Resonance based schemes for SPDEs
Jacob Armstrong-Goodall, Yvain Bruned
TL;DR
The paper develops resonance-based time-discretisation schemes for a broad class of SPDEs on $\mathbb{T}^d$, leveraging a Duhamel-based decorated-tree framework to encode iterated nonlinear and stochastic integrals. By discretising only the resonant deterministic parts exactly while treating the stochastic components via Taylor expansions, the authors achieve improved regularity requirements for low-order schemes, obtaining local errors up to $O(t^{3/2})$ and global strong convergence up to $O(t^{1/2})$. However, they prove a fundamental limitation: for higher-order schemes, stochastic integrals prevent explicit resonance discretisation, and regularity gains cannot extend beyond those of classical methods. The framework is demonstrated on stochastic NLS (with additive and multiplicative noise) and the Manakov system, with detailed local and global error analyses and stability results. Overall, the work provides a principled approach to constructing low-regularity integrators for SPDEs, guiding the design of efficient schemes when initial data regularity is limited and highlighting the boundary where resonance techniques cease to yield improvements.
Abstract
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 Schrödinger 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.