Modeling pulse propagation and scattering in a dispersive medium

Accurate modeling of pulse propagation and scattering is of great importance to the Navy. In a non-dispersive medium, a fourth order in time and space 2-D Finite Difference Time Domain (FDTD) scheme representation of the linear wave equation can be used However, when the medium is dispersive, one is required to take into account the frequency-dependent attenuation and phase velocity. Until recently, to include the dispersive effects, one typically solved the problem in the frequency domain and not in the time domain. The frequency domain solutions were Fourier transformed into the time domain. However by using a theory first proposed by Blackstock, the linear wave equation has been modified by adding an additional term (the derivative of the convolution between the causal time domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. In the cases we are examining, what makes the water environment dispersive is the presence of air bubbles that are present at only a very small set of grid points. This leads to a numerical scheme where the amount of work to be done on individual grid points can vary by two orders of magnitude, leading to load balancing problems. This paper will examine the challenges encountered in implementing this problem efficiently on High Performance Computer systems and the tools, algorithms, and code changes required to run this code efficiently on such machines.