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.

翻译：冯·诺依曼稳定性分析是一种基于傅里叶分析的经典方法，广泛应用于数值计算中的稳定性评估。然而，正如Smith在《偏微分方程数值解：有限差分方法》（1985年，第67-68页）中所述，该方法在应用于采用超过两个层级的格式的多层级方法时存在局限性。本研究旨在扩展冯·诺依曼稳定性分析对多层方法的适用性。与冯·诺依曼稳定性分析密切相关的另一种方法是近似色散关系（ADR）分析。本文不仅探讨了ADR分析，还介绍了多种ADR分析变体，并考察了其固有局限性，以便其他研究者在使用该方法前加以改进。此外，我们提出了一种创新的耗散减小策略，并通过进化算法对其进行优化。研究结果表明，在一维和二维空间条件下，与其它平流方程格式相比，我们的方法产生的误差最小。