Let us think on the Poisson equation $nabla^2 u(bf{x})=rho(x)$ with Neumann boundary conditions, with $bf{x}=it (x,y)$ in 2D.

Here is a stencil with central differences in both $x$ and $y$ (forgive my label $t$ on the vertical, it is supposed to be $y$) directions.

with $beta=Delta x/Delta y$. More specifically we have the

equation:

begin{eqnarray*}

w_{i+1 j} -2 w_{ij} left ( 1 + beta^2 right)

+ w_{i-1 j} + beta^2 (w_{i j+1} + w_{i j-1})

= Delta x^2 rho(x,y)

end{eqnarray*}

We can get a similar stencil in the (hyerbolic) wave equation in 1D where

time is the $y$ (vertical axis). We can always find, recursively

$w_{i j+1}$ in terms of all other $w_{lm}$ where $l=i,i+1,i-1$ and

$m=j,j-1$. The points 1 cell away from the boundary can be computed with

a ghost boundary condition (assuming Neumann BC with derivative equal to 0).

I have been searching for hours and all literature shows me an implicit method where I need to invert a huge $(n-1 times m-1)$ matrix. Is there a specific reason why the elliptic equation is not solved as we do the 1D wave explicit solution centered in space and time?

Or…if it could be solved, is there a place to look for that solution? Thanks.