Numerical Analysis/stability of RK methods/exercises

< Numerical Analysis < stability of RK methods

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{align} y_{n+1} &=y_n+h \lambda y_n+\frac {(h \lambda)^2}{2}y_n \\ \Rightarrow \end{align}

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 $r^i;r \ne 0$ 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

Re(z)\longrightarrow \infty

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

This article is issued from Wikiversity - version of the Saturday, April 02, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.