New answers tagged numerical-modelling
3
As mentioned in the comments, you are interested in solving a boundary value problem and not an initial value problem. Some of the most used methods are:
Finite Difference Method;
Finite Element Method.
For this particular problem the Ritz method might be a good choice, but in general you are better using the Finite Element Method.
1
Here is a commonly used alternative:
Let's assume that your probability density $f(x,y,v_x,v_y)$ lives in a domain $\Omega$ that is bounded and that you can subdivide into "cells" $\Omega_i$. In one of the comments, you mention that $\Omega$ is simply a 4-dimensional box, and then the $\Omega_i$ could simply be subdivision into a regular mesh ...
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Simply redoing my earlier comment with a bit more space ...
Depending on the particulars, a simple acceptance-rejection method might be a good place to start. Suppose you know $f$ never gets bigger than $f_{\text{max}}$. Generate $X$ uniformly between $x_{\text{min}}$ and $x_{\text{max}}$, then similarly $Y$ from between $y_{\text{min}}$ and $y_{\text{max}}$ ...
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