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 $y'=\lambda y$

we get ${\begin{aligned}y_{n+1}&=y_{n}+h\lambda y_{n}+{\frac {(h\lambda )^{2}}{2}}y_{n}\\\Rightarrow \end{aligned}}$

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

 

Ex:2

find the absolute stability region for RK2.

Solution:

by setting $z=h\lambda$

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

 

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 $r^{i};r\neq 0$ you get $r^{i+1}=(1+z+{\frac {z^{2}}{2}})r^{i}$

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

 

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:

$G(z)\longrightarrow 0$ as $Re(z)\longrightarrow \infty$

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