Abstract: Nonlinear acoustic propagation generated by a vibrating piston in a finite horn is numerically studied. A quasi-one-dimensional nonlinear model with varying cross-section uses high-order low-dispersion numerical schemes to solve the governing equation. Considering the nonlinear wave distortion and reflected sound waves at the mouth, broadband time-domain impedance boundary conditions are employed. The impedance approximation can be optimized to identify the complex-conjugate pole-residue pairs of the impedance functions, which can be calculated by fast and efficient recursive convolution. The numerical results agree very well with experimental data in the situations of weak nonlinear wave propagation in an exponential horn, it is shown that the model can describe the broadband characteristics caused by nonlinear distortion. Moreover, finite-amplitude acoustic propagation of a piston source in types of horns is simulated, including hyperbolic, conical, exponential and sinusoidal horns, it is found that the sound pressure level at the horn mouth are strongly affected by the driving velocity and frequency of the piston and the horn profiles, and the interplay between the waveform distortion and the horn geometry is discussed.