general relativity - How can one obtain the metric tensor numerically?

I am self-studying General Relativity.

Is there a method for obtaining the metric tensor exterior to a specified mass distribution numerically? In the simplest case of a spherical mass this should yield the Schwarzschild exterior geometry. I am primarily interested in such cases, without radiation fields (the simpler the better).

I realise that there is an ambiguity in the coordinate system chosen - presumably if such numerical methods exist they include a specification of the coordinate system.

My google-fu has been unable to find a simple answer to this question. The introductions to the topic of numerical GR that I found are dense, lacking in simple examples (if such things exist), and focus on gravitational waves. I realise that solving a system of non-linear coupled PDEs is not, in general, a simple task. I found a lot of literature talking about the so-called (3+1) method of foliating spacetime with space-like 3D hypersurfaces, but not much in the explicit sense of 'starting with these initial/boundary conditions and coordinate system, solve these equations using method x to obtain the metric tensor, here is some code for the simple example of a spherical mass'.

So basically: is it possible to start with a mass density function and obtain a numerical solution/approximation to the metric tensor exterior to this in some coordinate system, and if so, how?

If the answer to my question is 'no' or 'you are fundamentally misunderstanding something' I welcome being corrected.