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On Splitting Methods for the Dirac Equation in the Nonrelativistic Regime

Abstract: Splitting methods have wide applications. We study the second order Strang splitting for the Dirac equation in the nonrelativistic regime. In this regime, the solution admits high frequency waves in time. Error analysis and numerical examples show that the splitting methods have better resolution(w.r.t the high frequency waves) than the classical finite difference and exponential integrators. Surprisingly, when the external magnetic potentials are absent, we find that the splitting methods (Lie-Trotter splitting and Strang splitting) exhibit uniform convergence with half order rate for general time step size. In addition, if the time steps are non-resonant (w.r.t high frequency wave in time), better uniform convergence rates can be derived. Numerical tests show that the derived estimates are sharp.