WEBVTT 00:00:00.000 --> 00:00:04.000 It turns out that all positive values for the time step are okay no matter how high. 00:00:04.000 --> 00:00:07.000 This is how similar to the backward Euler method 00:00:07.000 --> 00:00:10.000 and now we're going to look into why that is the case. 00:00:10.000 --> 00:00:14.000 The trapezoidal weir turns this differential equation into the following. 00:00:14.000 --> 00:00:20.000 The next value of x equals the current value of x plus the time step 00:00:20.000 --> 00:00:27.000 times the average of two rates of change, the current rate of change (-kx₁) 00:00:27.000 --> 00:00:30.000 and the next rate of change (-kx₂). 00:00:30.000 --> 00:00:36.000 I bring the x₂ to the left-hand side and combined this x₁ with that x₁ 00:00:36.000 --> 00:00:48.000 and get (1+hk/2) * x2 = (1-hk/2) * x1. 00:00:48.000 --> 00:00:57.000 Let's check that, 1x₂, it's here. (hk/2)x₂ appears on the right-hand side with a - sign. 00:00:57.000 --> 00:01:05.000 1x₁ appears here minus (hk/2)x₁ - (hk/2)x₁ appears here. 00:01:05.000 --> 00:01:16.000 Correct. And that's easy to solve for x₂. x₂ = (1-hk/2)/(1+hk/2)*x₁. 00:01:16.000 --> 00:01:21.000 If you look at what happens with the next step. We of course, get a second factor of that sort. 00:01:21.000 --> 00:01:26.000 The next next step, another factor of that sort. So what happens is that we get powers of this factor. 00:01:26.000 --> 00:01:30.000 The question is what's going to happen about powers of that factor. 00:01:30.000 --> 00:01:36.000 So we see that to get it from x₁ to x₂, we have to multiple by this factor 00:01:36.000 --> 00:01:40.000 and this is going to occur every step if we want to go from x₂ to x₃, 00:01:40.000 --> 00:01:43.000 we again multiply by that factor and so on. 00:01:43.000 --> 00:01:46.000 So in the end, we get the powers of this factor. 00:01:46.000 --> 00:01:51.000 So the question is as the nth power of this expression converts to 0, 00:01:51.000 --> 00:01:56.000 as it intends to infinity to check that, one can analyze three different cases. 00:01:56.000 --> 00:01:59.000 Let's first check what happens if the step size is moderate. 00:01:59.000 --> 00:02:09.000 For instance, if this expression hk/2 equals 0.1 and we get (1-0.1)/(1+0.1) 00:02:09.000 --> 00:02:14.000 meaning (0.9)/(1.1), this is a positive number, <1. 00:02:14.000 --> 00:02:19.000 If you take higher and higher powers of that, this number is going to become 0. 00:02:19.000 --> 00:02:24.000 The numerator becomes 0.9. The denominator becomes 1.1. 00:02:24.000 --> 00:02:31.000 The fraction is a number between 0 and 1 and if we take higher and higher powers of such a number, 00:02:31.000 --> 00:02:34.000 the results converge to 0, so this is stable. 00:02:34.000 --> 00:02:36.000 So in this case, we have stability. 00:02:36.000 --> 00:02:43.000 If hk/2 happens to be equal to 1, we get a fraction with a 0 in the numerator, 00:02:43.000 --> 00:02:48.000 that's always 0, no problem with that, so we have stability in that case. 00:02:48.000 --> 00:02:55.000 And if you use a really high value of h, for instance, such that hk/2=9, 00:02:55.000 --> 00:03:02.000 we get something like 1-9=-8 divided by 1+9=10. 00:03:02.000 --> 00:03:05.000 This is a negative number with absolute value less than 1. 00:03:05.000 --> 00:03:11.000 So, if we take higher and higher powers, the result converges to 0, 00:03:11.000 --> 00:03:16.000 we also have stability in this case but you see that this convergence comes along with an oscillation 00:03:16.000 --> 00:03:23.000 that's time changes from minus to plus to minus to plus as we form even powers and odd powers. 00:03:23.000 --> 00:03:28.000 The result has a positive sign. For odd powers, the result has a negative sign. 00:03:28.000 --> 99:59:59.999 So we see some kind of decaying oscillation here.