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 stabilityfo Back Ward Euler method which is given by:
Solution:
applying this method to the test equation
- we get
- let
- since G(z) approaches 0, ans Re(z) approaches infinity,
then B.E.M is L-stable.