A generalized von Neumann analysis for multi-level schemes: Stability and spectral accuracy

Abstract 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.