Asymptotic global spectral analysis of physical and spurious modes in three-level time-integration methods applied to Fourier-discretised convection-diffusion dynamics

Global Spectral Analysis (GSA) is applied to the three-time-level second-order Adams–Bashforth (AB) and third-order Adams–Moulton (AM) time-integration methods to analyze the physical and spurious modes that occur in a spectrally discretized convection–diffusion equation. We develop an iterative procedure that enables GSA to be applied both (i) immediately after an initialization performed with a two-time-level method and (ii) at all subsequent time steps, until an asymptotically converged state (ACS) is reached. It is shown that, in ACS, the domains of influence of the physical and spurious modes on the stability plane (i.e., the distribution of amplification factors in Courant–Friedrichs–Lewy (CFL) / wave number coordinates) and their magnitudes differ significantly from those obtained immediately after initialization. In this regard, we show that: (a) Determining the stability limit solely from the stability map generated immediately after initialization leads to an incorrect estimate of the allowable CFL number. Long-term simulations performed with a CFL number close to the incorrectly predicted ”critical” value eventually may become unstable. The correct critical CFL number can only be assessed once ACS is reached. (b) The influence of the two-time-level initialization method on the physical and spurious modes is only temporary and completely vanishes in ACS. Thus, the particular choice of initialization has a negligible impact on the solution for sufficiently long simulation times. (c) The simultaneous influence of physical and spurious modes on specific solution length scales is also transient. In ACS, each length scale evolves under the dynamics of either the physical mode or the spurious mode, but not both. We demonstrate that the stability plane is divided into regions where the presence of one mode precludes the occurrence of the other. To investigate the influence of viscous effects on these characteristics, we introduce the Péclet number ( Pe ), which combines the viscosity coefficient, time step, and mesh spacing. Within the stability limits (i.e., for Péclet and CFL numbers below their critical values), three regions are identified. (1) Physical-mode dominated region where the spurious mode has no influence on the solution. The physical mode governs the dynamics at all length scales. This corresponds to small Péclet numbers, Pe < 2/(9 π 2) for the Adams-Bashforth method and Pe < 6/(7 π 2) for the Adams-Moulton method. (2) Conditional spurious-mode region (for AB: 2/(9 π 2) < Pe < 2/(3 π 2), for AM: 6/(7 π 2) < Pe < 3/(2 π 2)) where the influence of the spurious mode can be avoided by selecting a sufficiently small CFL number. Otherwise, small-scale features are governed by the spurious mode, while large-scale features remain governed by the physical mode. (3) Spurious-mode dominated region (AB: Pe > 2/(3 π 2), AM: Pe > 3/(2 π 2)) where the impact of the spurious mode cannot be avoided. It exclusively governs the evolution of small solution scales.

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