A Time-Splitting Fourier-Collocation Method for Computing the Nonlinear Dirac Equation with Perfectly Matched Layers

Abstract

This work is devoted to an efficient computation of the nonlinear Dirac (NLD) equation with perfectly matched layers (PML). When the PML is constructed, the Dirac operator in NLD equation has variable spatial coefficients. Direct application of the popular time-splitting Fourier spectral method to the equation with variable coefficients will cause numerical difficulties and lose the numerical accuracy. We introduce the time-splitting Fourier-collocation (TSFC) method to solve the problem. The Fourier collocation method chooses the Fourier series-based functions as basis but the expansion coefficients are computed at a set of collocation points. Extensive numerical experimen tal results show that the TSFC method with PML can absorb efficiently the reflections to the edges of the computational region. Furthermore, the novel method is success fully applied to simulate the binary collision in one-dimensional NLD equation and to show the conical diffraction in honeycomb lattices described by two-dimensional NLD equation.