2005-151.md

course	course_year	question_number	tags	title	year
Numerical Analysis	II	151	II 2005 Numerical Analysis	3.II.38A	2005

Consider the Runge-Kutta method

$$\begin{aligned} k_{1} &=f\left(y_{n}\right) \\ k_{2} &=f\left(y_{n}+(1-a) h k_{1}+a h k_{2}\right), \\ y_{n+1} &=y_{n}+\frac{h}{2}\left(k_{1}+k_{2}\right) \end{aligned}$$

for the solution of the scalar ordinary differential equation $y^{\prime}=f(y)$. Here $a$ is a real parameter.

(a) Determine the order of the method.

(b) Find the range of values of $a$ for which the method is A-stable.