1.3b Differencing in 2 dimensions
Let $f(x,y)$ be given for equidistant values of $x$ and $y$, respectively:
$
\begin{eqnarray}
f_{i,j} &\equiv& f(x_{0}+i \Delta x, y_{0}+j \Delta y)
\end{eqnarray}
$
First and second derivatives on a grid will be needed.
1.3.3 First Derivatives (2D)
We will use the short notation
$
f_{x} \equiv \frac{\textstyle \partial f(x,y)}{\textstyle \partial x}
$
etc. for the partial derivatives of the function $f$
with respect to its arguments.
Note: One of the arguments may be the time $t$:
$f = f(x,t)$ etc.
For the numerical treatment of partial differential
equations (PDEs) we again have to construct discrete
approximations to the partial derivatives at the base points
$(x_{i},y_{j})$.
Using the
DNGF, DNGB, or DST approximation of lowest order, we have
$
\begin{eqnarray}
{\left[ F_{x} \right]}_{i,j} & \approx & \frac{\textstyle 1}{\textstyle \Delta x}
\left[ f_{i+1,j} - f_{i,j} \right] + O[\Delta x]
\equiv \frac{\textstyle \Delta_{i} f_{i,j}}{\textstyle \Delta x} + O[\Delta x]
\end{eqnarray}
$
or
$
\begin{eqnarray}
{\left[ F_{x} \right]}_{i,j} & \approx & \frac{\textstyle 1}{\textstyle \Delta x}
\left[ f_{i,j} - f_{i-1,j} \right] + O[\Delta x]
\equiv \frac{\textstyle \nabla_{i} f_{i,j}}{\textstyle \Delta x} + O[\Delta x]
\end{eqnarray}
$
or
$
\begin{eqnarray}
{\left[ F_{x} \right]}_{i,j} & \approx & \frac{\textstyle 1}{\textstyle 2 \Delta x}
\left[ f_{i+1,j} - f_{i-1,j} \right] + O[(\Delta x)^{2}]
\\ && \\
& \equiv &
\frac{\textstyle \mu \delta_{i} f_{i,j}}{\textstyle \Delta x} + O[(\Delta x)^{2}]
\end{eqnarray}
$
Again, the central difference scheme is superior.
1.3.4 Second derivatives (2D)
By again fixing one of the independent variables
- $y$, say - and considering only $f_{xx}$, we obtain, in terms of the
Stirling (centered) approximation,
$
\begin{eqnarray}
[F_{xx}]_{i,j} & \approx &
\frac{\textstyle 1}{\textstyle (\Delta x)^{2}} [f_{i+1,j}-2f_{i,j}+f_{i-1,j}]
+ O[(\Delta x)^{2}]
\\ && \\
& \equiv & \frac{\textstyle \delta_{i}^{2}f_{i,j}}{\textstyle (\Delta x)^{2}}
+ O[(\Delta x)^{2}]
\end{eqnarray}
$
Analogous (and less accurate) formulae are valid within the
NGF- and NGB-approximations, respectively.
Mixed derivatives
Approximating $f_{xy}$ use the same kind of approximation with respect to both the
$x$- and the $y$-direction. (This may not hold if $x$ and $y$ have a different
character, e.g. one space and one time variable.) Again applying
Stirling's procedure we find
$
\begin{eqnarray}
[F_{xy}]_{i,j} & \approx &
\frac{\textstyle 1}{\textstyle 4\Delta x \Delta y}
\left[ f_{i+1,j+1}-f_{i+1,j-1} - f_{i-1,j+1}+f_{i-1,j-1} \right] +O[\Delta x \Delta y]
\\ && \\
& \equiv & \frac{\textstyle \mu \delta_{i}}{\textstyle \Delta x}
\left[\frac{\textstyle \mu \delta_{j} f_{i,j}} {\textstyle \Delta y} \right]
+ O[\Delta x \Delta y]
\end{eqnarray}
$
Curvature of a function f(x,y)
To find the local curvature at the grid point $(i,j)$ we have to apply the
nabla operator $\nabla$ twice.(*)
There are two ways:
Either "difference" along the grid axes,
$
\begin{eqnarray}
\nabla^{2} f(x,y) &\approx&
\frac{1}{(\Delta l)^{2}}
\left[ f_{i+1,j}+f_{i,j+1}+f_{i-1,j}+f_{i,j-1}-4f_{i,j}\right]
\end{eqnarray}
$
or apply "diagonal differencing", writing
$
\begin{eqnarray}
&& \nabla^{2} f(x,y) \approx
\frac{\textstyle 1}{\textstyle 2(\Delta l)^{2}}
\left[ f_{i+1,j+1}+f_{i-1,j+1} + f_{i-1,j-1}+f_{i+1,j-1}-4f_{i,j} \right]
\end{eqnarray}
$
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Axial vs. diagonal differencing
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(*) Note that the nabla operator $\nabla$ mentioned here is not
to be mixed up with the backward difference for which we
use the same symbol.
vesely
2005-10-10