# Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody!

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!

a) "but without calculating derivatives" This is not correct, unfortunately. You still need to calculate the derivatives at the given point to get the coefficients of $$x$$ terms. As can be seen in the definition of Taylor series $$f(x) = \sum^\infty_{n=0}\frac{f^n(a)}{n!}(x-a)^n$$
The main advantages of Runge-Kutta methods are that they are easy to implement, they are very stable, and they are self-starting'' (i.e., unlike muti-step methods, we do not have to treat the first few steps taken by a single-step integration method as special cases). The primary disadvantages of Runge-Kutta methods are that they require significantly more computer time than multi-step methods of comparable accuracy, and they do not easily yield good global estimates of the truncation error.