  A large toolbox of numerical schemes for dispersive equations has been
established, based on different discretization techniques such as discretizing
the variation-of-constants formula (e.g., exponential integrators) or splitting
the full equation into a series of simpler subproblems (e.g., splitting
methods). In many situations these classical schemes allow a precise and
efficient approximation. This, however, drastically changes whenever non-smooth
phenomena enter the scene such as for problems at low regularity and high
oscillations. Classical schemes fail to capture the oscillatory nature of the
solution, and this may lead to severe instabilities and loss of convergence. In
this article we review a new class of resonance-based schemes. The key idea in
the construction of the new schemes is to tackle and deeply embed the
underlying nonlinear structure of resonances into the numerical discretization.
As in the continuous case, these terms are central to structure preservation
and offer the new schemes strong properties at low regularity.
