Adhesives that bond well to polystyrene include:
Cyanoacrylates – Cyanoacrylates bond well to polystyrene. Use only enough adhesive to cover the joint, then assemble as soon as practical. For best appearance use a low odor grade.
Two component epoxies bond well to polystyrene.
UV Curable – UV Curable adhesives are an excellent choice for bonding polystyrene.
(Lifted from Permabond website and edited to remove product references.)
I would probably go with a two component epoxy. They are widely available and have a longer working time than the water like Cyanoacrylates. However the Cyanoacrylates may give you a cleaner and lighter weld.
Another advantage to the two part epoxy is that you can embed stuff in it. I do not think you would need to do that to make a stronger bond though as having another material in there may actually weaken the attachment.
Just check the product labels to be sure it will work with the specific materials you are using, and doing a test on some expendable material is always a good idea.