For travelling between Split and Dubrovnik by sea you can choose from two companies. One of them is Krilo which operates two different routes between these two coastal cities.
One catamaran departs from Split at 7:40 a.m. and sails via Brac (Milna), Hvar (Hvar town), Korcula, Mljet (Pomena) to the port of Dubrovnik (arrives at noon). It returns the same way from Dubrovnik in the afternoon hours (4:30 p.m.). For detailed sailing schedule, prices and online booking check Krilo official web site.
Another route run by Krilo company is the one departing from Split and sailing via Brac (Bol), Makarska, Korcula, Mljet (Sobra) to Dubrovnik. Sailing timetable can be checked here.
Another solution would be Jadrolinija's catamaran Novalja which runs daily from 8th June until 18th October.
Catamaran return from Split the same way in the afternoon. This line is convenient for making a day trip to Korcula, Hvar or Brac. Detailed sailing schedule can be found on Jadrolinija web site. Tickets for this catamaran can be booked easily online.
Both companies offer service with modern and comfortable passenger catamarans and journey time between Split and Dubrovnik is approximately 4 hours.
Note that there are no car ferries that sail between these two cities.