Convergence of 2nd order Runge-Kutta

Question

How can one show that the 2nd order Runge-Kutta method is convergent?

Or, more generally, how is the convergence of Runge-Kutta methods studied?

Answer 1

By showing that it has second order. For one-step methods you get convergence from showing that the local truncation order $p$ is 1 or greater. That is, for any exact solution $y$ and the method step function $\Phi_f$ you have that $$ y(x+h)-[y(x)+h\Phi_f(x,y(x),h)]=O(h^{p+1}) $$ with $p\ge 1$. See the argument in this answer to see how a bound on the local error translates into a bound on the global error of size $O(h^p)$ implying convergence for $p\ge 1$.