We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based on the numerical range of the underlying large matrices involved in such systems, and demonstrate its application with concrete examples of RK stability for hyperbolic methods of lines.

中文翻译：

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关于任意大型常微分方程组的 Runge-Kutta 方法的稳定性


我们证明了用于任意大型常微分方程组数值积分的 Runge-Kutta （RK） 方法是线性稳定的。标准稳定性参数 — 基于光谱分析、溶剂条件或强稳定性，无法确保任意大型系统的 RK 方法的稳定性。我们解释了不同方法的失败，基于此类系统中涉及的基础大矩阵的数值范围提供了一种新的稳定性理论，并通过 RK 稳定性的具体示例证明了其在双曲线方法中的应用。