I need to numerically solve the following second-order ODE in MATLAB

$$ 2ik\frac{\partial U(\vec{\rho},z)}{\partial z}+\left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right]U(\vec{\rho},z)=0 $$

which looks a lot like a 2-D time-dependent Schrödinger equation but with time replaced by $z$. Here, $\rho$ is the planar position vector. Any recommendations on which numerical technique to use would be helpful.

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    $\begingroup$ Your spatial domain completely determines the adopted numerical method. Describe it please. $\endgroup$ – HBR Dec 25 '18 at 12:32
  • $\begingroup$ For my project's purpose, I was going to use a box with z ranging from 0 to 1000km, and x and y ranging from 0 to 100km. The initial conditions will be known for the entire base of the box, i.e. U(x, y, 0) is known. $\endgroup$ – Cédric Arseneau Dec 27 '18 at 13:55

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