If I have a system of equations as follows (where $i = \sqrt{-1}$):
$$ \frac {\partial A}{\partial t} = iA^*B - A \tag{1} \\ $$ $$ \frac {\partial B}{\partial z} = AB^* - B \tag{2} $$
Using the method of lines, in general, I discretize space and solve the system for all time for a particular $z_k$—this transforms the PDEs into a system of ODEs. But in the above case, I have a spatial derivative and also a time derivative that determines the evolution of my system. Is the particular method of discretizing both time and space and solving over such a grid with different partial derivatives (e.g. $t, z$) given a particular name? Is such a method invalid for any particular reason when solving PDEs? I realize I will not have any intermediate values for $B$ when advancing in $t$, and lacking intermediate values for $A$ when advancing in $z$ when using a method such as a 4th order Runge-Kutta. This, I assume, will result in a poor approximation, but is such a method invalid for any reason? Does such a method have a particular name?