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Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime. (English) Zbl 1365.65208

The paper deals with the numerical solution of the nonlinear Dirac equation (NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter \(0 < \varepsilon \ll 1\), which is inversely proportional to the speed of light. The authors combine finite difference approximation with respect to the space and the Crank-Nicolson scheme for the time discretization. The scheme is analysed and a priori error estimates in terms of \(h\), \(\tau\) and \(\varepsilon\) are derived. Finally, the paper contains a list of numerical experiments investigating the experimental order of convergence in dependence on \(h\), \(\tau\) and \(\varepsilon\).

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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