Given the Cauchy problem
$$\left\{\begin{array}{ll} y(t_0)=y_0&&&&&&&&&&\\ y'(t) =f(t,y)\\ \end{array}\right.$$ The equivalent one-step method is of the form $$\left\{\begin{array}{ll} z_0=y_0&&&&&\\ z_{n+1}=z_n+h\phi(t_n, z_n, h)\\ \end{array}\right.$$
Here is my question
- How can I deduce the stability, consistency, convergence and its order of convergence of this method? Please, define and explain each of them in the order you deem as right.
I know I have to apply the Lipschitzien property of $\phi$ but I get really confused on where or how to apply it.
Another thing is the error, my prof uses two of 'em, one is $\tau_n$ and the other is $\epsilon_n $. What is the difference and relationship between this two error and their relationships with the stability, convergence et al?
I know how to develop all of the one-step methods that I've been taught, I just have trouble understanding the concept of stability, consistency, convergence and order of convergence.
Thank you for your time.