# Numerical Analysis/stability of RK methods/exercises

## Exercises

edit### Ex:1

editfind the stability function for RK2 which is given by:

Solution:

applying this method to the test equation

- we get

the stability polynomial

### Ex:2

editfind the absolute stability region for RK2.

Solution:

by setting

- the abs.stability region is given by

### Ex:3

editfind the characteristic polynomial for RK2.

Solution:

it is divide both sides of the equation by you get

### Ex:4

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

Solution:

you get by setting z=0,

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

### Ex:5

editDetermine the stability of Back ward Euler method.

Solution:

- by applying this method to the test equation
- then
- and so
- call and so:
- as
- and so this is L-Stable method when applied to stiff equation.