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A typical pattern of vehicle driving includes a series of frequent accelerations and decelerations. Fuel energy spent to accelerate the vehicle is later wasted during deceleration, when kinetic energy is converted into heat in friction brakes. New thermodynamic cycles have been conceived for automobile engines to capture the energy of braking in the form of compressed air, and reuse this energy during acceleration at a later time. They are applicable to all types of automotive engines. Each four-stroke cycle includes two power strokes, one with compressed air and a second one with combustion gas. It is also possible to switch the engine operation from a four-stroke to a two-stroke cycle during acceleration, which allows a reduction in engine displacement.