Low regularity integrators via decorated trees

TL;DR

This paper presents a unified framework for low regularity integrators capable of high-order approximation of diverse PDEs by embedding oscillations into the discretisation, using decorated trees inspired by singular SPDEs.

Contribution

It introduces a novel decorated tree formalism for constructing high-order low regularity integrators applicable to various PDEs, reducing regularity requirements.

Findings

Achieves high-order accuracy for parabolic, hyperbolic, and dispersive equations.

Reduces regularity assumptions compared to classical methods.

Provides a systematic approach inspired by singular SPDEs.

Abstract

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.