Low-regularity integrators for nonlinear Dirac equations

Abstract

In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose a ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in $H^r$ for solutions in $H^r$, i.e., without requiring any additional regularity on the solution. In contrast to classical methods, ULI overcomes the numerical loss of derivatives and is therefore more efficient and accurate for approximating low regular solutions. Convergence theorems and the extension of ULI to second order are established. Numerical experiments confirm the theoretical results and underline the favourable error behaviour of the new method at low regularity compared to classical integration schemes.

21 pages, 3 figures

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• Optimal convergence of a second order low-regularity integrator for the KdV equation
• Embedded exponential-type low-regularity integrators for KdV equation under rough data
• A semi-implicit low-regularity integrator for Navier-Stokes equations