This is a really interesting thought -- run faster and/or more frequent service to key stations (cities and park-n-rides) to attract leisure passengers on weekends. I think the conventional wisdom has always been that weekend demand is anemic and so you should just run the bare minimum anyway. But you're presenting an intriguing theory: latent demand might be higher but with a different profile and set of needs than weekday.
I'm not quite convinced that 4 round-trips would be enough to be more attractive than driving -- the time differences aren't going to be that enormous for rail vs driving, and 4 round-trips per day limits flexibility. So, the question is, how much more would the T need to do to make itself more attractive than driving?
Looking at the current schedule, it looks like the Providence Line is probably run with two sets simultaneously:
View attachment 46255
These may or may not be literally the same two sets all day -- that's what I'm attempting to indicate with the asterisks, in that these could be the same trains, but wouldn't have to be.
I played around in
my skip-stop generator and got the following:
View attachment 46267
An
All-Stops and E
xpress service, each every two hours, alternating to provide 2 trains per hour to Providence, Mansfield, and Route 128. Not perfectly clockfacing because the X trip is ~20 min faster (see below), but by synchronizing departures at :29 at Mansfield, the departures at PVD and RTE deviate from clockfacing by 8 min or less. (The corresponding outbound schedule is sketched out in the lower section: the top
finish layover indicates the departure from South Station,
finish return indicates the arrival in Providence [following the same stopping pattern as the set's inbound journey], and the bottom
finish layover indicates when the set is available for the next departure.)
By my read, this pattern would require 3 simultaneous sets, up from today's 2. It would also see all three sets in more or less constant service/turns, as opposed to today's schedule which definitely has "downtime" in it.
But yeah, looking at this schedule... I like. Those express times are comparable to driving. And with return trains leaving once an hour (give or take 5 minutes) all day, you can be flexible about when you decide to head home for the day.