Exercises

Answer these question(s).

1. Suppose that you are using a Runge-Kutta second order ODE solution method and you specify that you want to reduce the error from $\epsilon ={10}^{-3}$ to $\epsilon ={10}^{-5}$ .

   How many more time steps will you have to use to obtain the solution at the same point in time?

   You will have to use 10x more time steps because you have to reduce h by 10x.

2. Now, suppose that you are using a Runge-Kutta fourth order ODE solution method and you want to reduce your error from $\epsilon ={10}^{-3}$ to $\epsilon ={10}^{-5}$ .

   How many more time steps will you have to use to obtain the solution at the same point in time?

   You will have to reduce the time step by ${\left(\raisebox{1ex}{${10}^{-3}$}\!\left/ \!\raisebox{-1ex}{${10}^{-5}$}\right.\right)}^{1/4}$ or $\sqrt{10}\times$ so you will need approximately 3-4 times as many steps.