Low-regularity integrators for nonlinear Dirac equations
Katharina Schratz, Yan Wang, Xiaofei Zhao
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 an ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript r"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">H^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for solutions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript r"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">H^{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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.