There are many who have asked the question, whether coasting in neutral or engaging engine braking is more fuel efficient in coming to a stop. One such discussion led to the following top comments:

Engine braking. It shuts off fuel and lets the cars forwards motion turn the engine.
Coasting means you use fuel to keep the engine idling.
Next.

and

That assumes your car has deceleration fuel cut off.
But normally yeah, engine braking would technically use less fuel than idling/coasting.

Now I don’t know anything about cars but I do have a mental model of a mechanical engine. When no fuel is injected, engine braking should work by having the engine do mechanical work to compress and heat up air (then ejecting it). So RPM matters, as the higher the RPM the higher the energy absorbed per unit time.

Assuming the engine is at idle RPM when coasting in neutral — that isn’t always the case, sometimes neutral RPM is lower than the engine braking RPM — then the energy expended in the engine to do work should be exactly the same whether it is burning gasoline to keep idle or it is being turned by wheels, meaning, the kinetic energy lost when engine braking for n seconds should equal the energy consumed by the engine proper when coasting in neutral for n seconds. We’ll make one more assumption to idealize — that heat engines are 100% efficient (they’re not), but we’ll get back to this later. With that, we can say that the energy consumed in either case is the same.

Is that it? No. We did not end up in the same state after n seconds: In one case (engine braking) we have a lower speed and less distance traveled than in the other case (coasting in neutral). To properly evaluate which case uses less energy we need to end up in the same state. Now the evaluation becomes choice-dependent.

The typical end state is a complete stop at the same location like at a stoplight. Therefore we’ll need to match the speed trajectory of these two options. One way to match is to brake when coasting in neutral so that we exactly match the speed trajectory of engine braking, and evaluate energy with a complete stop after n seconds. In that case, clearly more energy has been expended in the case of coasting in neutral, because extra energy is used in friction braking, outside of what happens in the engine. Heat engines being less than 100% efficient makes it worse for coasting in neutral.

But that isn’t the only fair end state. Another end state is to be at the same speed, without any friction braking. This would be the situation if, for instance, one would engine brake in anticipation of coming to a stop, but while the car would still be moving after n seconds, it became clear that we should cancel the maneuver and recover the original speed. This happens if the stoplight turns green after being red, before the car makes it to the light under either option, so coasting in neutral should maintain the same speed throughout. The distances traveled in the two cases are different, but that’s okay, we’ll address that afterwards. It’s harder to see this but in terms of energy, the choice to engine brake then accelerate back to the original speed at the idle RPM, or coast in neutral at the same speed for the same amount of time, are identical, because you need to input the same amount of energy to recover the kinetic energy lost to engine braking, as it takes to idle the engine with gasoline.

Here’s the situation in summary: The engine is always running at idle RPM. We use n seconds to reduce speed or coast, then another n seconds to recover speed or coast. When coasting in neutral the car uses idle RPM gasoline for 2n seconds to compress and heat air. When engine braking, the car uses no gasoline for n seconds then uses twice the idle RPM gasoline for n seconds to recover speed, one time the idle RPM gasoline to keep the engine running, one more time the idle RPM gasoline to generate extra energy the amount of which was lost during engine braking compressing and heating air at idle RPM. So it’s exactly the same on both sides of the ledger.

Now finally, does the fact that heat engines do not have 100% efficiency matter here? Perhaps not… It depends on whether putting in twice the gasoline generates twice the energy at the same RPM. If that’s true, then even if the engine is not 100% efficient, generating 1x power for 2x the time is the same as generating 2x the power for 1x the time.

But coasting in neutral did cause you to travel farther, so in a flip of the earlier case, it wins slightly. In fact coasting in neutral should always win when there is no friction braking involved, because there is no reduction in speed and so reduces total driving time. If there is no need to slow down, then like intuition says, it’s most efficient to coast in neutral. The real culprit in the earlier case turns out to be friction braking. It is when the choice is between friction braking vs. engine braking that engine braking is more efficient, because it dissipates the kinetic energy in compressing air only and not in the brakes while also dissipating energy in compressing air.

What about the fact that engine braking typically runs at higher than idle RPM? In the first scenario of coming to a stop, nothing changes. Engine braking still uses no gasoline while friction braking does, so engine braking is more efficient. However, rerunning the second scenario of ending at the original speed, now the amount lost during engine braking compressing and heating air at high RPM is greater than idle RPM expenditure for the same amount of time, e.g. running 2x idle RPM expends 2x the energy of 1x idle RPM in n seconds. That extra 1x idle RPM worth of energy is lost forever and there is no possibility of recovering that vs. coasting in neutral. Coasting in neutral wins even without accounting for farther distance traveled.

All that to say, the best strategy seems to be to modulate engine braking and coasting in neutral to manage the speed trajectory desired. Use engine braking at the lowest possible RPM to slow down, use higher RPM only if it’s not enough, but if the lowest RPM is still is too much deceleration, go back to coasting in neutral, goal being to avoid any friction braking to manage speed.