A finite difference method for dispersive linear waves with applications to simulating microwave pulses in water

The EPd method, a finite difference method for highly dispersive linear wav e equations, is introduced and analyzed. Motivated by the problem of simula ting the propagation of microwave pulses through water, the method attempts to relieve the computational burden of resolving fast processes, such as d ipole relaxation or oscillation, occurring in a material with dynamic struc ture. 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 p roblem for a system of ordinary differential equations that, in an implemen tation 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 fo r the method are presented. The dispersion relation of the method exhibits a high degree of low-wavenumber asymptotic agreement with the dispersion re lation of the model to which it is applied. Comparisons with a finite diffe rence time-domain approach and an approach based on Strang splitting demons trate the potential of the method to substantially reduce the cost of simul ating linear waves in dispersive materials. A generalization of the EPd met hod for problems with variable coefficients appears to retain many of the a dvantages seen for constant coefficients. (C) 1999 Academic Press.