Accurate and efficient computation of Floquet multipliers and subspaces is essential for analyzing limit cycle in dynamical systems and periodic steady state in Radio Frequency (RF) simulation. This problem is typically addressed by solving a periodic linear eigenvalue problem, which is discretized from the linear periodic time-varying system using one-step methods. The backward Euler method offers a computationally inexpensive overall workflow but has limited accuracy. In contrast, one-step collocation methods achieve higher accuracy through over-sampling, explicit matrix construction, and condensation, thus become costly for large-scale sparse cases. We apply multistep methods to derive a periodic polynomial eigenvalue problem, which introduces additional spurious eigenvalues. Under mild smoothness assumptions, we prove that as the stepsize decreases, the computed Floquet multipliers and their associated invariant subspace converge with higher order, while the spurious eigenvalues converge to zero. To efficiently solve large-scale problems, we propose pTOAR, a memory-efficient iterative algorithm for computing the dominant Floquet eigenpairs. Numerical experiments demonstrate that multistep methods achieves high order accuracy, while its computational and memory costs are only marginally higher than those of the backward Euler method.

翻译：精确高效地计算Floquet乘子及其子空间，对于分析动力系统中的极限环以及射频（RF）仿真中的周期稳态至关重要。该问题通常通过求解周期线性特征值问题来处理，该问题由线性周期时变系统使用单步方法离散化得到。后向欧拉法提供了计算成本较低的整体工作流程，但精度有限。相比之下，单步配置法通过过采样、显式矩阵构建和凝聚实现了更高的精度，因此对于大规模稀疏情况成本较高。我们应用多步方法推导出周期多项式特征值问题，这引入了额外的伪特征值。在温和的光滑性假设下，我们证明随着步长减小，计算得到的Floquet乘子及其相关不变子空间以更高阶收敛，而伪特征值收敛于零。为了高效求解大规模问题，我们提出了pTOAR算法，这是一种用于计算主导Floquet特征对的内存高效迭代算法。数值实验表明，多步方法实现了高阶精度，而其计算和内存成本仅略高于后向欧拉法。