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Finite differences
Given a function $f$ of one variable. We are to "discretize" it and its derivatives.
First order
A forward difference is an expression of the form
$$\Delta_h[f](x) = f(x + h) - f(x). $$
The increment $h$ of the variable may be variable or constant.
A backward difference uses the function values at $x$ and $x-h$, instead of the values at $x+h$ and $x$:
$$ \nabla_h[f](x) = f(x) - f(x-h).$$
A central difference is given by
$$ \delta_h[f](x) = f(x+\tfrac12h)-f(x-\tfrac12h). $$
The limits of these as $h \rightarrow 0$ give you the derivative of $f$, if it exists.
Compare to 1st order discrete differential forms.
Second order
2nd order central difference: $$ \delta_h^2[f](x) = f(x+h) - 2 f(x) + f(x-h) . $$
Similarly we can apply other differencing formulas in a recursive manner.
2nd order forward difference: $$\Delta_h^2[f](x) = f(x+2h) - 2 f(x+h) + f(x) .$$
Compare to 2nd order discrete differential forms.
Several variables
These are analogous to partial derivatives.
For $f_{x}(x,y)$: $$f(x+h ,y) - f(x-h,y). $$
For $f_{y}(x,y) $: $$f(x,y+k ) - f(x,y-k). $$
For $f_{xx}(x,y)$: $$f(x+h ,y) - 2 f(x,y) + f(x-h,y). $$
For $f_{yy}(x,y)$: $$f(x,y+k) - 2 f(x,y) + f(x,y-k). $$
For $ f_{xy}(x,y)$: $$f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k). $$
These are used as replacements of the partial derivatives in order to discretize PDEs.