Designing high-order, time-domain numerical solvers for Maxwell´s equations

Considerable time and energy has been devoted to the design and development of second order, time-domain schemes for Maxwell´s equations. Schemes, for example, such as the finite-difference time-domain (FDTD) method have been applied to many diverse problems ranging from radar cross-section analysis to ionospheric radio wave propagation studies. Yet, the second-order nature of the schemes prevents one from obtaining accurate data for geometries or propagation distances that span tens of wavelengths. This property is clearly manifested by frequency domain analyses, which demonstrate the high accumulation of phase and dissipation errors over many time steps. To mitigate the effect of phase and dissipation errors, two key options exist: (1) decrease the size of the discretization cell and the time step or (2) increase the order of accuracy. We consider only option two. In essence, the design of a time domain solver is subdivided into three essential tasks: (1) spatial discretization, (2) temporal discretization and (3) evaluation of the fully discrete system.