Absolute Stability Region

Resources

LeVeque, Randall J. 2007. “Absolute Stability for Ordinary Differential Equations.”

Main Idea

In the numerical solution of Ordinary Differential Equations, a numerical method or Time Integrator is stable depending on the value of $λ \cdot d t$ , where $λ$ comes from the test equation

\frac{d y}{d t} = λ y .

For a (linear?) numerical integration method applied to this problem, we have

y_{k + 1} = ϕ (λ Δ t) y_{k},

and also

y_{n} = {(ϕ (λ Δ t))}^{n} y_{0} .

Taking $t \to \infty$ , $n \to \infty$ and for nonzero $y_{0}$ , in order to have finite $y$ , we must have

| ϕ (λ Δ t) | \leq 1 .

The absolute stability region is thus the values of $z = λ Δ t$ , such that this holds:

{z \in C : | ϕ (z) | \leq 1} .

Example

For forward Euler on the test equation, $y_{k + 1} = y_{k} + Δ t \cdot λ y_{k}$ , so $ϕ (z) = 1 + z$ . The region of stability is ${z \in C : | 1 + z | \leq 1}$ , which is a ball of radius $1$ , centered at $z = - 1$ .

Extensions

In the above, we took $λ \in C$ arbitrarily. Or in other words, the test equation seems arbitrary. Yet, $λ$ can be interpreted as an eigenvalue of a linear dynamical system / ODE for a specific mode. Then, if $λ$ is in the absolute stability region, the dynamical system is stable along this mode! Further, through the Method of Lines, we can pair a Partial Differential Equation with a spatial discretization such as Finite Differences to give a system of ODEs, and perform this same analysis.

Resources

Related

Main Idea

Extensions