Abstract: Many important developments in medical ultrasound are enabled by large-scale ultrasound simulations, which are essential for the design, evaluation, and optimization of new devices and applications. Despite the pervasiveness of computer simulations in medical ultrasound, numerous problems in diagnostic and therapeutic ultrasound remain unsolved, where better numerical models are expected to play a crucial role in the solution to these problems. In particular, significant improvements are needed in the available software for simulations of shock waves produced by histotripsy transducers to facilitate effective noninvasive treatments of liver, prostate, and brain tumors. Better, more effective simulation software is required to enable improvements in shear wave imaging of liver fibrosis, thyroid tumors, and breast tumors, and enhanced software tools are needed for simulations of ultrafast imaging and of other dynamic ultrasound imaging sequences. Improved models and methods are also needed for the attenuation of transient compressional and shear waves in soft tissues, which is significant for shear wave imaging, ultrasound microscopy, quantitative ultrasound, and ultrasound tomography. The main goal of this proposal is to create valuable new software resources that will enable new solutions of fundamental problems in medical ultrasound through the completion of three specific aims. In Aim 1, we propose to create new transient nonlinear full-wave ultrasound simulations for histotripsy using the discontinuous Galerkin (DG) method. Unlike finite difference, finite element, pseudo-spectral, and k-space methods, the discontinuous Galerkin method is ideal for full-wave models of shock waves evaluated on high performance computing systems. This is important because all of the existing nonlinear full-wave modeling tools for medical ultrasound that are based on these other methods encounter significant difficulties when attempting to model highly nonlinear pressure fields for histotripsy. In Aim 2, we propose to create transient full-wave simulations of shear waves with the discontinuous Galerkin method, which has similar advantages when applied to simulations of shear waves. We will also address another limitation of present shear wave simulations by augmenting the proposed shear wave simulations with a fractional calculus model that describes the attenuation and dispersion of shear waves in soft tissue. In Aim 3, we propose to create new programs for graphics processing units and compute clusters that further accelerate all of the existing programs in FOCUS, the `Fast Object-oriented C++ Ultrasound Simulator,' in order to facilitate effective numerical modeling of ultrafast imaging and other ultrasound imaging sequences. Aim 3 will also integrate a new numerical technique into FOCUS that models the power law attenuation and dispersion described by several different time-fractional and space-fractional models developed for medical ultrasound. Overall, we ...