Cool! That’s a neat feature, yet as far as I can tell not mentioned explicitly in the docs. Neither here How it Works · MethodOfLines.jl nor here MOLFiniteDifference · MethodOfLines.jl.
You’re right, I will add this.
Explains the instability. Indeed, bringing the Da term to the rhs solves the issue, i.e. this integrates fine:
Good catch! That is indeed a big problem, pushing a hotfix as soon as possible.