Optimum time step in ODE solvers

[adar]

Question 1 -

For solving nonlinear equations I use AdamsBDF solver. However, varying the number of discretization intervals (controlled by the variable "intvls") results in solutions quite different from one another. How can one determine the optimum number of intvls. I tried it by plotting the solutions by increasing the number of intervals till the two successive solution produce the same result. Is there any other way to find the optimum number of intervals?

Question 2 -

In oscillation problems, I am interested in knowing only the time required for a pendulum to reaach equilibrium (theta =0).

I do not want to get the entire solution but am interested only up to when the soultion reaches zero the first time. This is when the initial value of angle theta from vertical axis is given and I wish to find when it reaches zero the first time. How do I do this?

Any help is appreciated.

[adar]

I am answering my own question. 

Question 1: Please review my question on the same topic with the worksheet attached therein. I believe the ideal intvls=1000.  But I have yet to confirm it. 

Question 2: This is rather simpler. First obtain the solution matrix with three columns: time, rotation and angular velocity. Now in the second column, detect the change is the sign of theta. Note down the two values of time (first column): with positive and negative rotation. Consider the average of the two timings. This would be the time when the pendulum reaches zero rotation.

This prompts me to guess to get the answer to question 1 : In fact you can compare the time obtained in Q2 above with the natural period of the pendulum. If this is not the same then you may change the intvals to a different value till you get the natural period. Per Chopra et al, the time step should be 1/10th of the period of an oscillator. So, intvals=duration/timestep.  

[-MFra-]

Hi,

You should attach a file (.xmcd) related to your problem so that you get a comprehensive answer from the Mathcad community.