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- can someone show me how?

Hey there

I'm currently taking a course on numerical methods for solving differential equations, and atm we are working with the discrete laplacian operator. In particular the 9-point stencil:

However unlike the 5-point stencil, this one is getting to me. I have tried several things, in order to derivate it, but nothing works. Method of undetermined coefficients doesn't work, because no solution allows for the cancellation of all the undesired derivative terms of the Taylor expansions (All the even derivative terms, have the same coefficient composition). My book " Finite Difference Methods for Ordinary and Partial Differential Equations. " by Randall J. LeVeque merely states that is works, and searching the internet doesn't lead to any clarity either.

Can someone explain to me, how derivate it?

I'm currently taking a course on numerical methods for solving differential equations, and atm we are working with the discrete laplacian operator. In particular the 9-point stencil:

However unlike the 5-point stencil, this one is getting to me. I have tried several things, in order to derivate it, but nothing works. Method of undetermined coefficients doesn't work, because no solution allows for the cancellation of all the undesired derivative terms of the Taylor expansions (All the even derivative terms, have the same coefficient composition). My book " Finite Difference Methods for Ordinary and Partial Differential Equations. " by Randall J. LeVeque merely states that is works, and searching the internet doesn't lead to any clarity either.

Can someone explain to me, how derivate it?