Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

Abstract

We establish error bounds of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials, with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution propagates waves with O(ε 2 ) wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solutions accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). S 1 shows 1/2 order convergence uniformly with respect to ε, by establishing that there are two independent error bounds τ + ε and τ + τ /ε. Moreover, if τ is non-resonant, i.e. τ is away from certain region determined by ε, S 1 would yield an improved uniform first order O(τ ) error bound. In addition, we show S 2 is uniformly convergent with 1/2 order rate for general time step size τ and uniformly convergent with 3/2 order rate for non-resonant time step size. Finally, numerical examples are reported to validate our findings.

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

“…This suggests that the splitting methods may also be uniformly convergent. Superresolution and improved error bounds of the splitting methods have also been found and analyzed for Dirac and nonlinear Dirac equations in the nonrelativistic regime [6,7], and for the highly oscillatory nonlinear Schrödinger equation [18]. It should be also analysed in the massless nonrelativistic regime.…”

Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

“…For the analytical and numerical results in the classical regime, that is, $$$ \varepsilon =1 $$$, we refer to [14–20] and references therein. In the nonrelativistic/semiclassical regime, various numerical methods have been proposed and analyzed including the finite difference time domain (FDTD) methods [13, 21, 22], exponential wave integrator Fourier pseudospectral (EWI‐FP) method [13, 23], time‐splitting Fourier pseudospectral (TSFP) method [23–27], Gaussian bean method [20] and so on [9, 10, 28, 29].…”

“…For the analytical and numerical results in the classical regime, that is, 𝜀 = 1, we refer to [14][15][16][17][18][19][20] and references therein. In the nonrelativistic/semiclassical regime, various numerical methods have been proposed and analyzed including the finite difference time domain (FDTD) methods [13,21,22], exponential wave integrator Fourier pseudospectral (EWI-FP) method [13,23], time-splitting Fourier pseudospectral (TSFP) method [23][24][25][26][27], Gaussian bean method [20] and so on [9,10,28,29]. When 0 < 𝜀 ≪ 1 in the Dirac equation (1.1), that is, in the massless and nonrelativistic regime, the solution propagates waves with wavelength at O(1) in space and O(𝜀) in time.…”

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

“…For atomic processes in relativistic heavyion collisions, a treatment in momentum space was introduced in [39]. Additionally, there have been many studies on different regimes of the Dirac equation, such as the nonrelativistic regime [3,5,6,11], and the semiclassical regime [36].…”