take for example a pendulum which is a nonlinear oscillator. we can only solve this problem analytically in the limit that the pendulum sweeps out some small angle theta. why in the world is that even taught considering an iterative difference method can solve the nonlinear problem more accurately than the simplified model models reality?

if i'm wrong about this please correct me, i've obviously assumed the numerical soln to the nonlinear problem can be made arbitrarily accurate.