 Solution

At a steady state in pseudo-time, ${\partial }_{\tau }\Phi =0$ and the evolution becomes ${\Phi }_{xx}=4\pi G\rho$, the desired Poisson equation. Deviations from a steady solution obey a diffusion equation, so nonconstant error modes decay. This turns an elliptic boundary-value solve into repeated relaxation steps using familiar time-evolution methods. Boundary data and source compatibility still matter; pseudo-time does not remove a nonexistent steady solution.