Can You List the Names of Some Algorithms For Determining the Intersection of Two Context Free Grammars? Suppose we have two sets of strings XS and YS such that set XS is described by grammar GX and YS is described by grammar GY.

We want an algorithm which accepts GX and Gy as inputs. The algorithm will return true if the intersection of sets XS and YS is non-empty and will return false otherwise. We do not need to know exactly what the intersection is. It is possible that the intersection set is massive, so returning a container of all elements is unreasonable. Describing the intersection as a grammar is okay as long as it's apparent whether the grammar actually describes a non-empty set of string or not. 

Just the name of the algorithm is fine if a Google search using that name will bring up the result. If the algorithm is obscure, additional information would be helpful.
There is no such algorithm. Start by googling undecidability. The problem itself is called "language intersection/disjointness".