论文标题
通过装饰树的低规律性集成商
Low regularity integrators via decorated trees
论文作者
论文摘要
我们引入了一个低规则性集成符的一般框架，该框架使我们能够近似大类方程的时间动态，包括抛物线和双曲线问题以及分散方程，直至一般域上的任意高阶。新方案的局部误差的结构是由嵌套的换向器驱动的，嵌套的换向器通常需要（比经典方法要低得多的规律性假设）。我们的主要思想在于将非线性PDE的核心振荡嵌入数值离散化中。后者是由一种新型的装饰树形式主义实现的，该形式的灵感来自具有规律性结构的单数SPDE，并使我们能够控制系统中的非线性相互作用，直到无限尺寸（连续）以及有限的尺寸（离散）水平的任意高阶。
We introduce a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as dispersive equations, up to arbitrary high order on general domains. The structure of the local error of the new schemes is driven by nested commutators which in general require (much) lower regularity assumptions than classical methods do. Our main idea lies in embedding the central oscillations of the nonlinear PDE into the numerical discretisation. The latter is achieved by a novel decorated tree formalism inspired by singular SPDEs with Regularity Structures and allows us to control the nonlinear interactions in the system up to arbitrary high order on the infinite dimensional (continuous) as well as finite dimensional (discrete) level.
