Numerical Analysis/stability of RK methods/exercises
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Exercises
[edit | edit source]Ex:1
[edit | edit source]find the stability function for RK2 which is given by:
Solution:
applying this method to the test equation
- we get
the stability polynomial
Ex:2
[edit | edit source]find the absolute stability region for RK2.
Solution:
by setting
- the abs.stability region is given by
Ex:3
[edit | edit source]find the characteristic polynomial for RK2.
Solution:
it is divide both sides of the equation by you get
Ex:4
[edit | edit source]is 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
[edit | edit source]Determine 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.