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.

翻译：我们引入了低常态集成器的总体框架, 从而使我们能够将大量方程式的时间动态, 包括抛物线和双曲问题, 以及分散式方程式, 直至普通域的任意高顺序。 新方案本地错误的结构是由嵌套式通缩器驱动的, 通常需要( 大大) 较低的常态假设。 我们的主要想法在于将非线性 PDE 的中央振荡器嵌入数字分化中。 后者是通过由具有常态结构的单一SPDE 所启发的新颖的装饰树形形式实现的, 并使我们能够控制系统中的非线性互动, 直至无限维( 连续) 和 有限维( 差异) 水平的任意高顺序 。