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1.3.4 Second derivatives in 2 dimensions
By again fixing one of the independent variables
- , say - and considering only , we obtain, in terms of the
Stirling (centered) approximation,
Analogous (and less accurate) formulae are valid within the
NGF- and NGB-approximations, respectively.
How about
mixed derivatives?
Mixed derivatives
Approximating use the same
kind of approximation with respect to both the - and the -direction.
(This may not hold if and have a different character, e.g. one
space and one time variable.)
Stirling:
And now, fow for the
curvature of :
Curvature of a function f(x,y)
To find the local curvature at the grid point
we have to apply the nabla operator twice.(*)
There are two ways:
Either ``difference'' along the grid axes,
or apply ``diagonal differencing'', writing
| |
|
|
Axial vs. diagonal differencing
|
(*) Note that the nabla operator mentioned here is not
to be mixed up with the backward difference for which we
use the same symbol.
Next: 1.4 Sample Applications
Up: 1.3 Difference Quotients
Previous: 1.3.3 First derivatives in
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001