2007-149.md

course	course_year	question_number	tags	title	year
Numerical Analysis	II	149	II 2007 Numerical Analysis	1.II.38C	2007

(a) For a numerical method to solve $y^{\prime}=f(t, y)$, define the linear stability domain and state when such a method is A-stable.

(b) Determine all values of the real parameter $a$ for which the Runge-Kutta method

$$\begin{aligned} k_{1} &=f\left(t_{n}+\left(\frac{1}{2}-a\right) h, y_{n}+h\left[\frac{1}{4} k_{1}+\left(\frac{1}{4}-a\right) k_{2}\right]\right) \\ k_{2} &=f\left(t_{n}+\left(\frac{1}{2}+a\right) h, y_{n}+h\left[\left(\frac{1}{4}+a\right) k_{1}+\frac{1}{4} k_{2}\right]\right) \\ y_{n+1} &=y_{n}+\frac{1}{2} h\left(k_{1}+k_{2}\right) \end{aligned}$$

is A-stable.