I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel confused.
The Laplace operator in polar coordinates is $\nabla^2=\frac{1}{r}\partial_rr\partial_r+\frac{1}{r^2}\partial_\phi^2$
The derivatives should be approximated on the form
$y'(x_j)=\frac{y_{j+1}-y_{j-1}}{2\Delta x}$
$y''(x_j)=\frac{y_{j+1}-2y_j+y_{j-1}}{\Delta x^2}$
It's the $r$ between the partial derivatives in the Laplace operator that makes me confused. The formulas for approximating the derivatives with central difference assumes that the derivatives are operating on something. But since I have that $r$ and I don't know the r-dependence of the function the operator is operating on I can't see how I would approximate the first term in the Laplace operator.
How can I approximate this Laplace operator in polar coordinates with central difference?