Checking the river when you know you have villain beat is generally a mistake, You're basically giving him a chance to get away from a value bet if he has something to call you with. There are situations though where it can pay off.

First you need the right hole cards, the right board, the right flop+ turn action, and the right read on villain's tendencies based on your past history.

Hole cards: 

Preflop: Just smooth called button raiser with double suited premium AAxx 3 way pot and stacks are deep. Out of position. I normally reraise here but i also smooth call for deception. It really just depends on my mood and stack sizes as well. The deeper the stacks the more conservative you should be when playing AA. At this point Im kind of a half-empty kind of guy. A heavy reraise is a standard move here but if the flop is ugly what do you do? If EP 3bets though and button reraised I would have shoved.

Flop: 

Not bad at all. Overpair nutflush draw plus backdoor nutflush draw. I bet 75%. My bet sizes are basically bet slider settings 50%, 75% and pot size. Just varies depending on my mood, position and equity of course. Here I know he thinks I might have 2 pair but most likely he'll think i have KKK if he doesnt have that himself. If he reraises heavily of course that's what he has and I have to consider just calling or shoving as a slight underdog. He calls.

Turn: 

I pot. This further confirms his perception of me protecting my overset. Such is the power of betting draws. He floats.

River:

At this point the only hand that beats my A high flush is 79 hearts or 47 hearts. Very unlikely for him to have that heads up. Stacks are short. If I shove or value bet i know he wouldnt call. He's a reg and I knew he was too good for that based on our previous duels. So I chose to just check hoping one day this will keep him honest as well when Im just betting an overpair or 2pair and dont want any further river bets. He shoves. Strong move on his part repping the nut flush. If I only had KK or even a straight I most likely have to fold here. I make hero calls from time to time when I can smell the bluff but that's rare so I guess Im very exploitable heh. Well it turns out he was value betting as well with the nut straight. Too bad for him.