Grid Size Adapted Multistep Methods for High $Q$ Oscillators

The numerical calculation of the limit cycle of oscillators with resonators exhibiting a high-quality factor Q such as quartz crystals is a difficult task in the time domain. Time domain integration formulas, when not carefully selected, introduce numerical damping that leads to erroneous limit cycles or spurious oscillations. A novel class of adaptive multistep integration formulas based on finite difference (FD) schemes is derived, which circumvent the aforementioned problems. The results are compared with the well-known harmonic balance (HB) technique. Moreover, the range of absolute stability is derived for these methods. The resulting discretized system by FD methods is sparser than that of HB and, therefore, easier to solve and easier to implement.