Derivation of the order condition for the Implicit Runge-Kutta method

Question

I know how to derive the order condition for the explicit Runge-Kutta method by Taylor expansion, but do not know the implicit one. For instance, we list the two-stage implicit Runge-Kutta method for the autonomous ode $$ \frac{dy}{dx}=f(y), $$ $$ y^{(1)}=y_n+h(a_{11}f(y^{(1)})+a_{12}f(y^{(2)})), $$ $$ y^{(2)}=y_n+h(a_{21}f(y^{(1)})+a_{22}f(y^{(2)})), $$ $$ y_{n+1}=y_n+h(b_{1}f(y^{(1)})+b_{2}f(y^{(2)})). $$ Here $h$ is the step size. How to derive the condition for these coefficients if the order of accuracy is 2?

Answer 1

See https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/ for a short overview in form of slides.

There is not much difference, the Tayor method works the same way for implicit as for explicit methods. Only that the implicit method provides more degrees of freedom in the parameters.

Take care that the equations obtained from the B-tree calculus can be redundant.