A time-splitting spectral scheme for the Maxwell-Dirac system

Authors:
Zhongyi Huang;Shi Jin;Peter A. Markowich;Christof Sparber;Chunxiong Zheng
Affiliations:
ICMOR, Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China;ICMOR, Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China and Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA;Fakultät für Mathematik der Universität Wein, Nordbergstrasse 15, A-1090 Vienna, Austria;Fakultät für Mathematik der Universität Wein, Nordbergstrasse 15, A-1090 Vienna, Austria;ICMOR, Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China
Venue:
Journal of Computational Physics
Year:
2005

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Abstract

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition is unconditionally stable and highly efficient as our numerical examples show. In particular, we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter @d, which is the ratio of the characteristic speed and the speed of light.