Christof Sparber

Zhongyi Huang, Shi Jin, Peter Markowich et al.

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter $\dt$, which is the ratio of the characteristic speed and the speed of light.

Zhongyi Huang, Shi Jin, Peter Markowich et al.

We present a new numerical method for accurate computations of solutions to (linear) one dimensional Schrödinger equations with periodic potentials. This is a prominent model in solid state physics where we also allow for perturbations by non-periodic potentials describing external electric fields. Our approach is based on the classical Bloch decomposition method which allows to diagonalize the periodic part of the Hamiltonian operator. Hence, the dominant effects from dispersion and periodic lattice potential are computed together, while the non-periodic potential acts only as a perturbation. Because the split-step communicator error between the periodic and non-periodic parts is relatively small, the step size can be chosen substantially larger than for the traditional splitting of the dispersion and potential operators. Indeed it is shown by the given examples, that our method is unconditionally stable and more efficient than the traditional split-step pseudo spectral schemes. To this end a particular focus is on the semiclassical regime, where the new algorithm naturally incorporates the adiabatic splitting of slow and fast degrees of freedom.