Does anyone can tell me the form of 4th-order discretization of 2-order difference in cylindrical coordinate.

In Cartesian coordinate, it is (-F(i-2)+16F(i-1)-30F(i)+16F(i+1)-F(i+2))/(12*dx)

If you have mathematica/maple/maxima you can find the coefficients using it.Also look for the paper on compact scheme by sanjeev lele ( JCP 1992),it should give you the required finite difference discretization.

discretization is independent of co-ordinate system [d2u/dx2 or d2u/dr2 are same]. If, whatever u have wrote, is correct in Cartesian co-ordinate, it is correct in Cylindrical co-ordinate as well. So, enjoy!! you already have what r u looking for

What changes with change in co-ordinate system is terms itself, not the way it is discretized.

hope it helps.

You are right, but I want to discretize the Lapace of Phi in cylindrical coordinate. So the divergence of a gradient in cylindrical is different from the one in Cartetian coordinate.