1-D and 2-D numerical simulations of 800 MHz klystrons with efficiencies approaching 90% are presented. While traditional klystrons employ monotonic electron bunching along their lengths, the core oscillation method allows for an improved bunch shape at the output cavity, facilitating maximum energy extraction. The core oscillation bunching scheme proves an attractive method for attaining high efficiency operation in klystrons, which can be used to reduce the power consumption of future particle accelerators.