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1.3.3 First derivatives in 2 dimensions

Let $f(x,y)$ be given for equidistant values of $x$ and $y$, respectively:
$\displaystyle f_{i,j}$ $\textstyle \equiv$ $\displaystyle f(x_{0}+i   \Delta x, y_{0}+j   \Delta y)$  

We will use the short notation
$\displaystyle f_{x}$ $\textstyle \equiv$ $\displaystyle \frac{\partial f(x,y)}{\partial x}$  

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

$\displaystyle {\left[ F_{x} \right]}_{i,j}$ $\textstyle \approx$ $\displaystyle \frac{1}{\Delta x}
\left[ f_{i+1,j} - f_{i,j} \right] + O[\Delta x]
\equiv \frac{\Delta_{i} f_{i,j}}{\Delta x} + O[\Delta x]$  

or
$\displaystyle {\left[ F_{x} \right]}_{i,j}$ $\textstyle \approx$ $\displaystyle \frac{1}{\Delta x}
\left[ f_{i,j} - f_{i-1,j} \right] + O[\Delta x]
\equiv \frac{\nabla_{i} f_{i,j}}{\Delta x} + O[\Delta x]$  

or
$\displaystyle {\left[ F_{x} \right]}_{i,j}$ $\textstyle \approx$ $\displaystyle \frac{1}{2 \Delta x}
\left[ f_{i+1,j} - f_{i-1,j} \right] + O[(\D...
...onumber \\
\equiv \frac{\mu \delta_{i} f_{i,j}}{\Delta x} + O[(\Delta x)^{2}]$  


Again, the central difference scheme is superior.

But what about second derivatives? $\Longrightarrow$


next up previous
Next: 1.3.4 Second derivatives in Up: 1.3 Difference Quotients Previous: 1.3.2 Second Derivatives
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001