Resonance based numerical schemes are those in which cancellations in the oscillatory components of the equation are taken advantage of in order to reduce the regularity required of the initial data to achieve a particular order of error and convergence. We investigate the potential for the derivation of resonance based schemes in the context of nonlinear stochastic PDEs. By comparing the regularity conditions required for error analysis to traditional exponential schemes we demonstrate that at orders less than $ \mathcal{O}(t^2) $, the techniques are successful and provide a significant gain on the regularity of the initial data, while at orders greater than $ \mathcal{O}(t^2) $, that the resonance based techniques does not achieve any gain. This is due to limitations in the explicit path-wise analysis of stochastic integrals. As examples of applications of the method, we present schemes for the Schr\"odinger equation and Manakov system accompanied by local error and stability analysis as well as proof of global convergence in both the strong and path-wise sense.

翻译：基于共振的数值格式利用方程中振荡分量的抵消效应，以降低初始数据所需的正则性条件，从而获得特定阶数的误差与收敛性。本文探讨了在非线性随机偏微分方程中推导基于共振格式的可行性。通过将误差分析所需的正则性条件与传统指数格式进行比较，我们证明：当阶数低于$\mathcal{O}(t^2)$时，该技术是有效的，并能显著改善初始数据的正则性需求；而当阶数高于$\mathcal{O}(t^2)$时，基于共振的技术无法带来任何增益。这一局限性源于随机积分的显式路径分析固有的限制。作为方法应用实例，我们针对薛定谔方程和马纳科夫系统给出了具体格式，并附有局部误差与稳定性分析，以及强收敛和路径收敛意义上的全局收敛性证明。