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On numerical methods for the semi-nonrelativistic limit system of the nonlinear Dirac equation

Jahnke, Tobias 1,2; Kirn, Michael 1,2
1 Fakultät für Mathematik (MATH), Karlsruher Institut für Technologie (KIT)
2 Institut für Angewandte und Numerische Mathematik (IANM), Karlsruher Institut für Technologie (KIT)

Abstract:

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of 𝒪(ε$^{–2}$), where 0<ε≪1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of 𝒪(ε$^{2}$) by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly.


Verlagsausgabe §
DOI: 10.5445/IR/1000158458
Veröffentlicht am 10.05.2023
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Zeitschriftenaufsatz
Publikationsmonat/-jahr 06.2023
Sprache Englisch
Identifikator ISSN: 0006-3835, 1572-9125
KITopen-ID: 1000158458
Erschienen in BIT Numerical Mathematics
Verlag Springer
Band 63
Heft 2
Seiten 26
Vorab online veröffentlicht am 13.04.2023
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