Five point scheme Finite Difference Method

[sigh1342]

For possion equation $$u_{xx}+u_{yy}=f$$
I know the general five point scheme is in the form
$$a_{1}U_{i,j-1}+a_{2}U_{i-1,j}+a_{3}U_{i,j}+a_{4}U_{i+1,j}+a_{5}U_{i,j+1}=f_{i,j}$$
But , is there have the form
$$a_{1}U_{i-1,j-1}+a_{2}U_{i-1,j+1}+a_{3}U_{i,j}+a_{4}U_{i+1,j+1}+a_{5}U_{i+1,j-1}=f_{i,j}$$?

 

[AlephZero]

It would be easy enough to invent one. Just approximate $U_{i-1,j} = (U_{i-1,j-1}+U_{i-1,j+1})/2$ etc.

Whether that would be any good in practice is another question, of course.

Alternatively, imagine your grid is rotated through 45 degrees, and use your original formula with $h$ replaced by $h\sqrt 2$.

One feature of it would be: color the grid points red and black, in a pattern like a chess board. Except for the boundary conditions, you have one set of equations linking only the red points, and another set linking only the black points.

That may or may not be a good thing, depending on what use you make of it.