Accelerate means to speed up. A car accelerates when you step on the gas. You can accelerate the process of getting a visa if you happen to know someone who works in the consulate.
The fastest Amtrak trains are called the Acela line, a made-up word intended to suggest accelerate and therefore communicate that these trains are fast. If you are standing close to one when it zooms by, you may find yourself having an accelerated heart rate—they move pretty fast!