# Multidimensional first order linear PDE

Hello,

I have the following PDE :

$$v_y \frac{\partial f}{\partial y} + \Omega_x(y)\left(v_z\frac{\partial f}{\partial v_y} - v_y\frac{\partial f}{\partial v_z}\right) + \Omega_z(y)\left(v_y\frac{\partial f}{\partial v_x} - v_x\frac{\partial f}{\partial v_y}\right) = 0$$

(which is the steady state Vlasov equation in one spatial dimension and three velocity dimensions).

I know the analytic expression of the functions $$\Omega_{x}(y)$$ and $$\Omega_{z}(y)$$.

I also know the expression of the function f for y=0 and y=ymax, and know that f tends to 0 when v tends to (plus or minus) infinity.

is that enough to find a (numerical) solution to the PDE ? is there a method that should preferentially be used to solve it numerically? I've been looking a bit a the method of characteristics but I'm not sure I understand how that would help.

It does not look like an overly complicated equation (linear, first order) so I feel it should be possible to solve it.

I'd appreciate help like references, method names, explanations etc.

Thanks!