I am trying to solve an equation used to calculate ocean wave refraction gotten from the irrotationality of the wave number k.
$$ \frac{\partial \, k\sin\theta}{\partial x}-\frac{\partial \, k\cos\theta}{\partial y}=0 $$
I am using a lax-wendroff explicit approach of taking a half step along both the x and y axis, where the equation has been reduced to $$ \frac{\partial A}{\partial x}-\frac{\partial B}{\partial y}=0 $$ where $A = k\sin\theta$ and $B = k\cos\theta$. In the equation, there exist two dependent variables $A$ and $B$. My question is since i have two dependent variables, will i have two computational grids for the solution or just one computational grid where at every grid point both variables exist.
This problem is not well-posed, since it has more unknows than equations. Therefore, it is useless to derive a numerical method to solve it. One more equation on $A$, $B$ would be sufficient. One option could be to use $A^2 + B^2 = k^2$, but that won't be very useful if $k$ isn't constant.