[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.00,Default,,0000,0000,0000,,It turns out that all positive values for the time step are okay no matter how high.
Dialogue: 0,0:00:04.00,0:00:07.00,Default,,0000,0000,0000,,This is how similar to the backward Euler method
Dialogue: 0,0:00:07.00,0:00:10.00,Default,,0000,0000,0000,,and now we're going to look into why that is the case.
Dialogue: 0,0:00:10.00,0:00:14.00,Default,,0000,0000,0000,,The trapezoidal weir turns this differential equation into the following.
Dialogue: 0,0:00:14.00,0:00:20.00,Default,,0000,0000,0000,,The next value of x equals the current value of x plus the time step
Dialogue: 0,0:00:20.00,0:00:27.00,Default,,0000,0000,0000,,times the average of two rates of change, the current rate of change (-kx₁)
Dialogue: 0,0:00:27.00,0:00:30.00,Default,,0000,0000,0000,,and the next rate of change (-kx₂).
Dialogue: 0,0:00:30.00,0:00:36.00,Default,,0000,0000,0000,,I bring the x₂ to the left-hand side and combined this x₁ with that x₁
Dialogue: 0,0:00:36.00,0:00:48.00,Default,,0000,0000,0000,,and get (1+hk/2) * x2 = (1-hk/2) * x1.
Dialogue: 0,0:00:48.00,0:00:57.00,Default,,0000,0000,0000,,Let's check that, 1x₂, it's here. (hk/2)x₂ appears on the right-hand side with a - sign.
Dialogue: 0,0:00:57.00,0:01:05.00,Default,,0000,0000,0000,,1x₁ appears here minus (hk/2)x₁ - (hk/2)x₁ appears here.
Dialogue: 0,0:01:05.00,0:01:16.00,Default,,0000,0000,0000,,Correct. And that's easy to solve for x₂. x₂ = (1-hk/2)/(1+hk/2)*x₁.
Dialogue: 0,0:01:16.00,0:01:21.00,Default,,0000,0000,0000,,If you look at what happens with the next step. We of course, get a second factor of that sort.
Dialogue: 0,0:01:21.00,0:01:26.00,Default,,0000,0000,0000,,The next next step, another factor of that sort. So what happens is that we get powers of this factor.
Dialogue: 0,0:01:26.00,0:01:30.00,Default,,0000,0000,0000,,The question is what's going to happen about powers of that factor.
Dialogue: 0,0:01:30.00,0:01:36.00,Default,,0000,0000,0000,,So we see that to get it from x₁ to x₂, we have to multiple by this factor
Dialogue: 0,0:01:36.00,0:01:40.00,Default,,0000,0000,0000,,and this is going to occur every step if we want to go from x₂ to x₃,
Dialogue: 0,0:01:40.00,0:01:43.00,Default,,0000,0000,0000,,we again multiply by that factor and so on.
Dialogue: 0,0:01:43.00,0:01:46.00,Default,,0000,0000,0000,,So in the end, we get the powers of this factor.
Dialogue: 0,0:01:46.00,0:01:51.00,Default,,0000,0000,0000,,So the question is as the nth power of this expression converts to 0,
Dialogue: 0,0:01:51.00,0:01:56.00,Default,,0000,0000,0000,,as it intends to infinity to check that, one can analyze three different cases.
Dialogue: 0,0:01:56.00,0:01:59.00,Default,,0000,0000,0000,,Let's first check what happens if the step size is moderate.
Dialogue: 0,0:01:59.00,0:02:09.00,Default,,0000,0000,0000,,For instance, if this expression hk/2 equals 0.1 and we get (1-0.1)/(1+0.1)
Dialogue: 0,0:02:09.00,0:02:14.00,Default,,0000,0000,0000,,meaning (0.9)/(1.1), this is a positive number, <1.
Dialogue: 0,0:02:14.00,0:02:19.00,Default,,0000,0000,0000,,If you take higher and higher powers of that, this number is going to become 0.
Dialogue: 0,0:02:19.00,0:02:24.00,Default,,0000,0000,0000,,The numerator becomes 0.9. The denominator becomes 1.1.
Dialogue: 0,0:02:24.00,0:02:31.00,Default,,0000,0000,0000,,The fraction is a number between 0 and 1 and if we take higher and higher powers of such a number,
Dialogue: 0,0:02:31.00,0:02:34.00,Default,,0000,0000,0000,,the results converge to 0, so this is stable.
Dialogue: 0,0:02:34.00,0:02:36.00,Default,,0000,0000,0000,,So in this case, we have stability.
Dialogue: 0,0:02:36.00,0:02:43.00,Default,,0000,0000,0000,,If hk/2 happens to be equal to 1, we get a fraction with a 0 in the numerator,
Dialogue: 0,0:02:43.00,0:02:48.00,Default,,0000,0000,0000,,that's always 0, no problem with that, so we have stability in that case.
Dialogue: 0,0:02:48.00,0:02:55.00,Default,,0000,0000,0000,,And if you use a really high value of h, for instance, such that hk/2=9,
Dialogue: 0,0:02:55.00,0:03:02.00,Default,,0000,0000,0000,,we get something like 1-9=-8 divided by 1+9=10.
Dialogue: 0,0:03:02.00,0:03:05.00,Default,,0000,0000,0000,,This is a negative number with absolute value less than 1.
Dialogue: 0,0:03:05.00,0:03:11.00,Default,,0000,0000,0000,,So, if we take higher and higher powers, the result converges to 0,
Dialogue: 0,0:03:11.00,0:03:16.00,Default,,0000,0000,0000,,we also have stability in this case but you see that this convergence comes along with an oscillation
Dialogue: 0,0:03:16.00,0:03:23.00,Default,,0000,0000,0000,,that's time changes from minus to plus to minus to plus as we form even powers and odd powers.
Dialogue: 0,0:03:23.00,0:03:28.00,Default,,0000,0000,0000,,The result has a positive sign. For odd powers, the result has a negative sign.
Dialogue: 0,0:03:28.00,9:59:59.99,Default,,0000,0000,0000,,So we see some kind of decaying oscillation here.