A General Framework of Low Regularity Integrators

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 discretization technique not only with spectral methods, but rather with various spatial discretizations. We apply our general theory to the time discretization of various concrete PDEs, such as the nonlinear heat equation, the nonlinear Schrödinger 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.