East Asian J. Appl. Math., 14 (2024), pp. 79-103.
Published online: 2024-01
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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} }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.