Digital fuel injector, injection and hydraulic valve actuation module and engine and high pressure pump methods and apparatus primarily for diesel engines. The digital fuel injectors have a plurality of intensifier actuation pistons allowing a selection of up to seven intensified fuel pressures. The disclosed engine operating methods include using at least one cylinder for a compression cylinder for providing compressed intake air to at least one combustion cylinder. A pressure sensor in each combustion cylinder may be used to indicate temperature in the combustion cylinder to limit combustion temperatures to below temperatures at which substantial NOX is generated. A re-burn cycle may be used to complete the burning of hydrocarbons, providing a very low emission engine. A compression cylinder may be provided with a pump actuated by the piston in the compression cylinder. Various aspects and embellishments of the invention are disclosed.