Low-regularity integrators for nonlinear Dirac equations
Copyright policy )Low-regularity integrators for nonlinear Dirac equations
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 an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in H r H^r for solutions in H r H^{r} , i.e., without requiring any additional regularity on the solution. In contrast to classical methods, a 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 a 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.
- Sorbonne Paris Cité France
- Heriot-Watt University United Kingdom
- Laboratoire Jacques-Louis Lions France
- UNIVERSITE PARIS DESCARTES France
- Central China Normal University China (People's Republic of)
Smoothness and regularity of solutions to PDEs, splitting schemes, nonlinear Dirac equation, [MATH] Mathematics [math], Numerical Analysis (math.NA), Fourier integral operators applied to PDEs, optimal convergence, exponential-type integrator, Error bounds for initial value and initial-boundary value problems involving PDEs, Time-dependent Schrödinger equations and Dirac equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Spectral, collocation and related methods for boundary value problems involving PDEs, Mathematics - Numerical Analysis, 35Q41, 65M12, 65M70, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, low regularity, Dirac-Poisson system, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Smoothness and regularity of solutions to PDEs, splitting schemes, nonlinear Dirac equation, [MATH] Mathematics [math], Numerical Analysis (math.NA), Fourier integral operators applied to PDEs, optimal convergence, exponential-type integrator, Error bounds for initial value and initial-boundary value problems involving PDEs, Time-dependent Schrödinger equations and Dirac equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Spectral, collocation and related methods for boundary value problems involving PDEs, Mathematics - Numerical Analysis, 35Q41, 65M12, 65M70, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, low regularity, Dirac-Poisson system, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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