基于装饰树的色散方程共振方案
Resonance based schemes for dispersive equations via decorated trees
AI总结：
提出一种将共振结构嵌入离散化的数值框架，利用装饰树形式控制非线性频率，以高阶精度和较低正则性假设求解色散方程。并给出新的Birkhoff因式分解。
摘要：
我们引入了一个用于色散方程的数值框架，该框架将其潜在的共振结构嵌入到离散化中。这使我们能够解决偏微分方程的非线性振荡，并以高阶精度逼近一大类方程。其所需的正则性假设低于经典方法的要求。控制系统中任意高阶非线性频率的关键思想在于一种量身定制的装饰树形式体系。我们的代数结构与为带有正则性结构的奇异SPDEs所开发的结构相似。我们通过使用一类新的装饰{它编码了主导频率}，将其适应于色散PDE的语境。本文提出的结构是新的，并给出了装饰树上Butcher-Connes-Kreimer Hopf代数的一个变体。我们观察到与SPDEs和微扰量子场论中类似的Birkhoff因式分解。这种因式分解允许我们分离出振荡，并通过映射解的正则性来优化局部误差。与文献相比，本文的Birkhoff因式分解是新的。奇异SPDE领域采用了数值方法。微扰量子场论通过装饰树和Taylor展开式使用了其结构
英文摘要：
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.
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