A direct injection engine includes an ignition assistance section applying energy to fuel injected into a cylinder using an injector to assist auto-ignition combustion of the fuel when the engine is within an auto-ignition combustion operation range. A start time of fuel injection is set within a period from a terminal stage of a compression stroke to a compression top dead center. The energy is applied to the fuel injected into the cylinder in a period from start of the fuel injection to an initial stage of an expansion stroke such that a time of a specific crank angle when an increase rate of in-cylinder pressure, which is a ratio of a change in the in-cylinder pressure to a change in a crank angle in motoring the engine, reaches a negative maximum value overlaps a combustion period when a combustion mass percentage of the fuel ranges from 10% to 90%.