Numerical methods for axisymmetric and 3D nonlinear beams

Abstract

Time domain algorithms that solve the Khokhlov- Zabolotzskaya-Kuznetsov (KZK) equation are described and implemented. This equation represents the propagation of finite amplitude sound beams in a homogenous thermoviscous fluid for axisymmetric and fully three dimensional geometries. In the numerical solution each of the terms is considered separately and the numerical methods are compared with known solutions. First and second order operator splitting are used to combine the separate terms in the KZK equation and their convergence is examined. I. INTRODUCTION Linear theory provides a suitable approximation for small amplitude waves and short propagation lengths. Although in many cases this approximation is sufficient, a higher order description is necessary where large amplitudes or long prop- agation lengths and small attenuation is involved. Examples of such waves are beams with an amplitude large enough to produce shock waves, such as those used in therapeutic ultrasound for lithotripsy(1) or for harmonic imaging(2). Several numerical methods have been implemented to solve equations that describe nonlinear wave propagation(3-6). Al- though these methods are useful, significant challenges remain in modeling certain pulses and geometries at the initial con- dition surface. Lee et al.(4) solved the axisymmetric Khokhlov- Zabolotzskaya-Kuznetsov (KZK) equation in the time domain using implicit centered differences and the Crank- Nicolson scheme(7) for both the integral form of the diffraction operator and the absorption, and a distorted time solution for the nonlinearity. The propagation step was combined for pulsed unfocused waves using a first order operator split. Yang directly extended these methods to three dimensions(6). Here, in addition to these methods, we consider solutions of the KZK equation in axisymmetric and cartesian coordinates with a number of different numerical techniques. The diffraction is solved using the differential form of the operator for pulsed unfocused waves and focused plane waves. The nonlinearity is solved with the Lax- Friedrichs and Lax-Wendroff methods(7). These are applied separately to each term in the equations and compared to known solutions for pulsed unfocused waves and continuous focused waves or to frequency domain solutions. Then the combined effect for first and second order operator splitting are determined and compared. II. BASIC EQUATIONS The nonlinear parabolic KZK wave equation describes the effects of diffraction, absorption, and nonlinearity. Its axisym- metric form in terms of pressure can be written as(8) ∂ 2 p ∂z∂t � = c0 2 ∂ 2 p ∂r 2 + 1 r ∂p ∂r + δ 2c 3 ∂ 3 p ∂t � 3 + β