I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me feedback if I understand everything correctly:
There are two numeric approaches for solving differential equations:
a) Based on Taylor Series Approximation: Euler, Runge Kutta, etc. Goal: to have similar accuracy as with Taylor series but without calculating derivatives. Work-around was developed, where you only evaluate functions at certain points without calculating derivatives.
b) Based on Interpolation Polynomials: Multi-Step Methods, Collocation methods: Make use of past information; no intermediate calculations (as in Runge-Kutta) . General idea: fit a polynomial using this past data + extrapolate from tn to tn+1
Are ther caes, where Runge-Kutta methods are better compared to Multi-step methods and vice versa?
Thank you very much for your help!