1.3.3 First derivatives in 2 dimensions

Let $f(x,y)$ be given for equidistant values of $x$ and $y$, respectively:

$$f_{i,j} \equiv f(x_{0}+i \Delta x, y_{0}+j \Delta y)$$

We will use the short notation

$$f_{x} \equiv \frac{\partial f(x,y)}{\partial x}$$

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

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

or

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

or

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

Again, the central difference scheme is superior.

But what about second derivatives? $\Longrightarrow$