We introduce a unified framework of symmetric resonance based schemes which preserve central symmetries of the underlying PDE. We extend the resonance decorated trees approach introduced in arXiv:2005.01649 to a richer framework by exploring novel ways of iterating Duhamel's formula, capturing the dominant parts while interpolating the lower parts of the resonances in a symmetric manner. This gives a general class of new numerical schemes with more degrees of freedom than the original scheme from arXiv:2005.01649. To encapsulate the central structures we develop new forest formulae that contain the previous class of schemes and derive conditions on their coefficients in order to obtain symmetric schemes. These forest formulae echo the one used in Quantum Field Theory for renormalising Feynman diagrams and the one used for the renormalisation of singular SPDEs via the theory of Regularity Structures. These new algebraic tools not only provide a nice parametrisation of the previous resonance based integrators but also allow us to find new symmetric schemes with remarkable structure preservation properties even at very low regularity.

翻译：我们提出了一种统一的对称共振型格式框架，该框架能够保持底层偏微分方程的中心对称性。通过探索迭代杜阿梅尔公式的新方法，以对称方式插值共振低阶部分的同时捕获主导部分，我们将arXiv:2005.01649中引入的共振装饰树方法扩展至更丰富的框架中。这产生了一个通用类别的全新数值格式，相较于arXiv:2005.01649的原始格式具有更多自由度。为封装中心对称结构，我们发展了包含前述格式类别的新森林公式，并推导其系数需满足的条件以获得对称格式。这些森林公式与量子场论中重整化费曼图所用的公式，以及通过正则结构理论重整化奇异随机偏微分方程所用的公式相呼应。这些新型代数工具不仅为前述共振型积分器提供了优雅的参数化方法，更使得我们即使在极低正则性条件下也能发现具有显著结构保持特性的新对称格式。