The influence of parasitic modes on "weakly'' unstable multi-step Finite Difference schemes

Abstract: Numerical analysis for linear constant-coefficients Finite Difference schemes was developed approximately fifty years ago. It relies on the assumption of scheme stability and in particular -- for the $L^2$ setting -- on the absence of multiple roots of the amplification polynomial on the unit circle. This allows to decouple, while discussing the convergence of the method, the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that multi-step methods can be studied essentially as they were one-step schemes. In other words, the global truncation error can be inferred from the local truncation error. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we show that in the case of weakly'' unstable schemes with multiple roots on the unit circle, although the schemes may remain stable, the consideration of parasitic modes is essential in studying their consistency and, consequently, their convergence. Otherwise said, the lack of genuine stability prevents bounding the global truncation error using the local truncation error, and one is thus compelled to study the former on its own. This research was prompted by unexpected numerical results on lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of a fourth-order multi-step scheme. However, this assumption proved incorrect forweakly'' unstable schemes. This borderline scenario underscores the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle non-linear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulae.