Lecture 16: Derivation of Implicit Runge-Kutta methods
The general R-stage implicit Runge-Kutta method is defined by
(6.1)
where
with
(6.2)
and
,
(6.3)
The functions are no longer defined explicitly but by a set of implicit equations and thus making the derivation of such methods tedious. We shall consider here only the two-stage method obtained by setting in (6.1)-(6.3). Let us also recall the expansion for which is given by
, 
are no longer defined explicitly but by a set of 
implicit equations and thus making the derivation of such methods tedious. We shall consider here only the two-stage method obtained by setting 
in (6.1)-(6.3). Let us also recall the expansion for 
which is given by 
