Examples

1. Suppose that you are using Euler's method and you estimate that your solution at a certain time point f(t₂₉) is $\epsilon ={10}^{-3}$ .

   How much would you have to reduce the time step h in the solution algorithm to reduce the error at this point to $\epsilon ={10}^{-5}$ ?

   You would reduce it by 100x because Euler's method is a first order method.

2. Suppose that you are using a second order method and you wish to reduce the error in your solution from 10^-3 to 10^-5.

   How much would you have to reduce the time step h?

   You would reduce h by 10x because a second order method reduces error such that $\epsilon \sim h^2$ .