  Modern problems in magnetization dynamics require more and more the numerical
determination of the spin-wave spectra and -dispersion in magnetic systems
where analytic theories are not yet available. Micromagnetic simulations can be
used to compute the spatial profiles and oscillation frequencies of spin-waves
in magnetic system with almost arbitrary geometry and different magnetic
interactions. Although numerical approaches are very versatile, they often do
not give the same insight and physical understanding as analytical theories.
For example, it is not always possible to decide whether a certain feature
(such as dispersion asymmetry, for example) is governed by one magnetic
interaction or the other. Moreover, since numerical approaches typically yield
the normal modes of the system, it is not always feasible to disentangle
hybridized modes. In this manuscript, we build a bridge between numerics and
analytics by presenting a methodology to calculate the individual contributions
to general spin-wave dispersions in a fully numerical manner. We discuss the
general form of any spin-wave dispersion in terms of the effective (stiffness)
fields produced by the modes. Based on a special type of micromagnetic
simulations, the numerical dynamic-matrix approach, we show how to calculate
each stiffness field in the respective dispersion law, separately for each
magnetic interaction. In particular, it becomes possible to disentangle
contributions of different magnetic interactions to the dispersion asymmetry in
systems where non-reciprocity is present. At the same time, dipolar-hybridized
modes can be easily disentangled. Since this method is independent of the
geometry or the involved magnetic interactions at hand, we believe it is
attractive for experimental and theoretical studies of magnetic systems where
there are no analytics yet, but also to aid the development of new analytical
theories.
