A Finite Difference Method for Dispersive Linear Waves with Applications to Simulating Microwave Pulses in Water

The EP_d method, a finite difference method for highly dispersive linear wave equations, is introduced and analyzed. Motivated by the problem of simulating the propagation of microwave pulses through water, the method attempts to relieve the computational burden of resolving fast processes, such as dipole relaxation or oscillation, occurring in a material with dynamic structure. This method, based on a novel differencing scheme for the time step, is considered primarily for problems in one spatial dimension with constant coefficients. It is defined in terms of the solution of an initial value problem for a system of ordinary differential equations that, in an implementation of the method, need be solved only once in a preprocessing step. For certain wave equations of interest (nondispersive systems, the telegrapher's equation, and the Debye model for dielectric media) explicit formulas for the method are presented. The dispersion relation of the method exhibits a high degree of low-wavenumber asymptotic agreement with the dispersion relation of the model to which it is applied. Comparisons with a finite difference time-domain approach and an approach based on Strang splitting demonstrate the potential of the method to substantially reduce the cost of simulating linear waves in dispersive materials. A generalization of the EP_d method for problems with variable coefficients appears to retain many of the advantages seen for constant coefficients.