A fullwave model of the nonlinear wave equation with multiple relaxations and relaxing perfectly matched layers for high-order numerical finite difference solutions

Abstract

Objective. Large-scale acoustic simulation underpins the development of ultrasound imaging and therapy, but modeling nonlinearity, frequency-dependent attenuation, and absorbing boundaries in heterogeneous tissue remains computationally demanding. We present Fullwave 2, a unified time-domain formulation that represents arbitrary power-law tissue attenuation and perfectly matched layers (PMLs) within a single framework suited to high-order finite difference (FD) solution. Approach. Attenuation and dispersion are encoded directly into complex coordinate-stretched spatial derivatives through multiple relaxation mechanisms. Because the same mechanism describes both interior tissue attenuation and the absorbing boundary, the convolutional PML (C-PML) becomes a special case of the domain-wide model and adds no extra computational burden. The formulation preserves the structure of the d’Alembertian operator, which allows high-order staggered-grid FD stencils optimized for long-distance propagation, and a two-stage C-PML with a transition region is introduced to ensure numerical stability in the presence of multiple relaxations. Main results. The domain-wide multiple relaxation model reproduces power-law attenuation with less than 5% attenuation error and less than 0.5% phase-velocity error over a 1–20 MHz bandwidth. The two-stage C-PML reaches reflection coefficients below −49 dB with a compact 4λ footprint. Nonlinear propagation is validated against a 1D Burgers solution, with agreement up to the 7th harmonic. The framework is demonstrated on 2D abdominal wall imaging and 3D transcranial rat skull simulations, where it accurately captures complex scattering and aberration artifacts. Significance. Fullwave 2 unifies nonlinear propagation, arbitrary power-law attenuation, and absorbing boundaries in a single, computationally efficient time-domain formulation, providing an accurate and scalable wave propagation tool for medical ultrasound research.