In Runge - Kutta - Fehlberg methods, sometimes and in some cases the answer depends on the method we define the error and also on the magnitude of the error.
In the case I am working on, there are several "zero crossings"; which is problematic for 'scaling' in the definition of precision. Besides, the solution depends on the steplength; surprisingly, not its precision, but its behavior! So, the numerical method is not reliable at all.
How can I define error and control the sensitivity of the numerical method to be reliable?
I saw in the literature some different methods for defining the error and controlling the steplength. Kash and Carp happens to be the most famous. But the problem is the way all these methods 'scale' the precision. I need the step correction method to be 'robust' enough.
The most weird case is that in some time-steps the error (i.e., the difference between the 4th order estimation and the 5th order solution) becomes zero! The zero error causes the corrected steplength to become Inf!
How can I write the steplength correction to overcome this problem? Also, if anyone knows any useful reference please announce me.
Thanks in advance
