On the use of Gegenbauer reconstructions for shock wave propagation modeling

In therapeutic ultrasound, the presence of shock waves can be significant due to the use of high intensity beams, as well as due to shock formation during inertial cavitation. Although modeling of such strongly nonlinear waves can be carried out using spectral methods, such calculations are typically considered impractical, since accurate calculations often require hundreds or even thousands of harmonics to be considered, leading to prohibitive computational times. Instead, time-domain algorithms which generally utilize Godunov-type finite-difference schemes are commonly used. Although these time domain methods can accurately model steep shock wave fronts, unlike spectral methods they are inherently unsuitable for modeling realistic tissue dispersion relations. Motivated by the need for a more general model, the use of Gegenbauer reconstructions as a postprocess tool to resolve the band-limitations of the spectral methods are investigated. The present work focuses on eliminating the Gibbs phenomenon when representing a steep wave front using a limited number of harmonics. Both plane wave and axisymmetric 2D transducer problems will be presented to characterize the proposed method.