Von Neumann Analysis

The matrix

$$A \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left[\begin{array}{cc} 2\cos\omega\Delta & -1 \\ [2pt] 1 & 0 \end{array}\right]$$

can be called the state transition matrix corresponding to the state-space description determined by the choice of state vector

$${\boldmath {x}}(n) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\left[\begin{array}{c} U^{n}(\omega) \\ [2pt] U^{n-1}(\omega) \end{array}\right]$$

and the state update can be written more simply in vector form as ${\boldmath {x}}(n+1) = A {\boldmath {x}}(n)$ . Note that the state-space description is indexed by frequency $\omega$ , regarded as fixed.

• From linear systems theory, we know that such a system will be asymptotically stable if the eigenvalues $\lambda$ of the matrix $A$ are both less than 1 in magnitude.

• It is easy to show that the eigenvalues of $A$ are $\lambda_{+}=e^{j\omega\Delta}$ and $\lambda_{-}=e^{-j\omega\Delta}$ . Thus, $\left\vert\lambda_{\pm}(\omega)\right\vert=1, \forall\omega$ .

• While we are not guaranteed asymptotic stability, $\vert\lambda(\omega)\vert = 1$ does imply that, in some sense, our solution is not getting larger with time at any spatial frequency. This can be defined as marginal stability.

• Note that we should expect the eigenvalues to have unit modulus, because the wave equation we started with corresponds to a lossless medium (an ideal gas). The original PDEs were derived without any loss mechanisms.

• A lossless discrete-time simulation can be highly desirable, particularly as a modeling starting point.

• This kind of ``Von Neumann analysis'' can be applied to any constant-coefficient FDS which is linear in its spatial directions.