The so-called von Neumann analysis is a well-established approach used for stability analysis of numerical methods. The crux of this analysis is to bound the amplification factor by unity to ensure stability. However, an implicit but commonly unverified assumption in this approach is that the amplification factor does not vary with time, which as we show here is not always true for multi-level schemes. We propose a generalized von Neumann analysis wherein we take into account the temporal variation of the amplification factor and thus overcome the limitations of the standard analysis. We express this time-varying amplification factor as a continued fraction and obtain exact conditions for the applicability of the standard von Neumann approach. We define stability in terms of product of the amplification factor at all times that allows the instantaneous amplification to be larger than unity. This is indeed observed in simulations though the scheme remains stable which makes it then, unexplainable with the standard von Neumann analysis. We use the proposed generalized analysis and stability definition to assess the stability of asynchrony-tolerant schemes with periodic coefficients. The degrading effect of temporal scheme on the spectral accuracy of spatial schemes at large CFL values is also discussed.

所谓的冯·诺依曼分析是一种成熟的方法，用于数值方法的稳定性分析。该分析的关键是统一限制扩增因子以确保稳定性。但是，这种方法的一个隐含但通常未经验证的假设是，放大系数不会随时间变化，正如我们在此处所示，对于多级方案而言，并非总是如此。我们提出了广义冯·诺依曼分析，其中考虑了放大因子的时间变化，因此克服了标准分析的局限性。我们将此时变放大因子表示为连续分数，并获得标准冯·诺依曼方法适用性的确切条件。我们始终根据放大因子乘积来定义稳定性，以使瞬时放大倍数大于1。尽管该方案保持稳定，但在仿真中确实观察到了这一点，这使得它无法用标准冯·诺依曼分析来解释。我们使用提出的广义分析和稳定性定义来评估具有周期系数的异步容忍方案的稳定性。还讨论了时间方案对大CFL值下空间方案的频谱准确性的降低作用。我们使用提出的广义分析和稳定性定义来评估具有周期系数的异步容忍方案的稳定性。还讨论了时间方案对大CFL值下空间方案的光谱精度的降低作用。我们使用提出的广义分析和稳定性定义来评估具有周期系数的异步容忍方案的稳定性。还讨论了时间方案对大CFL值下空间方案的频谱准确性的降低作用。