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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. enter image description here

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.

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    $\begingroup$ Others might be able to provide some deeper insight, but generally the nature of Poisson's problem is such that the values of the solution at all grid points cannot be decoupled. For example, if you change the right-hand side $\rho$ in some area of your domain, it will affect the value of $u$ everywhere. This distinguishes it from the wave equation, where some initial data is propagated with finite speed. $\endgroup$ – cthl Oct 22 '18 at 21:22
  • $\begingroup$ Explicit methods are for problems with time dependence. The elliptic equation has no time dependence. $\endgroup$ – Paul Oct 22 '18 at 21:55
  • $\begingroup$ @Paul : This is the point. The time independent problems call for implicit implementations. Why? Is this due to what cthl indicates above? Thanks. $\endgroup$ – Herman Jaramillo Oct 22 '18 at 22:39
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    $\begingroup$ Time and space are fundamentally different for the problems that you mentioned. When solving the wave equation for some time interval $(0,T)$, information about the wave at times $t\le t_0\in (0,T)$ is sufficient to predict its behavior at time $t_0$. It does not depend on future events. However, when solving Poisson's problem on an interval $(0,L)$ (in space), the solution at $x_0\in(0,L)$ does depend on information for $x>x_0$, e.g., the values of $\rho$. $\endgroup$ – cthl Oct 22 '18 at 23:02
  • $\begingroup$ @cthl : I guess I follow you. Causality is a time dependence issue. Thanks. $\endgroup$ – Herman Jaramillo Oct 22 '18 at 23:04

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