B737 CREW RPTS ENG FLAMEOUT AT FL390 AND SUBSEQUENT RESTART.

Narrative

ENRTE TO RDU WE BEGAN TO SHOW GREEN RETURNS ON OUR RADAR AND ASKED FOR RIDE RPTS. ZJX SAID THEY HAD RPTS OF LIGHT CHOP ONLY; SO WE CONTINUED ON COURSE. AS WE BEGAN TO GET LIGHT CHOP THE CAPT SELECTED THE ENG IGNITION TO CONTINUOUS. AS WE CONTINUED THE TURB TURNED TO MODERATE AND THE CAPT SELECTED THE ENG IGNITION TO FLT AND I REQUESTED A CLB TO FL410. AS THE CAPT BEGAN TO CLB HE STATED THAT THE #1 ENG FLAMED OUT. I ASKED FOR A LOWER ALT AND WE WERE CLRED DOWN. I RAN THE ENG FAILURE/SHUTDOWN CHKLIST AND THE ENG RESTART CHKLIST AND GOT THE ENG RUNNING AGAIN. AFTER CAREFUL REVIEW OF THE WX BEHIND; BELOW; AND AHEAD WE FELT IT WAS BEST TO CONTINUE N TOWARDS RDU. WHEN WE FELT COMFORTABLE WITH THE WX TO DSND; RDU WAS AS CLOSE AS ANY OTHER ARPT TO LAND SO WE CONTINUED TO RDU AND MADE A NORMAL LNDG. SELECTION OF ENG ANTI-ICE MAY HAVE PREVENTED THE ENG FLAME-OUT EVEN THOUGH THE OUTSIDE AIR TEMP WAS -54 DEGS C. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE ACFT WAS NEAR SAV WHEN THE FLAMEOUT OCCURRED AND AFTER THE ENG WAS RESTARTED THE DECISION WAS MADE TO CONTINUE TO RDU; NOW 150 NM AHEAD; DUE TO BETTER WX AHEAD. MAINT AT RDU FOUND FAULTS WITH THE FUEL CTL.