Understanding Stability Regions in Numerical Methods

[TheCanadian]

I am reading about the stability of different numerical methods, although never exactly came across a good definition of what stability regions are. For example, considering this plot, what exactly is the figure representing? What do the unlabelled axes mean in this context? Are these stability regions the same for a particular numerical method regardless of the problem considered?

 

[FactChecker]

The plot is showing the values of z in the complex plane where the Euler method is stable. The horizontal axis are the real values and the vertical axis are the imaginary values. z is allowed to be complex for the reason that complex analysis provides methods for determining the stable regions. But in the example shown, the real values z = hk are the ones that apply. So as long as -2 < hk < 0, z = hk is in the Euler method stable region. The paragraph just above gives some explanation of the Runge-Kutta methods in general and you can see in the first line where ∅(hk) is defined and that leads to studying ∅(z).

 

[TheCanadian]

The plot is showing the values of z in the complex plane where the Euler method is stable. The horizontal axis are the real values and the vertical axis are the imaginary values. z is allowed to be complex for the reason that complex analysis provides methods for determining the stable regions. But in the example shown, the real values z = hk are the ones that apply. So as long as -2 < hk < 0, z = hk is in the Euler method stable region. The paragraph just above gives some explanation of the Runge-Kutta methods in general and you can see in the first line where ∅(hk) is defined and that leads to studying ∅(z).

Thank you for the explanation! If hk corresponds to a step size, what is the meaning of a negative region of stability? Also, what exactly are the units for -2 < hk < 0? Wouldn't this be simply defined based on how I scale my domain?

 

[FactChecker]

looking closer at the link explanation, I see that k can be a complex number and h looks like a step size. So hk is not necessarily a real number. The entire stability circle of the diagram, including complex values of hk applies.

 

[TheCanadian]

looking closer at the link explanation, I see that k can be a complex number and h looks like a step size. So hk is not necessarily a real number. The entire stability circle of the diagram, including complex values of hk applies.

So do all methods always have their stability based on its properties when testing the equation: $y' = ky$ specifically? If so, isn't this kind of arbitrary and likely not a very practical definition when considering more complex cases (e.g. hyperbolic PDEs)? And are we assuming then that $h$ must be a step on a real-valued domain?