1 00:00:00,000 --> 00:00:04,040 Included one of the terms out of the two. So, let's look at the other one. 2 00:00:07,960 --> 00:00:12,600 Okay. Now, the other term is here. 3 00:00:21,340 --> 00:00:25,813 A non-linearity. Now, it's not that often you can solve 4 00:00:25,813 --> 00:00:29,955 non-linear ODEs exactly, sometimes you can, okay? 5 00:00:29,955 --> 00:00:33,517 But this one here, you can solve it exactly. 6 00:00:33,517 --> 00:00:37,428 In fact, The solution to this is just simply here. 7 00:00:37,428 --> 00:00:50,220 Let me write it down for you. There's this solution. 8 00:00:51,200 --> 00:00:54,609 Pretty nice. Give it the initial value. 9 00:00:54,609 --> 00:00:57,660 Multiply by the space factor. Done. 10 00:00:59,320 --> 00:01:03,250 So, what you see here is the two pieces, each individually, 11 00:01:03,250 --> 00:01:07,870 I can actually solve exactly. I don't have to put this ODE45 either, 12 00:01:07,870 --> 00:01:10,214 Right? What is the big cost for us? 13 00:01:10,214 --> 00:01:15,523 Part of the big cost for us is when we take time steps, is we have to develop 14 00:01:15,523 --> 00:01:20,695 these time stepping algorithms, right? We have to say, okay, ODE45 to march me 15 00:01:20,695 --> 00:01:25,177 forward into the future. I already have the exact solution in the 16 00:01:25,177 --> 00:01:28,205 future. So, if you think about it, you could say, 17 00:01:28,205 --> 00:01:31,011 well, This is a perfect place for using the 18 00:01:31,011 --> 00:01:36,492 split step method cuz in the split step method, what I could do is take a delta t 19 00:01:36,492 --> 00:01:40,800 of this and a delta t of that, And I have exact solutions for both. 20 00:01:41,260 --> 00:01:45,176 So, I not even, I'm not even doing time stepping anymore. 21 00:01:45,176 --> 00:01:48,310 I'm just taking, taking my initial condition, 22 00:01:48,310 --> 00:01:53,295 Ha, take a delta t of each one. And I know the exact expressions for them, 23 00:01:53,295 --> 00:01:58,280 and I got myself into the future. So, let's look at how this might look. 24 00:02:00,340 --> 00:02:06,160 Alright. So, here's an algorithm for doing this. 25 00:02:13,160 --> 00:02:17,406 Alright, did I spell that right? [laugh] Okay good, I think I did. 26 00:02:17,406 --> 00:02:20,777 Alright. Sometime when you look at words they just, 27 00:02:20,777 --> 00:02:24,080 see that's why I told you not to spell in public. 28 00:02:25,160 --> 00:02:29,301 Alright. Here's the algorithm. 29 00:02:29,301 --> 00:02:38,906 Start off with the initial condition. U0. 30 00:02:39,769 --> 00:02:46,300 So, this your initial function at, let's say, time zero. 31 00:02:47,260 --> 00:02:52,060 Okay. Now, we can do one of two things. We could either take a step in the 32 00:02:52,060 --> 00:02:55,860 dispersion first or in the nominary second or vice versa. 33 00:02:55,860 --> 00:02:59,860 But, let's go ahead a take a step first in the dispersion, 34 00:03:00,160 --> 00:03:01,448 Okay? Step two, then. 35 00:03:01,448 --> 00:03:04,838 Let's come back and tell, see what this says to us. 36 00:03:04,838 --> 00:03:10,127 This just says, okay, I need to take a delta t forward here and what I'll do is, 37 00:03:10,127 --> 00:03:12,703 here's my formula. This is exact, right? 38 00:03:12,703 --> 00:03:17,450 All I've got to do is take my initial condition before you transform it, 39 00:03:17,450 --> 00:03:22,400 multiply it by this factor. That gives me u hat and I want u which is 40 00:03:22,400 --> 00:03:25,882 just the inverse Fourier transformer bit. So. 41 00:03:26,072 --> 00:03:27,274 Delta t [inaudible]. Yes. 42 00:03:27,464 --> 00:03:32,841 And that'll be delta t, exactly. Okay? So, what we're going to do here is 43 00:03:32,841 --> 00:03:37,240 calculate, let's call it the dispersion effect. 44 00:03:37,660 --> 00:03:42,760 And it's going to give us a solution you want which is, what we're going to do is 45 00:03:42,760 --> 00:03:47,820 we're going to take the FFT of this u0, multiply it. 46 00:03:48,100 --> 00:04:01,000 Times i, k squared, delta t over two. And then, just inverse Fourier transform 47 00:04:01,000 --> 00:04:11,363 It cost me N log N to do this, Okay? So, it's very, very fast. 48 00:04:11,363 --> 00:04:14,191 I'm not, I'm not using an ODE solver, right? 49 00:04:14,191 --> 00:04:18,207 So, when you first look at that equation, you think, oh, like, I've got, I'm 50 00:04:18,207 --> 00:04:21,883 stepping forward into the future. I've got a, you know, Fourier transform and 51 00:04:21,883 --> 00:04:25,898 everything. ODE45 forward into time. No, not in this case. 52 00:04:25,898 --> 00:04:30,197 I'm taking advantage of the fact that, pretty much, I know the exact solution. 53 00:04:30,197 --> 00:04:34,320 Okay? And this highlights something more 54 00:04:34,320 --> 00:04:39,144 important that's there. Anytime you have information about your 55 00:04:39,144 --> 00:04:45,371 solutions and you're not making use of the fact that you know something, you're being 56 00:04:45,371 --> 00:04:49,013 dumb, okay? Alright. You always take advantage, every 57 00:04:49,013 --> 00:04:53,746 single time you can, anything you know, you bring into the problem. 58 00:04:53,746 --> 00:04:57,824 If you know it, make use of it, okay? Always, always, always. 59 00:04:57,824 --> 00:05:03,723 By the way, you don't have to make use of it, you can do the slow way that doesn't 60 00:05:03,723 --> 00:05:08,349 take advantage of anything. But, this is what you really want to do 61 00:05:08,349 --> 00:05:12,236 when you program. Take advantage of the things you know. 62 00:05:12,236 --> 00:05:16,600 I just took a delta t step forward, just using the dispersion. 63 00:05:16,600 --> 00:05:23,571 And now, I want to take, instead of the dispersion, 64 00:05:23,571 --> 00:05:29,739 Now I want to take the non-linearity. So, the non-linearities has a formula 65 00:05:29,739 --> 00:05:33,401 there, and here it is. It just says, if I want to go forward, 66 00:05:33,401 --> 00:05:38,567 If I have some initial condition to get delta t into the future, I just, there's 67 00:05:38,567 --> 00:05:41,837 my formula. Now, that becomes a delta t. And usually 68 00:05:41,837 --> 00:05:44,453 this turns out u1. U1 is the initial condition for this. 69 00:05:44,453 --> 00:05:49,180 Okay? Alright. So, let's call this u2. 70 00:05:49,580 --> 00:05:59,994 U2 now is you just take this u1 and multiply it by exponent i u1 squared, 71 00:05:59,994 --> 00:06:04,661 delta t, Okay? 72 00:06:04,661 --> 00:06:11,195 This is the exact solution to this thing. And I've taken advantage of that. 73 00:06:11,195 --> 00:06:17,200 Three, step four, I guess, then you just say, alright, u0 is equal to u2. 74 00:06:17,200 --> 00:06:17,200 I'm sorry. Delta t or deta t, or delta t with delta 75 00:06:17,200 --> 00:06:24,682 t? been in [inaudible] What's your? Delta t over two. 76 00:06:25,069 --> 00:06:28,633 But that's the, the increment is delta t, right? 77 00:06:28,865 --> 00:06:32,815 Mm-hm. And then, number three increase delta t 78 00:06:32,815 --> 00:06:33,822 again. Yep. 79 00:06:34,055 --> 00:06:35,526 So, is t2(delta equal to two delta t?? Or is it. 80 00:06:35,759 --> 00:06:37,695 No, no. No, just delta t. 81 00:06:37,928 --> 00:06:43,273 I'm taking a delta t step forward. So, it's kind of like you're saying, 82 00:06:43,273 --> 00:06:47,920 basically, I want to evolve the physics from zero to delta t. 83 00:06:47,920 --> 00:06:49,160 Yep. Um-hm, um-hm. 84 00:06:49,819 --> 00:06:51,969 Yeah, that's right, that's right. Yeah. 85 00:06:52,148 --> 00:06:55,194 Oh, one parenthesis missing. So, is everybody okay with that? 86 00:06:55,194 --> 00:06:58,180 There's your little algorithm, boom, boom, boom, boom. 87 00:06:58,180 --> 00:07:02,898 And, oftentimes if you can take advantage of that kind of structure, if you have 88 00:07:02,898 --> 00:07:07,616 your pieces of your dynamics that are sitting there in your PDE that are kind of 89 00:07:07,616 --> 00:07:12,333 being separated and you know how to solve some of those pieces, this is awesome. 90 00:07:12,333 --> 00:07:17,111 Couple of things that are nice about it. One, we talked about numerical stiffness, 91 00:07:17,111 --> 00:07:20,309 right? Here, totally just gets rid of numerical 92 00:07:20,309 --> 00:07:23,929 stiffness. Why? Because you have actually used exact 93 00:07:23,929 --> 00:07:25,349 solutions. Fantastic. 94 00:07:25,349 --> 00:07:26,555 It's great, Right? 95 00:07:26,555 --> 00:07:32,092 So, when you have things like that, in fact, when you think about what we talked 96 00:07:32,092 --> 00:07:38,769 about in terms of the integrating factor method with removing numerical stiffness, 97 00:07:38,769 --> 00:07:43,171 it's kind of something like this but this taking it one step further. 98 00:07:43,171 --> 00:07:47,702 And you're saying, just take steps independently. If delta t is small enough, 99 00:07:47,702 --> 00:07:49,680 you'll be in pretty good shape.