the FD formula for the second derivative accurate at IV order is

[-f(i-2)+16*f(i-1)-30*f(i)+16*f(i+1)-f(i+2)]/(8*h^2)

An alternative formula which solves for 2D Poisson equations with non-constant inhomogeneous terms (albeit with Dirichlet boundary conditions) may be seen at https://people.bath.ac.uk/ensdasr/COMPACT/dasr.compact.pdf

This link is for some teaching notes I made for a workshop 18 months ago.

The fourth derivative is the second derivative of the second derivative.

So, just as Laplacian (f)(n)=((sum_neighbors(f)(m)-d.f(n))a^2,

Laplacian4(f)(n)=((sum_neighbors(Laplacian(f)(m)-d.Laplacian(f)(n))a^4

(In d dimensions).

Stam Nicolis

maybe you have misunderstood the original question, he asked for a forth order accurate formula for the second derivative in the Laplacian equation, not the forth derivative in the bi-harmonic operator

Oops! Sorry. A good reference for avoiding discretisation artefacts is this one:Article The theoretical background and properties of perfect actions

Leveque’s book is good resoorce fir high order scheme in FDM. In favt, he uses the laplace equation for demonstration purposes. Hirsh’s book is alao a good reference.

Just to add a further choice: the use of compact implicit IV order formula to avoid large stencil.

My earlier link gives a derivation of the 4th order 9-point compact stencil for a 2D Cartesian Poisson equation with a non-constant inhomogeneous term.

Dear Sir D. Andrew S. Rees ,

I am greatly thankful for sharing your worth experiences and notes.