Check Harry Bateman’s book, Partial Differential Equations, section 2.24, pg. 133. The book is on the internet archive web site. He gives an equation for the characteristics for a second order PDE in any number of variables. The formula for the characteristics follows his equation II for the PDE. I won’t give the derivation because it takes one page and I don’t understand it anyway.
However, his formula works for the simple case in two variables for the PDE
$$
U_{xx} - U_{yy} = 0,
$$
where $U$ is the unknown function and $U_{xx}$, $U_{yy}$ refer to the partial derivatives with respect to the variables $x$ and $y$. The characteristic surface in that case is the lines
$$
x^2 – y^2 = (x – y)(x + y) =0.
$$
The basic idea is that if the function $U$ is calculated on the characteristic surface, only the function $U$ and its first derivatives $U_x$ and $U_y$ are needed to find the function $U$ at other points. This is seen in his equation I for the PDE for $U$ because the term in equation I for the second derivatives of $U$ is zero on the characteristic surface. E. H. Twizell’s book, Computational Methods for Partial Differential Equations, section 4.1, makes this idea clear. Also see Bateman's book, Differential Equations, also on the internet archive, pg. 171, the equation following equation 9. When the differentials $dp$ and $dq$ in the equation following equation 9 are replaced by finite differences in the first derivatives $p=U_x$ and $q=U_y$, then the solution $U$ at any point $(x, y)$ can be found from the boundary values for $U_x$ and $U_y$ by working along the characteristic lines to get to the point $(x, y)$. The second and higher order differences are not needed.