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

EAST ASIAN JOURNAL ON APPLIED MATHEMATICS(2024)

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摘要
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 < epsilon <= 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 epsilon-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 epsilon is an element of (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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关键词
Nonlinear Dirac equation,uniformly accurate,finite difference method,time-splitting method,exponential integrator
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