A multi-time stepping integration method for the ultrasound heating problem

A two-dimensional thermo-acoustic problem consisting of the Pennes bioheat transfer model and a nonlinear wave equation with a temperature dependent speed of sound is considered. The large discrepancy in time scales of the wave propagation and the heat conduction processes requires a multi-time stepping method allowing to solve both equations on different time scales. However, the standard approach considers a sequential solution process without any error control. We propose a multi-time stepping scheme based on a fixed-point approach resulting in a thermo-acoustic coupling controlled by a stopping criterion. Furthermore, in most high-intensity focused ultrasound applications the problem has to deal with absorbing boundary conditions (ABCs) in a bounded region. The use of a Lagrange multiplier based technique allows us to efficiently incorporate the second order Engquist-Majda ABC into the weak formulation of the original problem. The efficiency and robustness of the proposed multi-time stepping method as well as improved accuracy compared to the widely used standard scheme for modeling high-intensity focused ultrasound is demonstrated through a series of numerical examples which are typical for ultrasound heating. A two-dimensional thermo-acoustic problem consisting of the Pennes bioheat transfer model and a nonlinear wave equation with a temperature dependent speed of sound is considered. The large discrepancy in time scales of the wave propagation and the heat conduction processes requires a multi-time stepping method allowing to solve both equations on different time scales. The authors propose a multi-time stepping scheme based on a fixed-point approach resulting in a thermo-acoustic coupling controlled by a stopping criterion. The use of a Lagrange multiplier based technique allows to efficiently incorporate the second order Engquist-Majda ABC into the weak formulation of the original problem.