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Vibration energy harvesting systems pose significant modeling and design challenges due to their mixed-technology nature, extremely low levels of available energy and disparate time scales between different parts of a complete harvester. An energy harvester is a complex system of tightly coupled components modeled in the mechanical, magnetic, as well as electrical analog and digital domains. Currently available design tools are inadequate for simulating such systems due to prohibitive CPU times. This paper proposes a new technique to accelerate simulations of complete vibration energy harvesters by approximately two orders of magnitude. The proposed technique is to linearize the state equations of the system's analog components to obtain a fast estimate of the maximum step-size to guarantee the numerical stability of explicit integration based on the Adams-Bashforth formula. We show that the energy harvester's analog electronics can be efficiently and reliably simulated in this way with CPU times two orders of magnitude lower than those obtained from two state-of-the-art tools, VHDL-AMS and SystemC-A. As a case study, a practical, complex microgenerator with magnetic tuning and two types of power-processing circuits have been simulated using the proposed technique and verified experimentally.