In this work, a novel approach for the reliable and efficient numerical integration of the Kuramoto model on graphs is studied. For this purpose, the notion of order parameters is revisited for the classical Kuramoto model describing all-to-all interactions of a set of oscillators. First numerical experiments confirm that the precomputation of certain sums significantly reduces the computational cost for the evaluation of the right-hand side and hence enables the simulation of high-dimensional systems. In order to design numerical integration methods that are favourable in the context of related dynamical systems on network graphs, the concept of localized order parameters is proposed. In addition, the detection of communities for a complex graph and the transformation of the underlying adjacency matrix to block structure is an essential component for further improvement. It is demonstrated that for a submatrix comprising relatively few coefficients equal to zero, the precomputation of sums is advantageous, whereas straightforward summation is appropriate in the complementary case. Concluding theoretical considerations and numerical comparisons show that the strategy of combining effective community detection algorithms with the localization of order parameters potentially reduces the computation time by several orders of magnitude.

在这项工作中，研究了一种在图上可靠和有效地对 Kuramoto 模型进行数值积分的新方法。为此目的，为描述一组振荡器的全部交互的经典 Kuramoto 模型重新审视了阶参数的概念。第一个数值实验证实，某些和的预计算显着降低了右侧评估的计算成本，因此能够模拟高维系统。为了设计适合网络图上相关动力系统背景的数值积分方法，提出了局部阶参数的概念。此外，复杂图的社区检测和底层邻接矩阵到块结构的转换是进一步改进的重要组成部分。已经证明，对于包含相对较少的等于零的系数的子矩阵，求和的预计算是有利的，而直接求和在互补情况下是合适的。总结的理论考虑和数值比较表明，将有效的社区检测算法与顺序参数的本地化相结合的策略可能会将计算时间减少几个数量级。而直接求和在互补情况下是合适的。总结的理论考虑和数值比较表明，将有效的社区检测算法与顺序参数的本地化相结合的策略可能会将计算时间减少几个数量级。而直接求和在互补情况下是合适的。总结的理论考虑和数值比较表明，将有效的社区检测算法与顺序参数的本地化相结合的策略可能会将计算时间减少几个数量级。