Modeling of shock wave propagation in large amplitude ultrasound

The Rankine-Hugoniot relation for shock wave propagation describes the shock speed of a nonlinear wave. This paper investigates time-domain numerical methods that solve the nonlinear parabolic wave equation, or the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, and the conditions they require to satisfy the Rankine-Hugoniot relation. Two numerical methods commonly used in hyperbolic conservation laws are adapted to solve the KZK equation: Godunov's method and the monotonic upwind scheme for conservation laws (MUSCL). It is shown that they satisfy the Rankine-Hugoniot relation regardless of attenuation. These two methods are compared with the current implicit solution based method. When the attenuation is small, such as in water, the current method requires a degree of grid refinement that is computationally impractical. All three numerical methods are compared in simulations for lithotripters and high intensity focused ultrasound (HIFU) where the attenuation is small compared to the nonlinearity because much of the propagation occurs in water. The simulations are performed on grid sizes that are consistent with present-day computational resources but are not sufficiently refined for the current method to satisfy the Rankine-Hugoniot condition. It is shown that satisfying the Rankine-Hugoniot conditions has a significant impact on metrics relevant to lithotripsy (such as peak pressures) and HIFU (intensity). Because the Godunov and MUSCL schemes satisfy the Rankine-Hugoniot conditions on coarse grids, they are particularly advantageous for three-dimensional simulations.