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 Sch\"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是一类利用方程振荡分量中的相消性来降低达到特定误差与收敛阶所需的初值正则性的方法。本文探讨了在非线性随机偏微分方程背景下推导基于共振格式的潜力。通过将误差分析所需的正则性条件与传统指数格式进行对比，我们证明：在阶数低于$\mathcal{O}(t^2)$时，该技术能够成功实现并显著提升初值正则性；而在阶数高于$\mathcal{O}(t^2)$时，基于共振的技术则无法获得任何增益。这一局限性源于随机积分路径显式分析的内在约束。作为方法应用实例，我们针对薛定谔方程和马纳科夫系统提出了相应格式，并附带了局部误差与稳定性分析，以及在强意义和路径意义下的全局收敛性证明。