You can model the non-linear circuit by taking it in stages: the switch is open (mechanical or transistor) - now carry out a linear analysis. The switch closes - use Laplace transform theory to find the transient signal.
This plus the final state of the previous analysis (states of capacitors and inductors) allows your to carry out an analysis of the next stage. Repeat ad nauseum. This is what a good Spice program does. The limitations are based on the models for the individual properties of the components being used - wrong specs, wrong results.
Oh - and if you want perfection, then you must know the lengths of wires, the resistivity, and the qualities of all of the connections, soldered and unsoldered.