Parallel algorithm for LU-decomposition

Question

I need to implement LU-decomposition in Kaira. In Kaira the programmer writes the "parallel part" as the diagram similar to Petri Nets.

So, could you, please, recommend me some parallel algorithms for LU-decomposition which are really easy to understand and implement? The low difficulty of the implementation has the highest priority for me, because I'm not very familiar with Kaira and I'm in hurry a little.

I looked at a Fined-grained LU-factorization, but I'm curious if some other algorithms are used.