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Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime. (English) Zbl 1440.35286

Summary: We establish error bounds of the Lie-Trotter splitting \((S_1)\) and Strang splitting \((S_2)\) for the Dirac equation in the nonrelativistic regime in the absence of external magnetic potentials, with a small parameter \(0 < \varepsilon \leq 1\) inversely proportional to the speed of light. In this regime, the solution propagates waves with \(O(\varepsilon^2)\) wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, i.e., the methods can capture the solutions accurately even if the time step size \(\tau\) is independent of \(\varepsilon\), while the wavelength in time is at \(O(\varepsilon^2).\ S_1\) shows \(1/2\) order convergence uniformly with respect to \(\varepsilon\), by establishing that there are two independent error bounds \(\tau + \varepsilon\) and \(\tau + \tau /\varepsilon\). Moreover, if \(\tau\) is nonresonant, i.e., \(\tau\) is away from a certain region determined by \(\varepsilon,\ S_1\) would yield an improved uniform first order \(O(\tau)\) error bound. In addition, we show \(S_2\) is uniformly convergent with \(1/2\) order rate for general time step size \(\tau\) and uniformly convergent with \(3/2\) order rate for nonresonant time step size. Finally, numerical examples are reported to validate our findings.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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