2002 Ford Taurus Dies

I purchased a 2002 Ford Taurus. 102K and a clean Carfax. Local one owner with a good service history. Took for a test drive and ran fine. Had to be jumped to start in the lot and dealer replaced the battery. When I picked it up it died before I got off the lot. Then died again 30 minutes later, when warm. Then took for a drive and died after 2 miles at 50mph. Started a few minutes later, but not immediately. Any chance the battery replacement/jump affected the computer? What would make all systems fail, then restart? How is the computer reset?

Two details we need to know first.

One: when the engine stopped, was it as though you had turned the key off?

Two: when the engine stopped, was it as though it ran out of fuel? Like a stutter and stumble type of stop?

Hello, and thanks for your reply. Yes to #1…it was as though the key was turned off. Everything abruptly was gone (the last time it happened after 2 mi I just turned on the heater fan…probably just a coincidence?). No to #2…no stutter or stumble or warning of any type…just died in a split second after running just fine.

That indicates that it was probably spark that cut off, rather than fuel. So, troubleshooting begins with spark production.
First, one gets the wiring diagram. Then, check if power (12 volts) is getting to the spark production control units (the ignition module and ignition coils) from the ignition switch and fuse. Actuate the igition switch several times when checking for power on the ignition switch supplied power circuits. Check the other circuits, several times, in accordance with the Haynes, or Chilton’s.