What to do (if anything) about the first point of a backward Euler solution being discontinuous?

[moorepants]

If you have N nodes, then you only get N-1 constraints for the equations of motion due to the discretization:

$\frac{d\mathbf{y}}{dt} \approx \frac{\mathbf{y}i - \mathbf{y}{i-1}}{h}$

dy   y_i - y_{i-1}
-- = -----------------
dt   t_i - t_{i-1}

So for i = 2, ..., N we make N-1 constraint equations. When n = 1 there is no n-1=0 point to make a constraint with. Thus the n=1 node does not have to abide by the equations of motion relative to the prior non-existent node.

Examples:

https://opty.readthedocs.io/latest/examples/beginner/plot_betts_10_7.html

https://opty.readthedocs.io/latest/examples/beginner/plot_pendulum_swing_up_fixed_duration.html

Only on the input:

https://opty.readthedocs.io/latest/examples/beginner/plot_betts_10_50.html

What can we do:

1. We could return a solution for only n=1, ..., N.
2. Don't plot the first node.
3. Add an extra node when formulating the problem, but they remove it in the solution so the user gets the expected results back.
4. Use forward Euler on the first node.
5. other?

[Peter230655]

Now at least I understand where these 'jumps' come from. And it explains another issue I oncve had (forgot exatly where)
On some examples, I simply set, say, an instance_constraint like $q_1 - 0.0$ when I saw this jump to 0.0 on the plot - but of course the worst way.
Your point 2. would be the easiest solution to implement, I can do if that's the decision.
Would point 4. be the 'correct' one mathematically, or equivalent to point 3. ?

[moorepants]

It is not necessary to "quote reply" on every reply, it makes the threads unnecessarily long.

[moorepants]

I think my point 4. is a proper fix.

[Peter230655]

Just curiosity: Would 4. not be equivalent to 3.?

[moorepants]

I don't think so.

[Peter230655]

On practical terms: would this improvement help convergence?

[moorepants]

No, I don't think so.

[Peter230655]

So really just optical. Then changing the plotting would be simplest, I could do it. :-)

[moorepants]

But that only "fixes" it for our plots. The data still has this bad value.

[moorepants]

I'm fine if we fix it for our plots, but this is just a bandaid and we still need to actually fix it.

[Peter230655]

Bandaid I could do, rest is over my head.
When one does such a bandaid, would I mention this in the docstrings?

Curiosity, I never looked at it. Would integration_method=midpoint exhibit the same issues?

[moorepants]

I would like to think about this one more. We don't need an immediate fix.

[Peter230655]

Just let me know whenever you would want me to do the bandaid - only thing I could do. :-)

[moorepants]

We may not do the bandaid.