Hi, I would recommend cutting them down to the ground and then immediately brushing all exposed parts of the vine with a brush-killing herbicide. Don't use Round-up because it's not strong enough. Look for a herbicide designed to kill shrubs and trees. This should do the job, although with the trumpet vine, you still may have shoots popping up here and there. But, killing the main vine will help a lot. You can buy brush killers at most garden centers or places such as Lowe's, Home Depot or Walmart.