An explicit structure-preserving algorithm with symmetric and energy-preserving for the nonlocal Schrödinger equation with wave operator

Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are Oh4+τ2. The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.

A high-order structure-preserving difference scheme for generalized fractional Schrödinger equation with wave operator

Recent development of structure-preserving geometric particle-in-cell (PIC) algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential (or integro-differential) equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modern mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure-preserving geometric PIC algorithms in comparison with the conventional PIC methods.

An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation

Time domain propagation algorithms have the general advantages of full vectorial formulation, calculation of transmission characteristica in one step, and the automatic and accurate inclusion of reflections. The latter properties are particular important for the analysis of grating devices. Gratings are very important elements for key components for WDM lightwave transmission systems and WD photonic switching systems in direct detection scheme. Especially, semiconductor wavelength tunable optical filters for laser diodes, semiconductor optical switches and photodetectors are some examples for the application of grating filters. A time domain analysis is useful e.g. in connection with waveguide reflections, since beam propagation techniques, which have the ability to treat reflected waves, are very time consuming and to our knowledge combined with errors, such as lack of energy conservation in lossless waveguides. Within the time domain the algorithm described in [1] has been extensively used in microwave techniques and was introduced in integrated optics by [2]. Though a calculation with all relevant electric and magnetic field components is very time consuming. The other way to form a propagation algorithm in the time domain is to apply the wave equation including the time dependent wave operator [3,4], These methods are of 1st order accuracy in time, only, as we will show. We present a method which allows an accurate simulation of arbitrary lossless waveguide structures in integrated optics. This time domain method is based on an efficient explicit algorithm.