Abstract. We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar timesplitting 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 δ, which is the ratio of the characteristic speed and the speed of light.

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 nonperiodic potentials describing external electric fields. Our approach is based on the classical Bloch decomposition method, which allows us to diagonalize the periodic part of the Hamiltonian operator. Hence, the dominant effects from dispersion and periodic lattice potential are computed together, while the nonperiodic potential acts only as a perturbation. Because the split-step communicator error between the periodic and nonperiodic 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 pseudospectral 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. Introduction.One of the main problems in solid state physics is to describe the motion of electrons within the periodic potentials generated by the ionic cores. This problem has been studied from a physical as well as a mathematical point of view in, e.g., [1,8,28,29,33], resulting in a profound theoretical understanding of the novel dynamical features. Indeed one of the most striking effects, known as Peierls substitution, is a modification of the dispersion relation for Schrödinger's equation, where the classical energy relation E free (k) = 1 2 |k| 2 has to be replaced by E m (k), m ∈ N, the energy corresponding to the mth Bloch band [7]. The basic idea behind this replacement is a separation of scales which is present in this context. More precisely one recognizes that experimentally imposed, and thus called external, electromagnetic fields typically vary on much larger spatial scales than the periodic potential generated by the cores. Moreover, this external field can be considered weak in comparison to the periodic fields of the cores [2].To study this problem, consider the Schrödinger equation for the electrons in a

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Abstract3D printing with a high degree of spatial and compositional precision could open new avenues to the design and fabrication of functional composites. By combining the direct ink writing and microfluidics, a multimaterial 3D printing system for fabricating textured composites with liquid inclusions of programmable spatial distribution and compositions is reported here. Phase diagrams for the rational selection of desired printing parameters are determined through a combination of simple theoretical analysis and experimental studies. 1D, 2D, and 3D structures programmed with desired inclusion patterns and compositions are fabricated. Moreover, the versatility of this 3D printing framework in fabricating layered composite beams of tunable thermal property and self‐healing materials is demonstrated. The proposed multimaterial microfluidic 3D printing framework could be broadly applicable for structural composites and soft robotic devices.