Low-regularity integrators for nonlinear Dirac equations

2019/06/22 by Katharina Schratz, Yan Wang, Schratz, Katharina +3 · 4 citations
Engineering · Mathematics · #35Q41 #65M12 #65M70 #Advanced Mathematical Physics Problems #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · doi:10.48550/arxiv.1906.09413

openalex publication_date 2019/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Numerical integration of ordinary differential equations based on trigonometric polynomials
• Error estimates of a Fourier integrator for the cubic Schr "odinger\n equation at low regularity

Cited by

• Numerical analysis of 2D Navier--Stokes equations with nonsmooth initial value in the critical space
• L²--convergence of the time-splitting scheme for nonlinear Dirac equation in 1+1 dimensions
• Resonances and computations
• A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation

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