Numerical Analysis/stability of RK methods/exercises

Exercises

Ex:1

find the stability function for RK2 which is given by: $y_{n+1}=y_n+hf(t+\frac{h}{2},y_n+\frac{hf(t_n,y_n)}{2})$

Solution:

applying this method to the test equation $y^{\prime }=\lambda y$

we get $\begin{array}{cc}{y}_{n+1}& =y_n+h\lambda y_n+\frac{(h\lambda {)}^2}{2}{y}_n\\ \Rightarrow \end{array}$

the stability polynomial $\varphi (z)=1+h\lambda +\frac{(h\lambda {)}^2}{2}$

 

Ex:2

find the absolute stability region for RK2.

Solution:

by setting $z=h\lambda$

$\Rightarrow$ the abs.stability region is given by $\{z\in ℂ||1+h\lambda +\frac{(h\lambda {)}^2}{2}|<1\}$

 

Ex:3

find the characteristic polynomial for RK2.

Solution:

it is $r^{i+1}=(1+z+\frac{z^2}{2})r^i$ divide both sides of the equation by $r^i;r\ne 0$ you get $r^{i+1}=(1+z+\frac{z^2}{2})r^i$

$\Rightarrow \varphi (r,z)=r-1+z+\frac{z^2}{2}$

 

Ex:4

is RK2 stable, if it is what type of stability.

Solution:

you get $r^{i+1}=(1+z+\frac{z^2}{2})r^i$ by setting z=0,

$\Rightarrow r=1$

so the method is strongly stable since r=1, is the only root, and has a value of 1.

 

Ex:5

Determine the stability of Back ward Euler method.

Solution:

$y_{n+1}=y_n+hf(t_{n+1},y_{n+1})$

by applying this method to the test equation

then $y_{n+1}=y_n+h\lambda y_{n+1}$

and so $y_{n+1}=\frac{1}{1+h\lambda }{y}_n$

call $z=h\lambda$ and so:

$G(z)⟶0$ as $Re(z)⟶\mathrm{\infty }$

and so this is L-Stable method when applied to stiff equation.