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 - <p style="text-align: justify;">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 $0 &lt;\varepsilon≤ 1,$ 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.</p>