Scaling method for the numerical solution of the strong-field ionization problem in the relativistic regime

The coordinate scaling method, previously developed for the numerical solution of the time-dependent Schrodinger equation, is generalized for the numerical treatment of the atomic ionization problem in relativistically strong laser fields, developing the prototype of the method for a one-dimensional case. To enable the use of the scaling method in relativistic settings, the Foldy-Wouthuysen transformation is employed in Silenko's form within the quasiclassical approximation, reducing the one-dimensional time-dependent Dirac equation (TDDE) to the square root Klein-Gordon-like equation. We demonstrate the computational advantage of the relativistic scaling method over the standard direct implementation of the TDDE solution, especially in the case of an applied non-uniform mesh.