Modelling the time-dependent (2+1)D Dirac equation has recently gained importance since this equation effectively describes multiple condensed matter systems. To avoid the large dispersion errors of second-order real space schemes, a highly accurate method is presented here instead. The method utilises a fourth-order central difference on a staggered grid and an explicit symplectic Partitioned Runge–Kutta (PRK) time integrator. In contrast to traditional Runge–Kutta (RK) time stepping, no unphysical dissipation is introduced into the simulation. Moreover, it is demonstrated, both theoretically via Poisson maps and numerically, that the novel scheme has excellent conservation properties. Furthermore, the proposed numerical method is provably stable and the dispersion error is low and isotropic. Several interesting numerical examples are presented. Besides validating the advocated method, they also showcase its computational efficiency and low memory consumption.
An accurate and efficient near-field intensity shaping method is proposed, capable of reproducing sharp patterns while simplifying the design requirements of the array’s feeding network.