  We develop a Gr\"obner basis theory for a class of algebras that generalizes
both PBW-algebras and rings of differential algebras on smooth varieties.
Emphasis lies on methods to compute filtrations and graded structures defined
by weight vectors. The approach is tailored for bifiltered D-modules satisfying
properties of mixed Hodge modules. As a key ingredient in functors of such
modules our theory applies to compute the order filtration on pieces of a
V-filtration.
