Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.
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