You can save much money [(by) using coupons].
You can save much money [thus].
"(T)hus" is an answer to "How?". "How?" triggers the adverbial interpretation. Be careful not to be hasty in judging things, however. Trickery is involved here! There is an elided preposition "by" in the sentence whose complement is the -ing clause, the status of which engages our interest. Prepositional complements are nominal in nature. Gerunds are nominal -ing forms. By now it should be painfully obvious to everyone that we have a gerund-clause here.
Student or Learner