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In class today was a highlight of stability methods for linear multistep methods. To motivate the methods used, it is helpful to take a step back and review stability concepts for LDE systems.

By way of example, consider a second order LDE homogeneous system defined by

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:20}

\frac{d^2 x}{dt^2} + 3 \frac{dx}{dt} + 2 = 0.

\end{equation}

Such a system can be solved by assuming an exponential solution, say

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:40}

x(t) = e^{s t}.

\end{equation}

Substitution gives

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:60}

e^{st} \lr{ s^2 + 3 s + 2 } = 0,

\end{equation}

The polynomial part of this equation, the characteristic equation has roots \( s = -2, -1 \).

The general solution of \ref{eqn:stabilityLDEandDiscreteTime:20} is formed by a superposition of solutions for each value of \(s\)

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:80}

x(t) = a e^{-2 t} + b e^{-t}.

\end{equation}

Independent of any selection of the superposition constants \( a, b \), this function will not blow up as \( t \rightarrow \infty \).

This “stability” is due to the fact that both of the characteristic equation roots lie in the left hand Argand plane.

Now consider a discretized form of this LDE. This will have the form

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:100}

\begin{aligned}

0 &=

\inv{\lr{\Delta t}^2}

\lr{ x_{n+2} – 2 x_{n-1} + x_n } + \frac{3}{\Delta t} \lr{ x_{n+1} – x_n } + 2

x_n \\

&=

x_{n+2} \lr{

\inv{\lr{\Delta t}^2}

}

+

x_{n+1} \lr{

\frac{3}{\Delta t}

-\frac{2}{\lr{\Delta t}^2}

}

+

x_{n} \lr{

\frac{1}{\lr{\Delta t}^2}

-\frac{3}{\Delta t}

+ 2

},

\end{aligned}

\end{equation}

or

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:220}

0

=

x_{n+2}

+

x_{n+1} \lr{

3 \Delta t – 2

}

+

x_{n} \lr{

1 – 3 \Delta t + 2 \lr{ \Delta t}^2

}.

\end{equation}

Note that after discretization, each subsequent index corresponds to a time shift. Also observe that the coefficients of this discretized equation are dependent on the discretization interval size \( \Delta t \). If the specifics of those coefficients are ignored, a general form with the following structure can be observed

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:120}

0 =

x_{n+2} \gamma_0

+

x_{n+1} \gamma_1

+

x_{n} \gamma_2.

\end{equation}

It turns out that, much like the LDE solution by characteristic polynomial, it is possible to attack this problem by assuming a solution of the form

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:140}

x_n = C z^n.

\end{equation}

A time shift index change \( x_n \rightarrow x_{n+1} \) results in a power adjustment in this assumed solution. This substitution applied to \ref{eqn:stabilityLDEandDiscreteTime:120} yields

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:160}

0 =

C z^n

\lr{

z^{2} \gamma_0

+

z \gamma_1

+

1 \gamma_2

},

\end{equation}

Suppose that this polynomial has roots \( z \in \{z_1, z_2\} \). A superposition, such as

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:180}

x_n = a z_1^n + b z_2^n,

\end{equation}

will also be a solution since insertion of this into the RHS of \ref{eqn:stabilityLDEandDiscreteTime:120} yields

\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:200}

a z_1^n

\lr{

z_1^{2} \gamma_0

+

z_1 \gamma_1

+

\gamma_2

}

+

b

z_2^n

\lr{

z_2^{2} \gamma_0

+

z_2 \gamma_1

+

\gamma_2

}

=

a z_1^n \times 0

+b z_2^n \times 0.

\end{equation}

The zero equality follows since \( z_1, z_2 \) are both roots of the characteristic equation for this discretized LDE.

In the discrete \( z \) domain stability requires that the roots satisfy the bound \( \Abs{z} < 1 \), a different stability criteria than in the continuous domain. In fact, there is no a-priori guarantee that stability in the continuous domain will imply stability in the discretized domain.
Let's plot those z-domain roots for this example LDE, using \( \Delta t \in \{ 1/2, 1, 2 \} \). The respective characteristic polynomials are
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:260}
0 = z^2 - \inv{2} z = z \lr{ z - \inv{2} }
\end{equation}
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:240}
0 = z^2 + z = z\lr{ z + 1 }
\end{equation}
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:280}
0 = z^2 + 4 z + 3 = (z + 3)(z + 1).
\end{equation}
These have respective roots
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:300}
z = 0, \inv{2}
\end{equation}
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:320}
z = 0, -1
\end{equation}
\begin{equation}\label{eqn:stabilityLDEandDiscreteTime:340}
z = -1, -3
\end{equation}
Only the first discretization of these three yields stable solutions in the z domain, although it appears that \( \Delta t = 1 \) is right on the boundary.

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