The matrix

can be called the

and the state update can be written more simply in vector form as . Note that the state-space description is indexed by frequency , regarded as fixed.

- From linear systems theory, we know that such a system
will be
*asymptotically*stable if the eigenvalues of the matrix are both less than 1 in magnitude. - It is easy to show that the eigenvalues of
are
and
. Thus,
.
- While we are not guaranteed asymptotic stability,
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.

Download NumericalInt.pdf

Download NumericalInt_2up.pdf

Download NumericalInt_4up.pdf

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