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We develop a numerical method for the Westervelt equation, an importantequation in nonlinear acoustics, in the form where the attenuation isrepresented by a class of non-local in time operators. A semi-discretisation intime based on the trapezoidal rule and A-stable convolution quadrature isstated and analysed. Existence and regularity analysis of the continuousequations informs the stability and error analysis of the semi-discrete system.The error analysis includes the consideration of the singularity at $t = 0$which is addressed by the use of a correction in the numerical scheme.Extensive numerical experiments confirm the theory.

Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping