I've read about the Fast Marching Method described by Sethian in his book "Level Set Methods and Fast Marching Methods" and also in the paper "Fast Marching Methods", which can be found for free here: https://www.jstor.org/stable/2653069?seq=1#metadata_info_tab_contents. In the latter, on page 207 the update procedure is described. It starts with only one known point and solves the Eikonal Quadratic for the four neighboring points. Now my question is, how is this done?
I mean, if f.e., the one known point is $u_{ij}$, then how can we calculate $u_{i+1, j}$ by solving $\left[max^2(D_{i+1,j}^{-x}u, -D_{i+1,j}^{+x} u, 0 + max^2(D_{i+1,j}^{-x}u, -D_{i+1,j}^{+x} u, 0)\right] = F_{i+1,j}^2$?
I.e., aren't we missing the information about $u_{i+2,j}, u_{i+1, j+1}, u_{i+1, j-1}$ in order to do so? Since only $u_{ij}$ is known, the only difference quotient we are able to calculate for $u_{i+1, j}$ has to be the one involving $u_{ij}$. But only one difference quotient isn't enough information for solving the Quadratic?
I'd very much appreciate it, if anyone could bring light into this.