** Next:** One-step Schemes
** Up:** Finite Difference Schemes for
** Previous:** Finite Difference Schemes for

#

Von Neumann Analysis of Difference Schemes

In this section, we summarize the basics of Von Neumann analysis provided in [176]. Consider the (+1)D real-valued grid function
, defined for integer and for
, the set of all integer -tuples. Such a grid function will be used, in a finite difference scheme, as an approximation to the continuous solution
to some problem, at the location
, and at time , where is the grid spacing, and is the time step. Here, and henceforth in this appendix, we have assumed that the grid spacing is uniform in all the spatial coordinates, and that the spatial domain is unbounded. As in Chapters 3 and 4, we define the space step/time step ratio to be

The spatial Fourier transform of
is defined by

and is a periodic function of
, a vector of spatial wavenumbers. The transform can be inverted by

where
refers to the space enclosed by the intervals
, for
.
If, for a given grid spacing , we define the discrete spatial norm of
by

and the corresponding spectral norm of
by

then if
and
are in their respective spaces, *Parseval's relation* gives

**Subsections**

** Next:** One-step Schemes
** Up:** Finite Difference Schemes for
** Previous:** Finite Difference Schemes for
Stefan Bilbao
2002-01-22