Numerical Analysis/stability of RK methods/Exercises
Exercises edit
Ex:1 edit
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
find the absolute stability region for RK2.
Solution:
by setting
- the abs.stability region is given by
Ex:3 edit
find the characteristic polynomial for RK2.
Solution:
it is divide both sides of the equation by you get
Ex:4 edit
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
determine the stabilityfo Back Ward Euler method which is given by:
Solution:
applying this method to the test equation
- we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \begin{align} y_{n+1} &=\frac {1}{1-h \lambda}y_n \\ \Rightarrow &=\frac {1}{1-z}y_n; z=h\lambda\\ \end{align}}
- let
- since G(z) approaches 0, ans Re(z) approaches infinity,
then B.E.M is L-stable.