Improving Yee’s scheme with asymptotic dispersion correction for time-harmonic Maxwell’s equations

• We show how to reduce the dispersion error in Yee’s finite difference scheme for the time-harmonic Maxwell’s equations. • We introduce a shifted angular frequency depending on a free parameter in the Yee stencil. The optimal parameter is then explicitly computed by minimizing the dispersion error for a large enough number of grid points per wavelength. • Numerical experiments show that the relative error is reduced when using the optimal shifted angular frequency as soon as a large enough number of grid points per wavelength is considered. This figure shows the relative error for the 3d time-harmonic Maxwell’s equations discretized with the Yee scheme for several angular frequencies ω and a fixed meshsize hence varying the number of grid points per wavelength G . The curve in black represents the usual Yee scheme and the one in green is the Yee scheme with the dispersion correction presented in this paper. (Top left) ω = 1 ; (Top right) ω = 2.5 ; (Bottom left) ω = 5 ; (Bottom right) ω = 10 . In this paper, we show how to reduce the dispersion error associated to Yee’s finite difference scheme applied to time-harmonic Maxwell’s equations in one, two and three spatial dimensions. Our method, called asymptotic dispersion correction, is based on the introduction of a shifted angular frequency depending on a free parameter in the Yee stencil. The optimal parameter, called the asymptotically optimal shift, is next explicitly determined by minimizing the dispersion error for small enough meshsize or, equivalently, for large enough number of grid points per wavelength. Numerical experiments are provided and show that the relative error is reduced when using the optimal shifted angular frequency as soon as the number of grid points per wavelength is large enough.