Module 5: Consistency, Stability and Convergence of General Single – Step Methods
Lecture 14: General Single Step Methods
Remark: If, for the differential equation the function f is a continuous function of and satisfies the Lipschitz condition in in the region , then we can see that for all the method discussed earlier, the increment function will also satisfy these conditions for . For example, in the case of the mid-point rule, we have
which is continuous in and if is, and
Thus, the increment function satisfies the Lipschitz condition in for
0 . Also note that is continuous in if is continuous in and .
the function f
is a continuous function of 
and satisfies the Lipschitz condition in 
in the region 
, then we can see that for all the method discussed earlier, the increment function 
will also satisfy these conditions for 
. For example, in the case of the mid-point rule, we have 
is, and 
. Also note that 
if 
