A Low-Dispersion Time Differencing Scheme for Maxwell's Equations

A novel low-dispersion time differencing scheme for the integration of Maxwell's equations is developed. The method employs a staggered temporal grid with a time difference operator that controls the magnitude of dispersion errors. It is tested within the physical domains employing fourth-order finite difference approximations but can be combined with grids using any order of accuracy, including the spectral grids. Stability analysis demonstrates that the proposed method achieves third-and fourth-order accuracies for amplitude and phase errors, respectively, outperforming the second-order phase error produced by the finite-difference time-domain methods based on the Yee scheme. The order of accuracy accomplished by the method and its zone of stability are comparable to the third-order Runge-Kutta scheme with the advantage of being three times more efficient. The proposed method is proven to be effective and stable when employed within the perfectly matched layers and nonlinear Kerr media. Its performance and ability to reduce the dispersion errors are demonstrated by comparing the solutions computed by the method with those obtained from the Yee and other high-order schemes.