Well anyone who watches knows its both boring and completely pointless to have a session on saturday.
However having just read the new format for the Indy 500 i have adapted it to work with F1.
Bumping. Basically you go back to the old 60 minute 12 lap system. perhaps adjust it to 40 minute 9 laps to make it a bit more exciting. On saturday every car goes out to decide the top 10 places, those who make the top 10 will then remain in the holding cell like normal till the race. Those who didnt make the top 10 will then battle it out for the positions 11 through 20 in the sunday session again either a 40 min 9 lap session or 60 minute 12 lap session. The whole idea would make saturday exciting as you wouldn't be able to risk sitting around with an average time till the end hoping to get a top 10 spot and you would have cars out on track most of the time rather than 1 after another. And sundays which is rarely shown on tv anyway would also be more exciting with cars battling for the highest possible positions.