A general framework of low regularity integrators

TL;DR

This paper presents a unified framework for low regularity integrators applicable to various evolution equations, overcoming previous limitations related to boundary conditions, equation types, and nonlinearities, and enabling coupling with different spatial discretizations.

Contribution

The authors develop a general, Fourier-free approach for low regularity integrators that applies to a broad class of PDEs, including non-periodic and nonlinear cases, unifying and extending existing methods.

Findings

Applicable to nonlinear heat, Schrödinger, Ginzburg-Landau, Klein-Gordon equations

Handles non-polynomial nonlinearities and non-periodic boundary conditions

Compatible with various spatial discretization methods

Abstract

We introduce a new general framework for the approximation of evolution equations at low regularity and develop a new class of schemes for a wide range of equations under lower regularity assumptions than classical methods require. In contrast to previous works, our new framework allows a unified practical formulation and the construction of the new schemes does not rely on any Fourier based expansions. This allows us for the first time to overcome the severe restriction to periodic boundary conditions, to embed in the same framework parabolic and dispersive equations and to handle nonlinearities that are not polynomial. In particular, as our new formalism does no longer require periodicity of the problem, one may couple the new time discretisation technique not only with spectral methods, but rather with various spatial discretisations. We apply our general theory to the time discretization of various concrete PDEs, such as the nonlinear heat equation, the nonlinear Schr\"odinger equation, the complex Ginzburg-Landau equation, the half wave and Klein--Gordon equations, set in $\Omega \subset \mathbb{R}^d$, $d \leq 3$ with suitable boundary conditions.