  We introduce a numerical framework for dispersive equations embedding their
underlying resonance structure into the discretisation. This will allow us to
resolve the nonlinear oscillations of the PDE and to approximate with high
order accuracy a large class of equations under lower regularity assumptions
than classical techniques require. The key idea to control the nonlinear
frequency interactions in the system up to arbitrary high order thereby lies in
a tailored decorated tree formalism. Our algebraic structures are close to the
ones developed for singular SPDEs with Regularity Structures. We adapt them to
the context of dispersive PDEs by using a novel class of decorations {which
encode the dominant frequencies}. The structure proposed in this paper is new
and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated
trees. We observe a similar Birkhoff type factorisation as in SPDEs and
perturbative quantum field theory. This factorisation allows us to single out
oscillations and to optimise the local error by mapping it to the particular
regularity of the solution. This use of the Birkhoff factorisation seems new in
comparison to the literature. The field of singular SPDEs took advantage of
numerical methods and renormalisation in perturbative quantum field theory by
extending their structures via the adjunction of decorations and Taylor
expansions. Now, through this work, Numerical Analysis is taking advantage of
these extended structures and provides a new perspective on them.
