How to change DATABASE_URL for a heroku application

An alternative method which does not require detaching (which may not be a desired outcome of the switch) is to simply attach the new database and then promote it, which the Heroku Documents explicitly states as a way to set the DATABASE_URL.

heroku addons:attach heroku-postgresql -a <app_name>
heroku pg:promote heroku-postgresql -a <app_name>

After trying out most these answers, I came across an update in 2016, here: the database needs to be detached first, then update the variable of the DATABASE_URL.

heroku addons:attach heroku-postgresql -a <app_name> --as HEROKU_DATABASE
heroku addons:detach DATABASE -a <app_name>
heroku config:add DATABASE_URL=