manually run ode45 step by step - and impose filter to the half-way solution

Hi everybody~

I'm solving a ode:

%%code 1

t = [tt1 tt2 tt3 ... tt100];
[t, y] = ode45(@equation, t, y1, options);  % y(t=1)=y1

Now I wanna to run it step by step. I wanna verify the above code is equivalent to

%%code 2

t = [tt1 tt2 tt3 ... tt100];
for i = 1:99
      [t(i+1) y(i+1)] = ode45(@equation, [t(i) t(i+1)], y(i), options);
end

The reason I want step-by-step is, I want to impose filters to y in the middle. Say, at t=t(5), I do y(5) = y(5)*filter, and then continue to solve y(6).

Let me know whether code 1 is equivalent to code 2.

Any thoughts of other ways to do this work is appreciated. I think this is called "numerical filtering" - but didn't find anything to read about it. Any recommended reading is appreciated. if true % code endThanks~

Answer 1

[Jan]

The calculations are equivalent, but not exactly identical: While in the first version ODE45 uses the stepsize from the former step after reaching one of the intermediate points, the 2nd method restarts the integrations with a new estimated stepsize. Due to rounding and local discretization errors, the results wil be slightly different - except if the solution is instable and the small deviations are amplified, which can cause large differences. But then the "result" of the integration is doubtful at all.

[Yuji Zhang]

Hi Jan~

Thank u for your help.

I learn from "help" that, t= t(i) is just the "record points" I want y(i) to be recorded. code45 automatically determines internal step-length of delta_t. You were talking about how the step-length is determined. Is there something I can read about it?

Let me know. Thank u!

[Jan]

This is a perfect question for an internet serach engine: Simply ask your favorite engine for "Runge Kutta stepsize control"