this one will be a minus r over 2 times e to the -- 762 00:53:16,880 --> 00:53:22,510 so it's back one step, so there will be an e to the minus i*k 763 00:53:22,510 --> 00:53:26,310 delta x. 764 00:53:26,310 --> 00:53:27,480 So what's with this now? 765 00:53:30,380 --> 00:53:35,970 G is 1 plus this quantity, r over 2 times e to the i*k*x 766 00:53:35,970 --> 00:53:38,980 minus e to the minus i*k*x. 767 00:53:38,980 --> 00:53:41,760 What am I seeing there? 768 00:53:41,760 --> 00:53:49,290 I'm seeing 1 plus r, and everybody recognizes this minus 769 00:53:49,290 --> 00:54:03,950 this over 2 as i sine, i*r sine k delta x. 770 00:54:03,950 --> 00:54:07,720 That's the amplification factor, and is it smaller than 1? 771 00:54:10,300 --> 00:54:11,800 No way. 772 00:54:11,800 --> 00:54:12,300 No way. 773 00:54:12,300 --> 00:54:14,730 Let me raise it up here. 774 00:54:14,730 --> 00:54:19,400 So I don't care whether r is less than 1 or not, I'm lost. 775 00:54:23,270 --> 00:54:25,670 This is a pure imaginary number. 776 00:54:25,670 --> 00:54:26,910 This is a real number. 777 00:54:26,910 --> 00:54:30,490 So I have the sum of squares to get the magnitude, 778 00:54:30,490 --> 00:54:36,220 it will be the square root of 1 plus r sine k delta x squared. 779 00:54:36,220 --> 00:54:41,370 So if I draw the bad picture then, what's the bad picture? 780 00:54:41,370 --> 00:54:48,180 Is now, it goes to 1 and then it goes way up the imaginary axis. 781 00:54:48,180 --> 00:54:52,270 Well maybe not way up, but up. 782 00:54:52,270 --> 00:54:55,000 Sorry, I shouldn't have made it quite as bad as it was. 783 00:54:55,000 --> 00:55:00,720 If I reduce r, I don't go so far up, but no hope. 784 00:55:00,720 --> 00:55:02,620 So do you see why that one is bad, 785 00:55:02,620 --> 00:55:06,000 because the amplification factor -- 786 00:55:06,000 --> 00:55:09,560 von Neumann tells us what Courant did not tell us, 787 00:55:09,560 --> 00:55:13,300 that the amplification factor here has magnitude bigger than 788 00:55:13,300 --> 00:55:13,800 1. 789 00:55:13,800 --> 00:55:20,150 It's outside the circle, and every exponential is growing 790 00:55:20,150 --> 00:55:22,040 and there's no hope. 791 00:55:22,040 --> 00:55:22,540 OK. 792 00:55:22,540 --> 00:55:28,040 So the second last lecture on this section 5-2 of the notes 793 00:55:28,040 --> 00:55:33,540 will be about Lax-Friedrichs and Lax-Wendroff, 794 00:55:33,540 --> 00:55:39,750 order of accuracy, stability and actual behavior in practice. 795 00:55:39,750 --> 00:55:40,250 OK. 796 00:55:40,250 --> 00:55:41,100 See you Wednesday. 797 00:55:41,100 --> 00:55:42,350 Thanks.