Obviously, a short story and a movie are two totally different art forms with separate narrative (and commercial) demands. The way the story subtly conveys shifts in time comes from verb tenses -- sentences like "I remember once we'll be driving to the mall to buy some new clothes for you" make the brain reel -- but the visual vocabulary of cinema can often be hazier and more impressionistic. Arrival is elliptical, which allows Villeneuve to toy with audience expectations in compelling ways.
Through decades of watching movies, we've all internalized certain ideas about what a "flashback" or a "flash-forward" (or, in the case of Lost, a "flash-sideways") looks like, but clever filmmakers have long played with our sense of time, destabilizing viewers by using editing, music, and voice-over to tell non-linear stories. Think of Terrence Malick's Tree of Life, Shane Carruth's Upstream Color, Michel Gondry's Eternal Sunshine of the Spotless Mind, or Christopher Nolan's many po-faced puzzle films.