I'm sure that the detailed answers to that are highly classified. My answer to this would be to observe three things: a) that there have been no publicly-disclosed substantially new designs in a really remarkably long time (remember -- we're talking about 1950s-1960s technology!); b) the underlying physics is well understood; and c) the amount of computing power available now is many orders of magnitude higher than in the 1940s.
Practically, this means two things: i) the governing equations and scaling laws are relatively well-understood, and have been experimentally validated; ii) the parts of the simulation that require detailed numerical codes were developed quite a long time ago, and have also been experimentally validated -- this was literally one of the first ever applications of computing and I am sure that they all have detailed notes. Somewhere there is a tale about an old DEC machine being used as a timekeeping clock in an underground explosion by printing out its CPU timers over a high-speed serial link, and the experimenters noting when the signal stopped. The core "physics package" is something that has been highly optimised with literally trillions of $CURRENCY's worth of man-hours over the last half-century. The real difficulty -- as this website beautifully shows -- is not in understanding the undergraduate-level physics behind it; it's in getting the starting materials and actually making the thing work well. For that, we can all be grateful.
Separately, another answer to the "how do you test things without testing" is by looking for analogues in terms of the insane pressures, temperatures and timescales that go on in these things. There's a reason that the US has NIF [1], the national ignition [laser] fusion facility, which helps both basic science and...not. Similar experiments include hydrodynamic blast testing [2] in which the behaviours of materials under extreme strain rates, driven by high explosives, are performed. The idea again is to validate the core simulation model of "what goes down" in the ps-ns-µs-ms that the explosion lasts, with its distinct phases. It's hard to stress how different materials are under high strain rates -- e.g. unreinforced concrete, often thought of as 'hard', performs poorly and plastically under similar conditions; very high velocity (gas-gun fired) projectiles can deform it very easily and penetrate for literally tens of meters. The challenge of design comes down to taking an assembly, and instantaneously doubling (or more) the density at its core, without jets, or spalling. All of these have analogous problems elsewhere in physics. Subcritical testing is allowed, as are explosions/detonations with other radioactive compounds, which can be followed by high-speed gamma-cameras to permit a determination of the degree of compression. Historically these were used a lot -- see the RaLa experiment [3], which was required to ascertain the behavior of converging shock waves (needed to achieve the spherical implosion necessary for compression of the plutonium pit) -- but instead of using plutonium, they used a radiotracer (Radioactive Lanthanum) in a sphere made of an inert material.
> Questions for anyone who maybe knows answers: How likely is that there are nuclear weapons beyond listed in the article, something as Black projects? (https://en.wikipedia.org/wiki/Black_project)
I think this is unlikely, largely because there isn't a military need for them. Black projects have historically been for useful things. We're already at the point whereby the US or Russia alone could literally annihilate most humans on earth, with their existing, 1980s designs and materials. A lot of the focus -- I understand -- of this area of research is on detecting other nation's activities, and ensuring that "your" own stockpile is safe, and ready. MAD was mad, but it was effective. My gut feeling is that, if anything, US black projects are more likely to focus on asymmetric / information warfare, yet more low-observable aircraft or spacecraft, and, above all, intelligence gathering, cryptography, and similar. We have the doomsday machine. We showed that we could have the doomsday machine literally a lifetime ago. I hope above all that we never use it.
[1] https://en.wikipedia.org/wiki/National_Ignition_Facility [2] https://www.globalsecurity.org/wmd/intro/hydrodynamic.htm [3] https://en.wikipedia.org/wiki/RaLa_Experiment