I have been reading the Strogatz book on Nonlinear Ordinary Differential equations and I understand the graphical/qualitative method to solving these types of equations. However, Strogatz did not seem to address the role of numerical methods in solving nonlinear ODEs or systems of ODEs. I also looked at the book by Jordan and Smith on nonlinear ODEs, but they don't have a chapter on numerical methods for ODEs either.
So I was just trying to understand what is the role of numerical methods in looking at nonlinear odes? The basic approach of the qualitative methods seems to be curve sketching, so in a sense we are already using a numerical method--but we are drawing the nonlinear ODE itself instead of its integral curves right. But how does this approach relate to using numerical methods for linear ODEs like Euler, or Runge-Kutta, or Tsit?
Tsit stands for Tsitouras-5 ODE solver.