writing style - Dropping the subject from sentences

Consider this example:

He got into the car. Started the engine, checked the mirrors. Stepped on the gas and headed down Main Street.

Omitting the subject from a sentence isn't proper construction, strictly speaking, but it infuses the writing with a sense of breathless urgency. I've seen a few writers use this style (notably in fast-paced suspense novels).

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  How common is this style?

  
  


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  Is there a name for it? (I was thinking staccato style, but that refers to short, choppy sentences in general)

  
  

Answer

There is grammatical ellipsis, which in general case might not introduce the sense of accelerated time, but quite the reverse, depending on how hard is it to parse, for example:

The average person thinks he isn't.

The parsing here needs backtracking which slows the reading, but gives an effect of delayed and double, in-depth comprehension.

In terms of pure effect, look at diazeugma

The figure by which a single subject governs several verbs or verbal constructions (usually arranged in parallel fashion and expressing a similar idea):

The Romans destroyed Numantia, razed Carthage, obliterated Corinth, overthrew Fregellae.

Wikipedia defines Diazaugma Disjunction:

The subject appears at the beginning of the sentence and each verb follows in its respective clause.

EDIT:
As I said - for pure effect this is diazeugma, but the problem is that the text is agrammatical and the figures apply to grammatically correct texts. However, I think that is a narrow view. I find this text explains the matter in appropriate detail, from which I will only very shortly cite:

Almost 30 years ago, in An Alternate Style: Options in Composition (now out of print), Winston Weathers made a strong case for going beyond strict definitions of correctness when teaching style. Students should be exposed to a wide range of styles, he argued, including the "variegated, discontinuous, fragmented" forms used to great effect by Coetzee, Dickens, Mencken, and countless other writers.

Finally, thanks to @Cerberus, under the same relaxed rules, the figure can be considered asyndeton, from wikipedia example

"We must... hold them, as we hold the rest of mankind, Enemies in War, in Peace Friends."

Finally, the reason why I want to be tolerant of the fullstops is that I find it interesting and rewarding to interpret them as commas, but with certain dramatic pause of undetermined length and tend to read the text as an internal monologue of the storyteller.

_{For answer on how common is this (I assume in literature?) head to literature.}