  Large-scale simulations of the wave equation in electromagnetism, seismology,
and acoustics, can be solved efficiently by finite difference methods. The
accuracy of these numerical solutions usually depends on the minimization of
dissipation and dispersion error using optimized schemes. However, these
optimized schemes are often difficult to implement for more complex
formulations of the wave equation. Here we present a model of the quadratically
nonlinear wave equation with multiple relaxations as a function of space that
can model perfectly matched layers and is designed to be easily used with
optimized finite-difference stencils on a staggered grid. Solutions are
demonstrated in 2D and 3D for ultrasound imaging applications where the long
propagation lengths, heterogeneous media in the human body, large dynamic
range, and requirements for computational speed, demonstrate the utility of the
proposed methods.
