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Von Neumann stability analysis, a well-known Fourier-based method, is awidely 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 faceslimitations when applied to multi-level methods employing schemes with morethan two levels. In this study, we aim to extend the applicability of VonNeumann stability analysis to multi-level methods. An alternative methodclosely related to Von Neumann stability analysis is the Approximate DispersionRelation (ADR) analysis. In this work, we not only explore ADR analysis butalso introduce various ADR analysis variants while examining their inherentlimitations so that other researchers can improve the analysis before usingthat in their work. Furthermore, we propose an innovative strategy for reducingdissipation, optimizing it through the use of an evolutionary algorithm. Ourfindings demonstrate that our proposed method yields minimal errors whencompared to other advection equation schemes, both in one and two spatialdimensions.

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