We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the PDE and to approximate with high order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular SPDEs with Regularity Structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations {which encode the dominant frequencies}. The structure proposed in this paper is new and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, Numerical Analysis is taking advantage of these extended structures and provides a new perspective on them.
翻译:我们引入了一个分布式方程式的数字框架, 将其内在的共振结构嵌入离散状态中。 这将使我们能够解决PDE的非线性振动, 并且以高于经典技术要求的常规性假设下, 以高的高度精确度将一系列大方程相近。 控制系统中非线性频率互动的关键理念, 直至任意高秩序, 在于一个定制的标定树型化。 我们的代数结构接近于为具有规律结构的单型SPDE 开发出来的那些结构。 我们通过使用新颖的装饰性观点来调整它们, 从而适应分散式 PDE 的背景 。 本文中提议的结构是全新的, 并给出了布切尔- Kreimer Hopf 代数变方程的变量。 我们观察到一种类似于 SPDE 和 扭曲性量场理论的伯克霍夫式因式因子化因子化 。 这个因子化结构让我们单独列出振动和优化本地错误, 通过将它映射成特定的常规性分析 {标 。 本文中提议一个比德尔- Kreimmeralalalalalalalizalalizalizal 的模型, 。 将它们在新的的模型中, 的模型的伸展展展成新的 。