UNIFORM ERROR BOUNDS OF TIME-SPLITTING METHODS FOR THE NONLINEAR DIRAC EQUATION IN THE NONRELATIVISTIC REGIME WITHOUT MAGNETIC POTENTIAL

Superresolution of the Lie-Trotter splitting (S-1) and Strang splitting (S-2) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter 0 < epsilon <= 1 inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at O(epsilon(2)). The splitting methods surprisingly show superresolution, i.e., the methods can capture the solution accurately even if the time step size tau is much larger than the sampled wavelength at O(epsilon(2)). Similar to the linear case, S-1 and S-2 both exhibit 1/2 order convergence uniformly with respect to epsilon. Moreover, if tau is nonresonant, i.e., tau is away from a certain region determined by epsilon, S-1 would yield an improved uniform first order O(tau) error bound, while S-2 would give improved uniform 3/2 order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that superresolution is still valid for higher order splitting methods.