Von Neumann Stability Analysis for Multi-level Multi-step Methods

Von Neumann stability analysis, a well-known Fourier-based method, is a widely used technique for assessing stability in numerical computations. However, as noted in "Numerical Solution of Partial Differential Equations: Finite Difference Methods" by Smith (1985, pp. 67-68), this approach faces limitations when applied to multi-level methods employing schemes with more than two levels. In this study, we aim to extend the applicability of Von Neumann stability analysis to multi-level methods. An alternative method closely related to Von Neumann stability analysis is the Approximate Dispersion Relation (ADR) analysis. In this work, we not only explore ADR analysis but also introduce various ADR analysis variants while examining their inherent limitations so that other researchers can improve the analysis before using that in their work. Furthermore, we propose an innovative strategy for reducing dissipation, optimizing it through the use of an evolutionary algorithm. Our findings demonstrate that our proposed method yields minimal errors when compared to other advection equation schemes, both in one and two spatial dimensions.