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Volume 14, Issue 1
Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime

Ying He, Yan Wang, Jerry Zhijian Yang & Hongshuang Yin

East Asian J. Appl. Math., 14 (2024), pp. 79-103.

Published online: 2024-01

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  • 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.

  • AMS Subject Headings

35Q41, 65M12, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-14-79, author = {He , YingWang , YanYang , Jerry Zhijian and Yin , Hongshuang}, title = {Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime}, journal = {East Asian Journal on Applied Mathematics}, year = {2024}, volume = {14}, number = {1}, pages = {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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2023-004.200423 }, url = {http://global-sci.org/intro/article_detail/eajam/22320.html} }
TY - JOUR T1 - Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime AU - He , Ying AU - Wang , Yan AU - Yang , Jerry Zhijian AU - Yin , Hongshuang JO - East Asian Journal on Applied Mathematics VL - 1 SP - 79 EP - 103 PY - 2024 DA - 2024/01 SN - 14 DO - http://doi.org/10.4208/eajam.2023-004.200423 UR - https://global-sci.org/intro/article_detail/eajam/22320.html KW - Nonlinear Dirac equation, uniformly accurate, finite difference method, time-splitting method, exponential integrator. AB -

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.

He , YingWang , YanYang , Jerry Zhijian and Yin , Hongshuang. (2024). Numerical Methods for the Nonlinear Dirac Equation in the Massless Nonrelativistic Regime. East Asian Journal on Applied Mathematics. 14 (1). 79-103. doi:10.4208/eajam.2023-004.200423
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