  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 localised 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 localisation of order parameters
potentially reduces the computation time by several orders of magnitude.
