My 2011 VW Multivan has a rough idle and blows black smoke

First suspicion would be that the fuel injectors need either reconditioning or replacing. Rough running, hard starting and excess black smoke are all classic symptoms of worn injectors. That said, the modern turbo-diesel is a pretty complex animal, and you could also have a problem with the emissions system (EGR valve) or maybe a build-up of soot and oil in the intake system (which could also affect any number of sensors around the engine).

Don’t forget the basics, either; something as simple as a dirty air-filter element can also cause problems just like the ones you’re having.