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Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime

Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime

Year:    2024

Author:    Ying He, Yan Wang, Jerry Zhijian Yang, Hongshuang Yin

East Asian Journal on Applied Mathematics, Vol. 14 (2024), Iss. 1 : pp. 79–103

Abstract

Numerical methods for the nonlinear Dirac equation (NDE) in the massless nonrelativistic regime are considered. In this regime, the equation contains a small dimensionless parameter $0 <\varepsilon≤ 1,$ and its solution is highly oscillatory in time. We present and analyze traditional numerical schemes for the NDE, including finite difference methods, time-splitting methods and exponential integrators. Error analysis indicates that all these methods require an $\varepsilon$-dependent time-step size to achieve an optimal convergence order. Utilizing an operator splitting technique, we propose a uniformly accurate (UA) scheme. The scheme enables first-order convergence in time for all $\varepsilon ∈ (0, 1]$ without restrictions on time-step size. Error estimates for the UA scheme are rigorously established and numerical results confirm the properties of the method.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.2023-004.200423

East Asian Journal on Applied Mathematics, Vol. 14 (2024), Iss. 1 : pp. 79–103

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    25

Keywords:    Nonlinear Dirac equation uniformly accurate finite difference method time-splitting method exponential integrator.

Author Details

Ying He

Yan Wang

Jerry Zhijian Yang

Hongshuang Yin