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.

基于不同的离散化技术，例如离散化常数变分公式（例如指数积分器）或将整个方程分解为一系列更简单的子问题（例如分裂方法），已经建立了色散方程数值方案的大型工具箱）。在许多情况下，这些经典方案可以实现精确且有效的近似。然而，每当非平滑现象出现时，例如低规律性和高振荡的问题，这种情况就会发生巨大变化。经典方案无法捕捉解决方案的振荡性质，这可能会导致严重的不稳定和收敛损失。在本文中，我们回顾了一类新的基于共振的方案。构建新方案的关键思想是解决共振的潜在非线性结构并将其深深嵌入到数值离散中。与连续情况一样，这些项对于结构保存至关重要，并为新方案提供了低正则性的强大特性。