2014-61.md

course	course_year	question_number	tags	title	year
Numerical Analysis	IB	61	IB 2014 Numerical Analysis	Paper 3, Section II, C	2014

A Runge-Kutta scheme is given by

$$k_{1}=h f\left(y_{n}\right), \quad k_{2}=h f\left(y_{n}+\left[(1-a) k_{1}+a k_{2}\right]\right), \quad y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)$$

for the solution of an autonomous differential equation $y^{\prime}=f(y)$, where $a$ is a real parameter. What is the order of the scheme? Identify all values of $a$ for which the scheme is A-stable. Determine the linear stability domain for this range.